· Dit proefschrift is goedgekeurd door de promotor: Prof.dr.ir. R. de Borst Copromotor: Dr.ir....

Goal-Adaptive Discretizationof

Fluid–Structure Interaction

Kristoffer G. van der Zee

Goal-Adaptive Discretization

of Fluid-Structure Interaction

Goal-Adaptive Discretization

of Fluid-Structure Interaction

Proefschrift

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus, Prof.dr.ir. J.T. Fokkema,voorzitter van het College voor Promoties,

in het openbaar te verdedigen op vrijdag 5 juni 2009 om 10:00 uur

door

Kristoffer George VAN DER ZEE

ingenieur luchtvaart en ruimtevaartgeboren te Lelystad

Dit proefschrift is goedgekeurd door de promotor:Prof.dr.ir. R. de Borst

Copromotor:Dr.ir. E.H. van Brummelen

Samenstelling promotiecommissie:Rector Magnificus voorzitterProf.dr.ir. R. de Borst Technische Universiteit Eindhoven, promotorDr.ir. E.H. van Brummelen Technische Universiteit Delft, copromotorProf.Dr.-Ing. W.A. Wall Technische Universität MünchenProf.dr. R.P. Stevenson Universiteit van AmsterdamProf.dr.ir. C. Vuik Technische Universiteit DelftProf.dr.ir. D.J. Rixen Technische Universiteit DelftDr. S. Prudhomme University of Texas at Austin

© 2009 by Kristoffer G. van der Zee. All rights reserved.

ISBN 978-90-79488-54-4

Printed in the Netherlands by Ipskamp Drukkers B.V.

This research was supported by the Dutch TechnologyFoundation STW, applied science division of NWO and theTechnology Program of the Ministry of Economic Affairs.

Voor mijn ouders

Preface

The research presented in this doctoral thesis has been carried out at the Engi-neering Mechanics Group of the Faculty of Aerospace Engineering at the DelftUniversity of Technology.

It has given me much pleasure to work on a subject that allowed me to delvedeeply into fundamental matters. This would not have been possible withoutHarald van Brummelen, who has initiated the project I worked on and whohas supported me throughout the years. Thank you for your unconditionaldedication. You have been a most pleasant supervisor.

Next, I would like to thank Professor René de Borst for his continuous con-fidence in me. Gratitude goes to Carla Roovers, our secretary. I would also liketo acknowledge all of my (former) colleagues for the pleasant atmosphere. Inparticular, I would like to thank Christian Michler, Edwin Munts, Steve Hul-shoff, Wijnand Hoitinga and Peter Fick for the many stimulating scientific dis-cussions. Special thanks go to Ido Akkerman, my office-buddy, for his scientificsharpness and broad knowledge. Having spend the last 13 years together, I amcertain our personal and professional paths will cross again.

Also, I am thankful to all of my wushu friends in Delft and Leiden for themany joyful times that kept my mind away from research. In particular, I wouldlike to express my gratitude to Gabe, John-Sebastian and Ralph.

My family deserves special recognition. I thank my parents for their undi-vided love and support. As well as my sister Dolly and her soon to becomehusband Mervin. I wish them all the luck. Thanks also to my Philippine familyin Canada who have persuaded me to follow my dreams.

Finally, thank you, Nadine, for taking up a special place in my heart andhaving me in yours.

Kris van der ZeeVoorschoten, May 2009

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Numerical simulation of fluid–structure interaction . . . . 21.1.2 Goal-oriented adaptive discretization . . . . . . . . . . . . 21.1.3 Linearization of the domain-dependence . . . . . . . . . . 4

1.2 Aims and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Thesis overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

I Three Fundamentals 9

2 Free-Boundary Problems and Fluid–Structure Interaction 11

2.1 Bernoulli free-boundary problem . . . . . . . . . . . . . . . . . . . 112.1.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . 122.1.2 Weak form of the free-boundary problem . . . . . . . . . . 13

2.2 Steady incompressible fluid–structure interaction . . . . . . . . . 142.2.1 Fluid problem statement . . . . . . . . . . . . . . . . . . . . 142.2.2 Structure problem statement . . . . . . . . . . . . . . . . . 162.2.3 Coupled weak form with unbounded tractions . . . . . . . 162.2.4 Coupled weak form with weakly-enforced tractions . . . . 182.2.5 Existence of coupled fluid–structure solutions . . . . . . . 19

2.3 Goal quantities and discretization errors . . . . . . . . . . . . . . . 20

3 Goal-Oriented Error Analysis and Adaptivity 23

3.1 Abstract linear problems . . . . . . . . . . . . . . . . . . . . . . . . 243.1.1 Functional setting . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2 Galerkin approximations . . . . . . . . . . . . . . . . . . . 263.1.3 Duality, adjoints and errors in goal quantities . . . . . . . . 28

3.2 Linear boundary value problems . . . . . . . . . . . . . . . . . . . 303.2.1 Dirichlet problem for Laplace operator . . . . . . . . . . . 303.2.2 Convergence of uniform discretizations . . . . . . . . . . . 31

x Contents

3.2.3 Convergence of optimal discretizations . . . . . . . . . . . 343.2.4 Adaptive discretization procedure . . . . . . . . . . . . . . 38

3.3 Generalization to nonlinear problems . . . . . . . . . . . . . . . . 423.3.1 Differential calculus in Banach spaces . . . . . . . . . . . . 433.3.2 Nonsingular solutions . . . . . . . . . . . . . . . . . . . . . 443.3.3 Galerkin approximations . . . . . . . . . . . . . . . . . . . 453.3.4 Duality, linearized adjoints and errors in goal quantities . 47

3.4 Nontrivial nonlinearity in free-boundary problems . . . . . . . . . 513.4.1 Domain-map linearization approach . . . . . . . . . . . . . 513.4.2 Shape-linearization approach . . . . . . . . . . . . . . . . . 52

4 Shape Differential Calculus 55

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Parametrized families of domain transformations . . . . . . . . . 57

4.2.1 The velocity method and general transformations . . . . . 574.2.2 Perturbations of the identity . . . . . . . . . . . . . . . . . 604.2.3 Scalar perturbations . . . . . . . . . . . . . . . . . . . . . . 614.2.4 Overview of parametrizations . . . . . . . . . . . . . . . . 634.2.5 Shape continuity based on parametrized domains . . . . . 63

4.3 Shape derivatives of shape functionals . . . . . . . . . . . . . . . . 644.3.1 Velocity method and shape derivative . . . . . . . . . . . . 644.3.2 Perturbations of the identity and Gâteaux derivative . . . 664.3.3 Shape functional involving domain integral . . . . . . . . 674.3.4 Shape functional involving boundary integral . . . . . . . 70

II Scalar Free-Boundary Problems 73

5 Domain-Map Linearization Approach to Free-Boundary Problems 75

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2.1 Bernoulli free-boundary problem . . . . . . . . . . . . . . . 765.2.2 Parametrization of the unknown domain . . . . . . . . . . 775.2.3 Weak form of the free-boundary problem . . . . . . . . . . 785.2.4 Goal functionals and approximation errors . . . . . . . . . 78

5.3 Goal-oriented error estimation by domain-map linearization . . . 795.3.1 Domain-map linearization at reference domain . . . . . . . 795.3.2 Domain-map linearization at approximate domain . . . . 835.3.3 Equivalence of dual problems . . . . . . . . . . . . . . . . . 86

5.4 Analysis of the dual problem . . . . . . . . . . . . . . . . . . . . . 875.4.1 Specification of the dual problem . . . . . . . . . . . . . . . 885.4.2 Interpretation of the dual problem . . . . . . . . . . . . . . 90

Contents xi

5.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 925.5.1 One-dimensional application . . . . . . . . . . . . . . . . . 935.5.2 Two-dimensional application . . . . . . . . . . . . . . . . . 95

5.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 Shape-Linearization Approach to Free-Boundary Problems 101


6.2.1 Bernoulli free-boundary problem . . . . . . . . . . . . . . . 1036.2.2 Errors in goal quantities . . . . . . . . . . . . . . . . . . . . 1046.2.3 Very weak form of the free-boundary problem . . . . . . . 1046.2.4 Free-boundary perturbations . . . . . . . . . . . . . . . . . 105

6.3 Goal-oriented error estimation by shape linearization . . . . . . . 1076.3.1 Linearization of the free-boundary problem . . . . . . . . . 1076.3.2 Linearization of the goal functional . . . . . . . . . . . . . 1086.3.3 Dual problem and goal-error estimate . . . . . . . . . . . . 109

6.4 Extension to nonsmooth free boundaries . . . . . . . . . . . . . . . 1126.4.1 Shape linearization at nonsmooth free boundaries . . . . . 1126.4.2 Dual problem and goal-error estimate . . . . . . . . . . . . 114

6.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 1166.5.1 One-dimensional application . . . . . . . . . . . . . . . . . 1166.5.2 Two-dimensional application . . . . . . . . . . . . . . . . . 118


III Steady Incompressible Fluid–Structure Interaction 125

7 Domain-Map Linearization Approach to Fluid–Structure Interaction 127


7.2.1 Fluid–structure-interaction model . . . . . . . . . . . . . . 1297.2.2 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . 1307.2.3 Goal functionals and discretization errors . . . . . . . . . . 131

7.3 Goal-oriented error estimation by domain-map linearization . . . 1327.3.1 Reformulation to the approximate domain . . . . . . . . . 1327.3.2 Dual problem and error representation . . . . . . . . . . . 134

7.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 1377.4.1 Parabolic interface testcase . . . . . . . . . . . . . . . . . . 1377.4.2 Driven cavity with flexible bottom . . . . . . . . . . . . . . 1397.4.3 Backward-step with flexible bottom . . . . . . . . . . . . . 141


xii Contents

8 Shape-Linearization Approach to Fluid–Structure Interaction 147


8.2.1 Fluid–structure interaction model . . . . . . . . . . . . . . 1488.2.2 Weak formulation . . . . . . . . . . . . . . . . . . . . . . . . 1498.2.3 Goal functionals . . . . . . . . . . . . . . . . . . . . . . . . 1508.2.4 Very-weak fluid formulation . . . . . . . . . . . . . . . . . 150

8.3 Goal-oriented error estimation by shape linearization . . . . . . . 1518.3.1 Linearization with respect to fluid variables . . . . . . . . 1528.3.2 Shape linearization at smooth interfaces . . . . . . . . . . . 1528.3.3 Dual problem and goal-error estimate . . . . . . . . . . . . 1548.3.4 Extension to nonsmooth interfaces . . . . . . . . . . . . . . 156

8.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 1568.4.1 Parabolic interface testcase . . . . . . . . . . . . . . . . . . 1578.4.2 Driven cavity with flexible bottom . . . . . . . . . . . . . . 1588.4.3 Backward step with flexible bottom . . . . . . . . . . . . . 159


9 Conclusions and Future Prospects 165

Appendix 169

A.1 Elements of tangential calculus . . . . . . . . . . . . . . . . . . . . 169A.1.1 Tangential derivatives . . . . . . . . . . . . . . . . . . . . . 169A.1.2 Tangential integration by parts . . . . . . . . . . . . . . . . 170

A.2 Shape derivatives of shape functions . . . . . . . . . . . . . . . . . 173A.2.1 Extension approach to shape derivative . . . . . . . . . . . 173A.2.2 Material derivative approach to shape derivative . . . . . 175A.2.3 Shape function involving restriction . . . . . . . . . . . . . 176A.2.4 Shape function involving Dirichlet problem . . . . . . . . 177

A.3 Additional analysis of the dual problem of Sec. 6.4 . . . . . . . . . 180A.3.1 Boundedness of the tangential divergence term . . . . . . 180A.3.2 Wellposedness by coercivity . . . . . . . . . . . . . . . . . . 181

A.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183A.4.1 Equivalent inf-sup conditions . . . . . . . . . . . . . . . . . 183A.4.2 Linearization of the elevation goal . . . . . . . . . . . . . . 184A.4.3 Vanishing shape derivative of domain integral . . . . . . . 185

List of Theorems 187

Bibliography 189

Contents xiii

Summary 207

Samenvatting 209

Curriculum Vitae 213

Chapter 1

Introduction

One has to ask the question: What is the purpose of computation? Theanswer is not simple.

Ivo M. Babuška and Theofanis Stroubolis, 2001The Finite Element Method and its Reliability1

The simulation of complex physical phenomena, such as fluid–structure inter-action, appears to be within reach in view of the significant progress in com-puting power over the last decades. Yet, we are still far away from what isdesirable in an evermore-demanding, science-and-technology-based society. If,however, we are modest with what we need of a physical system in that weask for specific goal quantities instead of the entire solution, then we are able tosave tremendous amounts of computing effort.

1.1 Background

Fluid–structure interaction concerns the interaction of fluid flows (consisting ofliquids or gases) with flexible structures, for example, deformable solids, shellsor membranes. From a mathematical perspective, fluid–structure-interactionproblems are challenging problems as they constitute multiphysics, multiscale,free-boundary problems. Apart from the multiscale phenomena that can oc-cur in the fluid and structure subsystems separately, the intricate coupling ad-ditionally induces a wide range of length and time scales in the aggregatedproblem. Moreover, the interface at which the fluid and the structure interactconstitutes a free-boundary, i.e., its position is part of the solution and there-fore unknown a priori. This free-boundary character causes fluid–structure-interaction problems to be inherently nonlinear through the geometry of thefluid domain.

1Numerical Mathematics and Scientific Computation, Oxford University Press, 2001, [12, p. 1].

2 1. Introduction

1.1.1 Numerical simulation of fluid–structure interaction

Classically, the main motivation for the numerical simulation of fluid–structureinteraction has been the prediction of aeroelastic phenomena in aerospace andcivil engineering applications. Since the 1990s, many computational techniqueshave been developed and analyzed 1) to handle the moving domain underly-ing the fluid subproblem; see the early work by Farhat, Lesoinne and Maman[91], LeTallec and Mouro [153] and the overviews in [212, 234], 2) to impose thecoupling conditions in a conservative manner; see Farhat, Lesoinne and LeTal-lec [90] and also [217, 218], and 3) to time-integrate the coupled problem bysimple (loosely-coupled) partitioned procedures in a stable and accurate man-ner; see the overview by Farhat [88] and references therein and, in particular,[92, 184].

As of the late 1990s, another important motivation for the numerical simula-tion of fluid–structure interaction is hemodynamics (vascular fluid dynamics orblood flow) in biomechanical applications; see the early works in [173, 192, 232],and recent applications in [3, 20, 110]. For these problems, a particular dif-ficulty is the derivation of efficient iteration schemes to solve the nonlinearcoupled system. The popular loosely-coupled schemes (also called explicitschemes) used in aeroelastic applications are generally unstable. The originof this instability, appearing most prominently for incompressible fluids, is at-tributed to a significant so-called fluid added-mass effect. That is, typically, inthese applications the fluid and solid densities are comparable and/or the do-main is slender. A rigorous clarification of this numerical instability has onlyrecently been given by Causin, Gerbeau and Nobile [41]; see also [101, 215].The design of stable explicit coupling schemes in this context is still ongoingresearch; see [13, 62, 95, 109, 216] and, in particular, the recent result by Bur-man and Fernández [34]. On the other hand, efficient strongly-coupled schemes(also called monolitic or implicit schemes) have been devised by many authors,e.g., advanced fixed-point procedures [145], Newton-type methods [19, 94, 98],Newton-Krylov acceleration [164, 165, 231] and multigrid acceleration [220].

We are currently at a time where robust solvers can be devised for mostfluid–structure-interaction problems allowing, in principle, the computation ofsolutions to many complex practical problems. These computations need, how-ever, not be based on optimal discretizations.

1.1.2 Goal-oriented adaptive discretization

Numerical procedures for practical fluid–structure interaction problems re-quire substantial computational effort. Typically, most of the computationalresources are consumed by the fluid subsystem. Often, however, practical in-terest is restricted to a prescribed response quantity of the structure subsystem

1.1 Background 3

rather than full resolution of the complete coupled problem. In fact, one is oftennot so much interested in the solution itself, but uses the computed solution toquantify a certain goal. Example goals in aeroelastic computations are the struc-ture displacement at specific points, global forces acting on the structure suchas the lift or drag or the energy that is transferred from the fluid to the structure;see for instance [89, 92, 184]. Examples in biomechanical applications are thewall shear stress in the vicnity of aneurysms [199] or the volumetric flow ratesat arterial cross sections [210].

Goal quantities of interest can usually be written in the form of a functionalof the solution, the so-called goal functional, output functional or target func-tional. In computations, the accuracy of the goal quantity depends on both theaccuracy of the fluid approximation and the accuracy of the structure approxi-mation. In general, this dependence is non-obvious and any heuristic approachto constructing the meshes underlying the approximation spaces leads to aninefficient approximation of the goal quantity.

Finite-element discretization techniques employing goal-oriented adaptivestrategies can offer a significant efficiency improvement in such simulations.Starting with a coarse discretization, only those refinements are made whichsubstantially benefit to the accuracy of the goal quantity, in contrast to stan-dard, norm-oriented, adaptive strategies which make refinements that benefitthe accuracy of the solution in the full norm. If applied appropriately to fluid–structure interaction problems, goal-oriented adaptive strategies result in anoptimal discretization of both the fluid and structure variables for the goal func-tional under consideration, each with appropriate resolution.

Goal-oriented adaptive strategies rely on local refinement indicators toguide the adaptive procedure. These refinement indicators are obtained fromduality-based a-posteriori error estimates for the goal functional of interest.Such specific estimates for the error in the goal form the core of the adaptiveprocedure and they are referred to as goal-oriented error estimates. To computegoal-oriented error estimates, one requires the solution of a dual problem. For(non)linear problems, this dual problem is based on the (linearized) adjoint oper-ator. The corresponding dual solution, also called influence, extraction or gen-eralized Green’s function, indicates the precise spatial influence of the residualfunctional on the error in the goal. That is, it provides a residual weighting thatyields an estimate of the error in the goal.

(Norm-oriented) adaptive methods originate from the works of Babuskaand Rheinboldt [10, 11] in 1978, and are treated thoroughly in the books [1, 12,229]. Goal-oriented error estimation and goal-oriented adaptive methods havebeen developed in the late 1990s and as such are relatively new. They havemostly been applied to linear and nonlinear problems in solid and fluid me-chanics. Pioneering work in this field has been performed by Erikkson, Estep,

4 1. Introduction

Hansbo and Johnson [80], Becker and Rannacher [22], Giles, Süli and Hous-ton [112, 209], Prudhomme and Oden [191], Patera and Peraire [182], Stein andRüter [204], and Hoffman and Johnson [132]. This topic has also been includedin several recent books, see [14, 81, 133, 171]. An important recent develop-ment is the extension of goal-oriented error estimation and adaptivity to mul-tiphysics problems involving multiple coupled boundary value problems; seeLarson et al. [146–148], Estep et al. [36, 82, 83, 85] and Fick [99].

Although, in principle, the goal-oriented-error estimation framework ap-plies immediately to all (non-)linear problems that can be cast in canonical vari-ational form, fluid–structure-interaction problems elude this standard frame-work on account of their free-boundary character. Indeed, like many other free-boundary problems, the usual variational form of fluid–structure-interactionproblems is based on domain-dependent spaces, but the domain itself consti-tutes an unknown. It is exactly this unusual domain-dependent nonlinearitythat renders the derivation of the linearized-adjoint operator of fluid–structureinteraction problems highly nontrivial.

Several approaches have been suggested to bypass the derivation of the lin-earized adjoint. The first approach is by reformulating the fluid–structure in-teraction problem a priori such that the domain-dependence vanishes. This hasbeen pursued by Dunne [74, 75] who rewrites the structure to a Eulerian frame-work. The disadvantage being that one has to capture the interface afterwards.The second approach, by Grätsch and Bathe [119], is to replace the linearizedadjoint with a finite difference approximation. In this approach one, of course,has the problem of choosing a sufficiently small difference, but not too smallto affect conditioning. The third approach, which hasn’t been applied to fluid–structure interaction problems yet, is to compute the dual solution by a specific(infinite, defect-correction) iteration scheme between the subsystems, the limitof this iteration resulting in the solution of the (linearized) adjoint problem, cf.[85, 146].

We do not bypass the derivation of the linearized-adjoint, but aim to rigor-ously derive it. It can then be applied directly in the canonical goal-oriented-error estimation framework.

Let us allude to other fields that employ linearized-adjoints. Classical exam-ples are optimal control and inverse problems. A recent field is modeling-errorestimation and adaptivity [30, 179, 204]; see also [149, 177, 180, 183, 189, 205].

1.1.3 Linearization of the domain-dependence

To derive the linearized-adjoint operator of free-boundary problems, we needto linearize free-boundary problems with respect to the domain geometry. Thescientific field that deals with this type of linearization is called shape (differen-

1.2 Aims and scope 5

tial) calculus, and the corresponding derivatives are called shape derivatives. Itsorigins are attributed to Hadamard [123], who published a paper in 1907 onhow the solution of a biharmonic problem varies when the underlying domainis perturbed. Shape calculus has mostly been developed in the context of shapeoptimization by Simon [200], Pironneau [185, 186], and Zolésio [240]. Compre-hensive treatments can be found in the books of Sokołowski and Zolésio [202],Haslinger and Neittaanmäki [130], Delfour and Zolésio [59], and Moubachirand Zolésio [170]. Recent developments in shape derivatives under state con-straints can be found in [136, 137] and in shape derivatives for domains withcracks in [93, 102, 150]. Shape calculus has a wide range of recent applications,e.g., the computation of problems with stochastic boundaries [125].

In the context of shape optimization, there are two notions of derivativeswith respect to domain perturbations. The extension of these notions to asuitable linearization technique for free-boundary problems is not immediate.One notion is the so-called material derivative. Here, the perturbed domainsare mapped back to a fixed reference domain, and derivatives are obtained inthe reference domain by taking the derivative with respect to the correspond-ing domain map. We refer to the associated linearization technique as domain-map linearization. A more elegant notion is the so-called shape derivative. Thisderivative is independent of any domain map and can be interpreted as a localderivative, assuming functions have smooth extensions outside their domain.We refer to the associated linearization technique as shape linearization.

Domain-map and shape-linearization approaches have recently been in-vestigated for Newton-type iterative solution algorithms for free-boundaryproblems. In the context of fluid–structure-interaction problems, only thedomain-map linearization approach has been used; see [19, 94, 98]. The shape-linearization approach, on the other hand, has only been investigated for simplefree-boundary problems (in a formal sense); see Kärkkäinen and Tiihonen [140–142]. The use of domain-map and shape-linearization approaches to derivelinearized-adjoint operators in the context of goal-oriented error estimation hasto our knowledge not been investigated.

1.2 Aims and scope

The aim of this thesis is to develop techniques for the adaptive discretization offluid–structure interaction controlling the error in goal functionals of the solu-tion. In particular, we identify two primary objectives:

1. To investigate the linearization difficulties due to the free-boundary char-acter in fluid–structure interaction;

6 1. Introduction

2. To derive dual (linearized-adjoint) problems for fluid–structure interac-tion that lead to suitable goal-oriented error estimates.

We shall limit our scope to steady model problems. Specifically, we consider anincompressible fluid coupled with a low-order structural model, viz., a string(membrane) model. However, to focus entirely on the free-boundary character,we also consider a simpler free-boundary problem. For both of these problems,we show how they can be linearized using the domain-map linearization andshape linearization approaches. Moreover, based on these linearizations, wederive suitable dual problems. We demonstrate their applicability to generateadaptive discretizations employing established goal-oriented adaptive strate-gies.

1.3 Thesis overview

The thesis consists of three parts. The first part contains fundamental theoryand it essentially consists of known results unless indicated otherwise. Theother two parts contain applications and further developments of the theory toscalar free-boundary problems and fluid–structure interaction. These two partsconsist of essentially new results. Indeed, most of the results have not beenstudied before. Some related results may have been dealt with before but oftenwith less mathematical abstraction. We now briefly outline each part in moredetail.

Part I (Three Fundamentals) This part contains an introduction to three topicsthat are fundamental in this thesis.

Chapter 2 (Free-Boundary Problems and Fluid–structure Interaction) presents amathematical description of fluid–structure interaction in the context of free-boundary problems. It introduces the free-boundary model problem and theenvisaged fluid–structure interaction problem. For each problem, weak formu-lations are derived that are suitable for (Galerkin-type) discretization.

Chapter 3 (Goal-Oriented Error Analysis and Adaptivity) provides a brief ac-count of the theory behind goal-oriented adaptive methods. We present detailson the convergence behavior of goal errors in the form of a priori estimatesand show how a posteriori estimates are employed in adaptive discretizationprocedures. Since the theory is relatively new, we have tried to collect all thedirectly related results in the literature and aimed to present these results in ab-stract linear and nonlinear settings. In the specific case of linear boundary valueproblems, we present a definition of optimal adaptive discretizations from a re-cent approximation-theoretic perspective. Our motivation for the definition ofoptimal goal-adaptive discretizations appears to be new. In abstract nonlinear

1.3 Thesis overview 7

settings, we introduce the goal-oriented error estimation theory in the contextof nonsingular solutions. This natural starting point has to our knowledge notbeen taken before, but it puts the nonlinear framework on a more solid basis.We end the chapter with a brief discussion of the complications that are en-countered when applying the theory to free-boundary problems.

Chapter 4 (Shape Differential Calculus) presents an introduction to the differ-ential calculus that deals with varying domains. Most important are the deriva-tives of shape functionals, which are fundamental in free-boundary problems.

Part II (Scalar Free-Boundary Problems) In this part, we apply the fundamen-tals of Part I to study goal-oriented error estimation and adaptivity of the modelfree-boundary problem.

Chapter 5 (Domain-Map Linearization Approach to Free-Boundary Problems)presents the domain-map linearization approach. We introduce a domain mapand linearize with respect to this map to derive the dual problem. We demon-strate that the dual problem is essentially independent of the selected domainmap. Furthermore, we provide an interpretation of the dual problem by show-ing that it corresponds to a boundary value problem with a nonlocal boundarycondition.

Chapter 6 (Shape-Linearization Approach to Free-Boundary Problems) presentthe shape-linearization approach. We use the techniques of shape calculus tolinearize the domain-dependence and derive the dual problem. We show thatthe dual problem corresponds to a similar boundary value problem as before,but with a local Robin-type boundary condition which depends on the cur-vature of the boundary. In both chapters, we present numerical experimentsthat demonstrate the effectivity of the implied goal-oriented error estimates andtheir usefulness in goal-oriented adaptive mesh-refinement.

Part III (Incompressible Steady Fluid–structure Interaction) In this part, weapply the fundamentals of Part I to study goal-oriented error estimation andadaptivity of the fluid–structure-interaction model.

Chapter 7 (Domain-Map Linearization Approach to Fluid–structure Interaction)present the domain-map linearization approach. We linearize with respect tothe ALE (arbitrary Langrangian–Eulerian) map to obtain the dual problem. Weshow that this dual problem corresponds to a linear coupled fluid–structureproblem with a kinematically straightforward coupling. The other couplingcondition, however, depends in a nonstandard and nonlocal manner on thedual fluid. This nonlocality is similar to the one obtained for the model free-boundary problem in Chapter 5 and motivates a dual derivation using shapelinearization techniques.

Indeed, in Chapter 8 (Shape-Linearization Approach to Fluid–structure Interac-

8 1. Introduction

tion), we investigate the shape-linearization approach to fluid–structure interac-tion. We introduce a very weak form of the fluid–structure-interaction problemand show that it can be linearized using techniques from shape calculus. Thedual problem corresponds to the same linear fluid and structure subproblemswith the same kinematic coupling. The other coupling, however, although non-standard, is local.

We close the thesis with Chapter 9 (Conclusions and Future Prospect) whichpresents a discussion of the presented results, as well as some suggestions forfurther research.

Part I

Three Fundamentals

Chapter 2

Free-Boundary Problems and


If the facts don’t fit the theory, change the facts.

Albert Einstein (1879–1955)

From a mathematical perspective, fluid–structure interaction can be classifiedas a free-boundary problem. Such problems contain an unusual nonlinearityassociated with the free-boundary character. Accordingly, typical variationalforms of free-boundary problems are based on function spaces for which theunderlying domain is unknown a priori.

In this chapter, we give a mathematical description of fluid–structure inter-action in the light of free-boundary problems. First, in Sec. 2.1, we considera specific model free-boundary problem. This model problem serves as animportant guidance in later chapters, preceding our consideration of more in-volved fluid–structure-interaction problems. In Sec. 2.2, we consider the fluid–structure-interaction model problem. Finally, in Sec. 2.3, we introduce goalquantities and briefly consider their error under conforming discretizations.

2.1 Bernoulli free-boundary problem

Free-boundary problems1 arise in various applications such as free-surfaceflow, phase transitions (Stefan problems), electrochemical machining andevolving interfaces; see for example [50, 103, 122]. Its connection with fluid–structure interaction, however, is generally overlooked. Unlike ordinaryboundary value problems yet similar to fluid–structure interaction, in free-boundary problems a part of the boundary is unknown a priori, and has tobe found as part of the solution. For closure, an additional boundary conditionholds at the free boundary.

A standard model free-boundary problem is the Bernoulli (or Bernoulli-type) free-boundary problem; see [78, 100, 128], for instance. This model

1Unsteady free-boundary problems are also referred to as moving-boundary problems.

12 2. Free-Boundary Problems and Fluid–Structure Interaction

Ω

Γ

ΓD

Figure 2.1: Geometric set-up of a Bernoulli free-boundary problem: domain Ω, fixedboundary ΓD and free boundary Γ.

is preferred for its mathematical simplicity without losing the essential free-boundary character. In particular, it consists of the Laplace operator on a vari-able domain, with Dirichlet boundary conditions along the fixed boundary andDirichet and Neumann boundary conditions along the part corresponding tothe free boundary. In this section, we introduce the free-boundary problem andderive a straightforward weak formulation.

2.1.1 Problem statement

Let O denote the set of bounded open Lipschitz domains Ω ⊂ RN for which the

boundary ∂Ω consists of two complementary parts, viz., a fixed part ΓD anda variable part Γ, referred to as the free boundary; see Fig. 2.1. We formulatethe Bernoulli free-boundary problem as follows: Find a domain Ω ∈ O (orequivalently, its free boundary Γ) and a function u : Ω → R such that

−∆u = f in Ω ,

∂nu = g on Γ ,

u = h∣∣Γ

= 1 on Γ ,

u = h∣∣ΓD

on ΓD ,

(2.1a)

(2.1b)

(2.1c)

(2.1d)

where ∂n(·) := ∂(·)/∂n = ∇(·) · n is the normal derivative with n the out-ward unit normal, and f , g and h are sufficiently smooth functions defined onR

N. Moreover, g satisfies the lower bound g ≥ g0 > 0 and, in accordancewith (2.1c), h|Γ = 1 is required for all admissible free boundaries Γ. Note thata Dirichlet boundary condition is imposed on ΓD and both Dirichlet and Neu-mann boundary conditions are imposed on Γ.

Let us remark that for f = 0 and Γ ∩ ΓD = ∅ (typically, annular domains)this problem corresponds to the interior or exterior Bernoulli free-boundary prob-lem. A concise review of existence and regularity results as well as numericalsolution algorithms for this case is given by Flucher and Rumpf [100]. Othernumerical approaches can be found in, for instance, [29, 78, 128, 144, 219, 239].

2.1 Bernoulli free-boundary problem 13

To enable an interpretation of (2.1), we note that in two dimensions, thefunction u can be thought of as the stream function of a steady free-surfacepotential-flow problem. The constant Dirichlet condition at the free boundaryexpresses flow tangency and the Neumann boundary condition corresponds toa simplified version of Bernoulli’s equation (no surface tension); see for instance[141, 159]. Although the free-boundary problem is a problem in N dimensions,our interest is mainly N = 2, 3. Furthermore, since N = 1 is not trivial, we shalloccasionally also consider this setting.

2.1.2 Weak form of the free-boundary problem

For each Ω ∈ O, let H10,γ(Ω) denote the space of functions in H1(Ω) with zero

trace on γ ⊆ ∂Ω, i.e.,

H10,γ(Ω) :=

v ∈ H1(Ω) : v = 0 on γ

,

and let the (affine) space incorporating h be defined as

H1h(Ω) := h|Ω + H1

0,∂Ω(Ω) .

A weak formulation of (2.1) is obtained by multiplying (2.1a) with v ∈H1

0,ΓD(Ω), integrating over Ω, and integrating by parts the Laplacian. As v

is nonzero on Γ, we invoke (2.1b) to incorporate the Neumann boundary condi-tion weakly. Furthermore, the Dirichlet boundary conditions (2.1c) and (2.1d)can be imposed strongly. We then arrive at the variational formulation:

Find Ω ∈ O and u ∈ H1h(Ω) :

N((Ω, u); v

)= 0 ∀v ∈ H1

0,ΓD(Ω) .

(2.2)

where we have introduced the semilinear functional N defined as2,3

N(Ω, u; v

):=

∫

Ω

(∇u · ∇v − f v

)−

∫

Γg v .

Since for each domain Ω there is an associated free boundary Γ, we think of Γ

as a function of Ω, i.e., Γ = Γ(Ω). We note that a similar weak formulation hasbeen presented by Zhang and Babuska [239]. Of course, standard variationalarguments show that smooth solutions of (2.2) satisfy (2.1). In the sequel, we

2For semilinear functionals we use the convention that the functional is linear with respect tothe arguments after the semicolon “;”.

3For notational convenience we often neglect the integration measure in integrals. Domain andboundary integrals are to be integrated with respect to the usual volume and surface measure. Forexample, for φ ∈ L1(Γ), we write

∫Γ

φ instead of∫

Γφ dΓ.


assume there exists a (possibly nonunique) Lipschitz domain Ω ∈ O and acorresponding solution u ∈ H1

h(Ω) which solve (2.2).Because the solution consists of both Ω and u, the variational problem is of

mixed type. Note that the dependence on Ω yields an unusual nonlinearity.Furthermore, it is important to observe that (2.2) is a noncanonical variationalstatement in the sense that u and v reside in function spaces that depend on thesolution component Ω. This is a typical characteristic of free-boundary prob-lems.

2.2 Steady incompressible fluid–structure interac-

tion

Fluid–structure interaction has mostly been studied in the context of aeroe-lasticity and hemodynamics. In numerical applications of aeroelasticity, thefluid is typically modelled by the compressible Euler or Navier-Stokes equa-tions, and the structure consists mainly of lower-dimensional models such asbeams, plates and shells; see [88]. In hemodynamic computations, one mostlycombines viscous incompressible fluids with deformable solids or lower-dimensional structures such as membranes and shells. Lower-dimensionalmodels have also been preferred to simplify analysis [41, 109, 192].

In this work, we consider fluids governed by the incompressible Navier-Stokes equations or Stokes equations. Since our focus will be on the free-boundary character of the fluid subproblem, we shall mainly be interestedin simple lower-dimensional structure models for which the extension to de-formable solids is not too involved. To fix ideas, we shall describe a problem intwo dimensions, although the extension to three dimensions is straightforward.As a one-dimensional structure model we consider the vector-string equation;see for example [235]. String models have the computational advantage of be-ing low-order models. Moreover, we prefer a vector model over a scalar model,since in this case the traction coupling condition applies to the full fluid-tractionvector. A vector model is not only a more realistic counterpart of deformablesolids, but it also simplifies the ensuing analysis.

2.2.1 Fluid problem statement

Let Γ0 denote the reference configuration of the string. This also correspondsto the reference fluid–structure interface; see Fig. 2.2. Let θ : Γ0 → R

2 denotea (vector) displacement of the string. To each admissible θ, we associate thecurrent fluid–structure interface Γθ and the open bounded fluid domain Ωθ ⊂R

2. In general, the boundary of the fluid domain, ∂Ωθ , also consists of in- and

2.2 Steady incompressible fluid–structure interaction 15

Ωθ

ΓθΓ0

Γin Γout

θ

Figure 2.2: Geometric set-up of a fluid–structure-interaction problem: fluid domain Ωθ

and current fluid–structure interface Γθ for structure displacement θ, the reference inter-face Γ0, and in- and outflow boundaries Γin and Γout. The thick lines correspond to thewall boundaries Γwall.

outflow boundaries Γin/out and wall boundaries Γwall.Within the domain Ωθ , we consider the following fluid problem for the ve-

locity u : Ωθ → R2 and pressure p : Ωθ → R:

−ν ∆u + u · ∇u + ∇p = f

−div u = 0

in Ωθ ,

u = 0 on Γθ ,

(2.3a)

(2.3b)

where ν is the fluid kinematic viscosity and we assume f ∈ L2(R2) is a given

body force.4 The equations in (2.3a) are the incompressible Navier-Stokes equa-tions. The second equation in (2.3a) is the continuity equation expressing massconservation. The first equation in (2.3a) expresses momentum conservation,where the nonlinear convective term is defined as (u · ∇u)j = (u · ∇)uj =

∑i=1,2 ui (∂uj/∂xi) for j = 1, 2. If the convective term is omitted, then the equa-tions in (2.3a) reduce to the Stokes equations. Eq. (2.3b) is the kinematic cou-pling condition which in this steady case imposes a no-slip condition. On theother boundaries, suitable Dirichlet or Neumann boundary conditions hold. Atypical choice specifies no-slip on Γwall, nonzero inflow on Γin and zero tractionon Γout, i.e.,

u = 0 on Γwall ,

u = hin on Γin ,

−ν ∂nu + p n = 0 on Γout ,

where hin is a given inflow velocity. Let us denote these other boundaries col-lectively as ΓD (ΓN ) if Dirichlet (Neumann) boundary conditions are imposedon them. Note that ∂Ωθ = Γθ ∪ ΓD ∪ ΓN . We stipulate that both ΓD and ΓN arenonempty.

4For vector functions φ : G → RN, we write, for example, “φ ∈ L2(G)” meaning that each

component φj (j = 1, . . . , N) is in L2(G), where G ⊆ RN.


2.2.2 Structure problem statement

To state the structure subproblem, let Tθ := Id + θ : Γ0 → Γθ denote the (defor-mation) map that takes the points of the string in the reference configuration tothe current configuration, where Id : Γ0 → Γ0 is an identity map. The structuresubproblem for a string model is simply the traction coupling condition, wherethe fluid traction at the interface drives the string equation. That is, the string(vector) displacement θ : Γ0 → R

2 satisfies the (vector) equation

−E ∂2s θ = g +

((−ν ∂nu + p n) Tθ

)|∂sTθ | in Γ0 , (2.3c)

where the constant E is the string’s Young modulus, ∂s(·) := ∂(·)/∂s is thederivative along Γ0, and g : Γ0 → R

2 is a sufficiently smooth external (refer-ential) force on the string. Note that in (2.3c), the fluid traction is mapped tothe reference configuration by means of Tθ . The function |∂sTθ | : Γ0 → R isreferred to as a tangential Jacobian (or surface Jacobian) and corrects the fluidtraction for its evaluation in the reference configuration. For a horizontal Γ0such as depicted in Fig. 2.2, we have

|∂sTθ | =√

(1 + ∂sθ1)2 + (∂sθ2)2 .

Suitable boundary conditions need to be imposed on the string. We shall em-ploy homogeneous Dirichlet boundary conditions, i.e.,

θ = 0 on ∂Γ0 .

The presented vector-string model corresponds to the constitutive behav-ior of a so-called perfectly elastic material, in which tension is equal to E |∂sTθ |(rubber-like behavior); see [235] for a concise derivation.5

2.2.3 Coupled weak form with unbounded tractions

The specification of the envisaged fluid–structure-interaction problem in termsof partial differential equations for (θ, u, p) has now been given. The most im-portant equations are stated in (2.3a)–(2.3c). Let us emphasize the two-waycoupling of the subproblems: the fluid subproblem for (u, p) holds in the vari-able domain Ωθ for which θ is determined by the structure subproblem whichis driven by the fluid traction, −ν ∂nu + p n, at the interface Γθ .

A preliminary weak formulation of the coupled problem can be given byemploying standard weak formulations of the subproblems. To provide a suit-able functional setting for the string, let us denote by H1

0(Γ0) the space of

5We note that the derivation in [235] is not based on a small displacement assumption.


(vector-) functions in H1(Γ0) that vanish on the boundary ∂Γ0, i.e.,

H10(Γ0) :=

η ∈ H1(Γ0) : η = 0 on ∂Γ0

.

A suitable function space for fluid velocities is denoted by H10(Ωθ) which is

defined as the space of (vector-) functions in H1(Ωθ) with a zero trace on allboundaries where Dirichlet boundary conditions are imposed, i.e.,

H10(Ωθ) :=

v ∈ H1(Ωθ) : v = 0 on Γθ ∪ ΓD

.

To restrict our focus to the fluid–structure interface and not be distracted by theother boundaries, we assume that only homogeneous boundary conditions arespecified on ΓD and ΓN .

We can then state a preliminary weak formulation of the fluid–structure-interaction problem as follows: Find θ ∈ H1

0(Γ0) and (u, p) ∈ H10(Ωθ)× L2(Ωθ)

such that∫

Ωθ

(ν ∇u · ∇v + u · ∇u · v − p div v

)=

∫

Ωθ

f · v ∀v ∈ H10(Ωθ) , (2.4a)

∫

Ωθ

−q div u = 0 ∀q ∈ L2(Ωθ) , (2.4b)∫

Γ0

E ∂sθ · ∂sη =∫

Γ0

(g + (−ν ∂nu + p n) Tθ |∂sTθ |

)· η

∀η ∈ H10(Γ0) , (2.4c)

where ∇u · ∇v := ∑i=1,2 ∇ui · ∇vi. The first two equations (2.4a)–(2.4b) havebeen obtained by employing the standard weak formulation for the fluid prob-lem (2.3a)–(2.3b) with strongly enforced Dirichlet boundary conditions. In fixeddomains, many results on the existence6 and regularity of solutions (u, p), aswell as its numerical approximation based on such weak formulations, can befound in several books; see for example [113, 121, 151, 211] and for the Stokescase see also [5, 32]. The third equation (2.4c) has been obtained by employinga straightforward weak formulation for the elliptic problem in (2.3c), leavingthe right-hand side as is. Existence and regularity results of (2.4c) for givenright-hand sides independent of θ are elementary; see for example [31, 45, 81].

A fundamental shortcoming of the coupled weak formulation (2.4) is thatthe fluid traction appearing in the last equation is not necessarily defined forarbitrary velocities and pressures in the a priori indicated spaces. To addressthis issue, the associated term is rewritten in a consistent manner.7 Assuming

6For example, in the Navier-Stokes case, a solution (u, p) exists in the indicated spaces and isunique under a so-called small data condition (large enough viscosity). The pressure p is unique inL2(Ωθ) since we assumed that ΓN is nonempty. If ΓN = ∅, then p is unique up to a constant.

7The fluid traction term should actually be understood as a duality pairing. The consistentrewriting provides an interpretation of this pairing.


sufficient smoothness, note that we can transform the integral to Γθ by meansof T -1

θ , i.e.,∫

Γ0

((−ν ∂nu + p n) Tθ |∂sTθ |

)· η =

∫

Γθ

(−ν ∂nu + p n) · (η T -1θ ) .

The main idea is then to use a Green’s identity to replace this interface inte-gral by domain integrals and subsequently invoke the momentum equationin (2.3a). To make sense of η T -1

θ in Ωθ , we first introduce the operatorEθ : H1

0(Γ0) → H1(Ωθ) which maps a test function η to Γθ and then extendsit onto Ωθ , i.e., Eθ η := ExtΓθ

(η T -1θ ), where ExtΓθ

extends functions from Γθ

onto Ωθ such that they vanish on ∂Ωθ \ Γθ . Hence, we obtain∫

Γθ

(−ν ∂nu + p n) · (η T -1θ )

=∫

Ωθ

(f · (Eθ η) − ν ∇u · ∇(Eθ η) − u · ∇u · (Eθ η) + p div (Eθ η)

).

This final expression is indeed bounded for f , p ∈ L2(Ωθ) and u, Eθ η ∈H1(Ωθ). Note that this expression is equal to the residual form of the momen-tum equation, see (2.4a), with Eθ η substituted for v. The residual evaluatedat Eθ η is, in general, nonzero since Eθ η /∈ H1

0(Ωθ).Let us remark that this reformulation of the traction term is a well-known

procedure applied in finite-element discretizations of fluid–structure interac-tion, see for instance [19, 95, 111, 153]. In such discrete settings, the fluid trac-tion is essentially enforced in a weak sense. This procedure is also related to thepost-processing (or extraction) approach of outputs (such as stresses and fluxes)in the finite-element method as investigated in the early works of Babuska andMiller [7–9]; see also [12, p. 726] and [112, 161].

2.2.4 Coupled weak form with weakly-enforced tractions

To summarize the final coupled weak formulation, let us introduce the fol-lowing semilinear functionals. Let Rm and Rc denote the momentum- andcontinuity-equation residual functionals:

Rm(Ω, (u, p); v

):=

∫

Ω

(f · v − ν ∇u · ∇v − u · ∇u · v + p div v

),

Rc(Ω, u; q)

:=∫

Ωq div u ,

for all Lipschitz Ω, u, v ∈ H1(Ω) and p, q ∈ L2(Ω). Let us then define a semi-linear form for the total fluid problem, i.e.,

N(Ω, (u, p); (v, q)

):= −Rm(

Ω, (u, p); v)−Rc(Ω, u; q

).


Similarly, we define a bilinear form for the structure part as

S(θ, η

):=

∫

Γ0

E ∂sθ · ∂sη ,

for all θ, η ∈ H10(Γ0). The coupled weak formulation can then be condensed

into the following problem:

Find θ ∈ H10(Γ0) and (u, p) ∈ H1

0(Ωθ) × L2(Ωθ) :

N(Ωθ , (u, p); (v, q)

)= 0 ,

S(θ, η) −Rm(Ωθ , (u, p); Eθ η

)=

∫Γ0

g · η ,

∀(η, v, q) ∈ H10(Γ0) × H1

0(Ωθ) × L2(Ωθ) .

(2.5)

In principle, many other steady fluid–structure-interaction problems fit into theabove abstract form (with suitable spaces substituted), such as three dimen-sional extensions as well as more complicated structure models. The funda-mental two-way coupling can also be observed in this abstract form: The fluidsubproblem, the (v, q)-equation, is posed in the variable domain Ωθ , and thestructure subproblem, the η-equation, is driven by the (weak) fluid traction.

We have introduced the abstract semilinear forms in (2.5) to highlight theconnection of the fluid subproblem with free-boundary problems. Indeed, thedependence of the fluid part on the structure displacement θ is entirely via theunderlying fluid domain Ωθ . This dependence manifests itself through the firstargument of N as well as the function spaces upon which N

(Ωθ , (·, ·); (·, ·)

)

acts. This corresponds exactly to free-boundary problems; see Sec. 2.1.2.

2.2.5 Existence of coupled fluid–structure solutions

Even with the introduced weak traction term, the existence theory for a cou-pled solution (θ, u, p) of the fluid–structure-interaction problem (2.5) is highlynontrivial, and entails more than just the sum of existence theories for the indi-vidual subproblems. There are at least two problems that occur. The first issueis that for large structure displacements, collision of the structure with anotherfluid boundary can occur, for example. The problem of contact is, however, notaccounted for in the current setting. Hence, most existence theories for fluid–structure interaction require the structure solution to be small enough (enforcedby a suitable bound on the norm). This also ensures that displacements θ areadmissible in the sense that the corresponding structure deformation maps Tθ

(see Sec. 2.2.2) are one-to-one.The second issue is that in the current setting, structure displacements can

be rather nonsmooth, i.e., arbitrary in H10(Γ0). These displacements can give


rise to kinked fluid domains or, even worse, cusped fluid domains, which arenon-Lipschitz domains. In current existence theories, it is, however, preferredto work with Lipschitz domains. Therefore, structure displacements are a pri-ori assumed to be smoother. For example, a priori displacements in the non-integer Sobolev space H3/2+ǫ(Γ0) ∩ H1

0(Γ0) (with ǫ > 0 small) are used in thepioneering existence theory of Grandmont [115, 116]; see also [16]. Anotheroption would be to assume displacements in the non-Hilbertian Sobolev spaceW1,∞(Γ0) ∩ H1

0(Γ0) consisting of displacements for which their pointwise val-ues and derivatives are bounded almost everywhere. This space has been usedin the undervalued work of Saavedra and Scott [194], in the simpler setting of aLaplace fluid and scalar string, though. Various extensions of these works havebeen studied. For example, an extension to three dimensions and (non)linearsolids can be found in [117]. Extensions to unsteady fluid–structure interac-tion with various structural models are given in [44, 48, 49, 118] and referencestherein.

In the sequel, we assume the existence of an admissible displacement θ ∈W1,∞(Γ0) ∩ H1

0(Γ0), leading to a Lipschitz fluid domain Ωθ , and a correspond-ing velocity and pressure (u, p) ∈ H1

0(Ωθ) × L2(Ωθ) that solve (2.5).

2.3 Goal quantities and discretization errors

Given a solution (θ, u, p) of the fluid–structure-interaction problem (2.5), weshall be particularly interested in specific goal functionals Q of the solution. Anexample is the weighted structure displacement defined by

Q(θ) =∫

Γ0

qdisp · θ ,

where qdisp ∈ L2(Γ0) is a given vector function. By localizing qdisp, the goalcorresponds to local displacements. Another example is the weighted vorticityof the fluid:

Q(θ; u) =∫

Ωθ

qω ∇× u ,

where ∇× u = ∂u2/∂x1 − ∂u1/∂x2 and qω ∈ L2(R2) is a given function. Such a

goal can be useful in computing the circulation at local regions; see [187, p. 129].In general, we consider a bounded, possibly nonlinear, goal functional

(θ, u, p) 7→ Q(θ, u, p) : H10(Γ0) × H1

0(Ωθ) × L2(Ωθ) → R .

Let us consider an approximate solution (θh, uh, ph) to (2.5); see Fig. 2.3.Such an approximation can be obtained by employing a suitable discretization

2.3 Goal quantities and discretization errors 21

Ωθh

Γθh

Figure 2.3: A conforming approximation to the fluid–structure-interaction problem. Thedotted lines represent streamlines of the approximate velocity.

method; for example, a finite-element-type discretization based on the weakformulation (2.5); see, for instance, [19, 111]. In this work, we assume thatapproximations are conforming in the following sense: An approximate struc-ture displacement θh is in H1

0(Γ0), and a corresponding approximate velocitypressure pair (uh, ph) reside in H1

0(Ωθh) × L2(Ωθh) (hence, they live on Ωθh ).Note that, in general, we have that the corresponding value of the goal func-tional Q(θh, uh, ph) 6= Q(θ, u, p). The accuracy in Q can of course be improvedby suitably refining the discretization.

Recalling from Chapter 1 the aim of this thesis, we note that it correspondsto developing techniques for the goal-oriented adaptive discretization of (2.5),which leads to an optimal approximation of (θ, u, p) specifically for Q.

Chapter 3

Goal-Oriented

Error Analysis and Adaptivity

Whereas a mathematician is generally satisfied with a sufficient conditionfor convergence such as that of [Céa’s Lemma], this condition rightly ap-pears as a philosophical matter to many an engineer, who is much moreconcerned in getting (even rough) estimates of the error for a given spaceVh: For practical problems, one chooses often one, sometimes two, seldommore, subspaces Vh, but certainly not an infinite family. In other words, the

parameter h never approaches zero in practice!

Nevertheless, we found it worth examining such questions of conver-gence because [. . . ] the problem of estimating the error for a given h is atthe present time not solved in a satisfactory way.

Philippe G. Ciarlet, 1978The Finite Element Method for Elliptic Problems1

What is the approximation error in such simulations? How can the er-ror be measured, controlled, and effectively minimized? These ques-tions have confronted computational mechanics, practitioners, and theo-rists alike since the earliest applications of numerical methods to problemsin engineering and science.

Concrete advances toward the resolution of such questions have beenmade in the form of theories and methods of a posteriori error estimation,whereby the computed solution itself is used to assess the accuracy.

Mark Ainsworth and J. Tinsley Oden, 2000A Posteriori Error Estimation In Finite Element Analysis2

Galerkin’s method is a general discretization technique used for many prob-lems in fluid and solid mechanics. In this method, the approximation satisfies asuitable weak formulation of the problem on a discrete approximation space. Toobtain a more accurate approximation, one enlarges the approximation space,typically by uniformly refining the underlying mesh. Although this process of

1North-Holland, 1978, [45, p. 106]2Pure and Applied Mathematics, Wiley, 2000, [1, p. 1].

24 3. Goal-Oriented Error Analysis and Adaptivity

uniform refinement increases the accuracy, it is often inefficient with respect tothe accompanied increase in the number of degrees of freedom. A sequence ofapproximation spaces generated by uniform refinement (or any other heuristicrefinement procedure) is therefore nonoptimal.

Optimal discretizations can be obtained by adaptive refinement. A judiciouschoice of where to refine the approximation space is based on local refinementindicators which are usually extracted from a computable estimate of the error.These estimates are called a posteriori error estimates. Specific a posteriori esti-mates for the error in goal functionals of the solution are known as goal-orientederror estimates and the associated adaptive procedures are called goal-orientedadaptive methods.3

In this chapter, we provide a brief account of the theory behind goal-oriented adaptive methods. Our main concern are errors in goal quantities,but we also consider errors measured in the norm. In particular, we give de-tails on what one can expect of the convergence behavior of goal errors in theform of a priori error estimates. Furthermore, we show how goal-oriented er-ror estimates are obtained and how these estimates are used in an adaptivediscretization procedure. We start with abstract linear problems in Sec. 3.1, af-ter which we go into more detail discussing linear boundary value problems inSec. 3.2. In Sec. 3.3, we consider generalizations to abstract nonlinear problems.This chapter ends with a brief discussion of nonlinear complications pertainingto free-boundary problems in Sec. 3.4.

3.1 Abstract linear problems

For abstract linear problems, there are general results available on the conver-gence of Galerkin approximations. For a posteriori estimates, it is important torelate the errors to the residual. The results we present in this section can befound in many monographs; for a priori estimates we refer to [31, 45, 81] andfor a posteriori estimates see [1, 12, 22, 80, 112, 191], for example.

3.1.1 Functional setting

Let U denote a Banach space, V a reflexive Banach space and let B : U ×V → R

and L : V → R denote a continuous bilinear and linear functional, i.e., thereare constants cB, cL > 0 such that

|B(u, v)| ≤ cB ‖u‖U ‖v‖V ∀u ∈ U , ∀v ∈ V ,

|L(v)| ≤ cL ‖v‖V ∀v ∈ V .

3The “goal-oriented” terminology has been coined by Prudhomme and Oden; see [187, 190].

3.1 Abstract linear problems 25

Denoting by V∗ the dual space of V, i.e., the space of continuous linear func-tionals on V, we have L ∈ V∗. The abstract linear problem is then given by thefollowing linear variational problem:

Find u ∈ U :

B(u, v) = L(v) ∀v ∈ V .(3.1)

The existence of a unique solution to (3.1) can be established by a well-posedness theorem. The following is the most general theorem possible in thecurrent setting. It is a generalization of the celebrated Lax–Milgram Theorem.4

Theorem 3.A (Banach–Necas–Babuška Theorem on Well-Posedness) Let B

be defined as above. If and only if B satisfies the inf-sup conditions on U × V,that is, there exists a constant cB such that:5

infu∈U\0

supv∈V\0

B(u, v)

‖u‖U ‖v‖V

≥ cB > 0 , (3.2a)

∀v ∈ V ,(∀u ∈ U , B(u, v) = 0

)⇒

(v = 0

), (3.2b)

then, problem (3.1) has a unique solution. Moreover, the solution u of (3.1)satisfies the a priori estimate:

‖u‖U ≤ cL/cB . (3.3)

Proof See [196, p. 16] or [81, p. 85]. ¤

Note that the inf-sup conditions are not only sufficient but also necessary forwell-posedness. The constant cB in (3.2a) is referred to as the inf-sup constant.In view of the stability estimate (3.3), the solution depends continuously on thedata. Hence, if (3.2) holds, problem (3.1) is said to be well-posed in the sense ofHadamard.

The abstract problem can be written equivalently in operator form. Let B :U → V∗ be defined as the unique continuous linear map that satisfies

〈Bu, v〉V∗,V = B(u, v) ∀u ∈ U , ∀v ∈ V ,

where we introduced the notation 〈·, ·〉V∗,V to denote a duality pairing of V∗

with V. Note that we also have 〈L, v〉V∗ ,V = L(v). Problem (3.1) is then equiv-alent to finding u ∈ U such that

Bu = L in V∗ .

4Theorem 3.A is also referred to as the Generalized Lax–Milgram Theorem or Babuška–Lax–Milgram Theorem. We adopt the terminology introduced by Ern and Guermond [81].

5Throughout this thesis, we occasionally abuse the inf and sup notation by writing for example

supv∈V

f (v)‖v‖V

instead of supv∈V\0

f (v)‖v‖V

.


Regarding B, the inf-sup conditions imply that it is a bijective operator; in-deed (3.2a) implies injectivity of B with closed range and (3.2b) implies surjec-tivity of B; see [81, p. 85].

3.1.2 Galerkin approximations

Let us consider conforming Galerkin approximations of problem (3.1). We de-note by Uh ⊂ U a trial approximation subspace and by Vh ⊂ V a correspondingtest subspace. The Galerkin discretization of (3.1) is then given by:

Find uh ∈ Uh :

B(uh, vh) = L(vh) ∀vh ∈ Vh(3.4)

The approximate problem in (3.4) is well-posed if and only if B satisfies the inf-sup conditions on Uh × Vh, i.e., B satisfies (3.2a)–(3.2b) with U, V replaced byUh, Vh.6 We shall denote the corresponding inf-sup constant by ch

Band call it

the discrete inf-sup constant.An important statement of the error u − uh satisfied by Galerkin approxi-

mations is provided by the Galerkin orthogonality property:

B(u − uh, vh) = 0 ∀vh ∈ Vh . (3.5)

This follows easily from (3.1) and (3.4) using Vh ⊂ V. The convergence ofGalerkin approximations is implied by the following well-known theorem.

Theorem 3.B (Céa’s Lemma on Norm Convergence) Assume that prob-lems (3.1) and (3.4) are well-posed and have solutions u and uh, respectively.Then, the following a priori error estimate holds:

‖u − uh‖U ≤(1 + cB/ch

B

)inf

φ∈Uh‖u − φ‖U .

Proof Consider an arbitrary φ ∈ Uh. Applying the triangle inequality andsubsequently the necessary inf-sup condition of B on Uh × Vh, we obtain

‖u − uh‖U ≤ ‖u − φ‖U + ‖uh − φ‖U

≤ ‖u − φ‖U +1

chB

supvh∈Vh

B(uh − φ, vh)

‖vh‖V

.

6In the case that Uh and Vh are finite-dimensional with dim Uh = dim Vh, then (3.4) is well-posed if and only if B satisfies only (3.2a) on Uh × Vh; see [81, p. 91].


Next, invoking Galerkin orthogonality, see (3.5), and subsequently using conti-nuity of B, we obtain

‖u − uh‖U ≤ ‖u − φ‖U +1

chB

supvh∈Vh

B(u − φ, vh)

‖vh‖V

≤(1 + cB/ch

B

)‖u − φ‖U .

The proof follows by taking the infimum of all φ in Uh. ¤

Note that the smallest error possible that can be obtained in the approximationspace Uh is the so-called best-approximation error

infφ∈Uh

‖u − φ‖ . (3.6)

Theorem 3.B states that the error of the Galerkin approximation is proportionalto this best-approximation error. In other words, Galerkin approximations arequasi-optimal.

The inequality in Theorem 3.B is the starting point for a priori estimates ofthe error given a particular family Uhh of approximation spaces. Althoughthese estimates provide insight in the convergence rate with respect to Uhh,they are not very practical in providing computable quantitive estimates of theerror. This is the aim of a posteriori error estimates.

A particularly attractive class of a posteriori error estimates is based on es-timating the residual. To see why, let us denote the residual at uh by the contin-uous linear functional v 7→ R(uh; v) : V → R defined as

R(uh; v) := L(v) −B(uh, v) . (3.7)

If we denote the error by e := u − uh, then it is clear that the error satisfies

B(e, v) = R(uh; v) ∀v ∈ V . (3.8)

Hence, the residual contains all information of the error. This is corroboratedby the following theorem.

Theorem 3.C (Error–Residual Equivalence) Assume that problem (3.1) iswell-posed and has solution u. Then, for any approximation uh ∈ U the fol-lowing equivalence holds:

cB ‖u − uh‖U ≤ ‖R(uh; ·)‖V∗ ≤ cB‖u − uh‖U ,

where the dual norm ‖ · ‖V∗ is defined as

‖R(uh; ·)‖V∗ := supv∈V

R(uh; v)

‖v‖V

.


Proof From (3.8) it follows that

‖R(uh; ·)‖V∗ = supv∈V

R(uh; v)

‖v‖V

= supv∈V

B(u − uh, v)

‖v‖V

.

Then the upper bound follows using continuity of B. The lower bound followsfrom the inf-sup condition of B on U × V; see (3.2a). ¤

Theorem 3.C essentially shows that an estimate of the residual implies an esti-mate of the error.

3.1.3 Duality, adjoints and errors in goal quantities

Error estimates in norms that are weaker than the U-norm can be obtained byusing the classical Aubin–Nitsche Lemma; see [45, p. 137] or [81, p. 98]. Theseestimates rely on the introduction of a suitable dual problem. The same tech-nique underlies error estimates in goal quantities of the solution. Such esti-mates have been put forward in the early works of Babuška and Miller; see [7–9] or [12, p. 719].

Let us consider a goal functional

Q : U → R ,

which is assumed to be a continuous linear functional, i.e., Q ∈ U∗.7 The dualproblem is then given by:

Find z ∈ V :

B(δu, z) = Q(δu) ∀δu ∈ U .(3.9)

Compared with problem (3.1), the so-called primal problem, the test and trialspaces in the dual problem have reversed roles. Note that the dual problem isan adjoint problem. Indeed, let B∗ : V → U∗ denote the adjoint (dual or trans-pose) of B, which is defined as the unique continuous linear map that satisfies

〈δu, B∗v〉U,U∗ = 〈B δu, v〉V∗ ,V = B(δu, v) ∀δu ∈ U , ∀v ∈ V ;

see [193, p. 92] or [178, p. 471], for example. Then problem (3.9) is equivalent tofinding z ∈ V such that:

B∗z = Q in U∗ .

Let us from now on assume that the Banach space U is reflexive. The well-posedness of the dual problem then follows from Theorem 3.A applied to (3.9).

7We consider nonlinear functionals Q in Sec. 3.3.


Essentially, this requires that the adjoint inf-sup conditions are satisfied, i.e.,there exists a constant cB∗ such that

infv∈V

supδu∈U

B(δu, v)

‖δu‖U‖v‖V

≥ cB∗ > 0 , (3.10a)

∀δu ∈ U ,(∀v ∈ V , B(δu, v) = 0

)⇒

(δu = 0

). (3.10b)

These conditions are, however, equivalent to the inf-sup conditions (3.2a)–(3.2b) with cB∗ = cB. Since this equivalence result is not very well-knownin literature, we provide a proof in the Appendix; see Prop. A.9. Basically, whatthis means is that if the primal problem is well-posed, then so is the dual prob-lem (and vice versa).

The most basic relation showing the usage of the dual problem is the primal-dual equivalence identity:

Q(u) = B(u, z) = L(z) ,

where u and z are the solutions of the primal and dual problem, (3.1) and (3.9),respectively. This identity shows that the goal quantity of u (which is a functionof L) can be obtained without resorting to u, but directly from L (in particular,its value at z).

The dual problem (3.9) also provides the key to errors in the goal quantity:

Q(u) −Q(uh) = Q(u − uh) = B(u − uh, z)

= B(u − uh, z − ψ) ∀ψ ∈ Vh , (3.11)

where we invoked Galerkin orthogonality (3.5) in the last step. If we next usecontinuity of B, we obtain an a priori estimate. We summarize this result in thefollowing theorem.

Theorem 3.D (Babuška–Miller Theorem on Quantity Convergence) Let theassumptions of Theorem 3.B hold and let z denote the solution of the dual prob-lem (3.9). Then the following a priori estimate holds:

|Q(u) −Q(uh)| ≤ cB ‖u − uh‖U infψ∈Vh

‖z − ψ‖V .

Proof Apply continuity of B to (3.11) and take the infimum of all ψ in Vh. ¤

This theorem should be compared with Céa’s Lemma; see Theorem 3.B. Be-cause of the product of two norms in the estimate, goal functionals of the solu-tion generally have a higher rate of convergence than the solution itself (mea-sured in ‖ · ‖U). This is even more clear if we substitute the result of Céa’s


Lemma (Theorem 3.B):

|Q(u) −Q(uh)| ≤ c infφ∈Uh

‖u − φ‖U infψ∈Vh

‖z − ψ‖V , (3.12)

with c = cB (1 + cB/chB). Eq. (3.12) shows that the quantity error is bounded

by two best-approximation errors; see [112] for additional remarks.To estimate errors in goal quantities, it is advantageous to relate the quan-

tity error to the residual R(uh; ·) defined in (3.7). Remarkably, the error can beexpressed exactly:

Q(u) −Q(uh) = R(uh; z − ψ) ∀ψ ∈ Vh , (3.13)

which follows from (3.11) using (3.1) and (3.7). This equation is therefore calledan error representation formula. We note that for boundary value problems, theright-hand side can generally be expressed in terms of local residuals.

3.2 Linear boundary value problems

In this section, we consider the results of Sec. 3.1 in the light of boundaryvalue problems. In particular, we examine the precise advantage of optimaldiscretizations over uniform discretizations. Furthermore, we present the es-sentials of an adaptive procedure which aims at generating optimal discretiza-tions.

3.2.1 Dirichlet problem for Laplace operator

As a model boundary value problem, we consider the Poisson equation withhomogeneous Dirichlet boundary conditions. Let Ω ⊂ R

N denote a polyhedral(polygonal in R

2) domain. The boundary value problem for u : Ω → R is thenspecified by:

−∆u = f in Ω ,

u = 0 on ∂Ω ,

(3.14a)

(3.14b)

where f ∈ L2(Ω). The standard weak formulation of (3.14) is given by theabstract linear problem (3.1) with U = V = H1

0(Ω) and

B(u, v) =∫

Ω∇u · ∇v ,

L(v) =∫

Ωf v .

3.2 Linear boundary value problems 31

We note that the H1(Ω)-seminorm defines a norm on H10(Ω) (also referred to

as the energy norm), i.e.,

‖v‖H10 (Ω) := |v|H1(Ω) :=

√∫

Ω|∇v|2 .

It follows from the Cauchy-Schwarz inequality that B(u, v) is continuous withcB = 1. Furthermore, B is H1

0(Ω)-coercive, i.e.,

B(v, v) ≥ ‖v‖2H1

0 (Ω) ∀v ∈ H10(Ω) . (3.15)

Well-posedness follows by noting that (3.15) implies the inf-sup conditions (3.2)with cB = 1. Of course, well-posedness follows also directly from the Lax–Milgram Theorem.8

A Galerkin approximation based on a subspace Vh ⊂ H10(Ω) is defined

by (3.4) setting Uh = Vh. Since coercivity carries over to subspaces, such anapproximation is also well defined.

3.2.2 Convergence of uniform discretizations

Let us consider the convergence of Galerkin approximations by uniform finite-element discretizations. To describe such a discretization, we introduce a finite-element partition of Ω. Let τ := τ(Ω) denote a conformal, simplicial partitionof Ω, i.e., τ is a finite collection of open nonoverlapping elements K (trianglesin two and tetrahedra in three dimensions), such that

Ω = int( ⋃

K∈τK)

;

see [81, p. 34]. In two dimensions τ corresponds to a triangular mesh withouthanging vertices. Next, consider a quasi-uniform family τhh of partitions,where h > 0 denotes the diameter of the largest element in τh, i.e.,

h := max

diam K , K ∈ τh

.

Note that quasi-uniformity implies that the diameter of the inscribed ball ineach element is always larger than some constant times h; see [31, p. 107]. Inshort, τh consists of shape-regular elements approximately of size h. We shallsimply call τhh a uniform family of meshes and τh a uniform mesh.

8Nonstandard weak formulations for (3.14) are provided by discontinuous Galerkin methods;see [2, 46]. We stress that most of these formulations need not be coercive on infinite-dimensionalspaces. Well-posedness can then be guaranteed by the more general Theorem 3.A; see [6]. A non-conventional discontinuous Galerkin formulation which is coercive is presented in [222] though.


The finite-element discretizations are based on piecewise polynomial spacesSp(τ) of order p ≥ 1, i.e.,

Sp(τ) :=

v ∈ H10(Ω) : v|K ∈ P

p(K), ∀K ∈ τ

,

where Pp(K) is the space of polynomials up to order p on K. We consider p to

be fixed throughout. We denote by

n := dim Sp(τ) ,

the number of degrees of freedom in Sp(τ). Note that on uniform meshes thereis a c > 0 such that

n -1/c ≤ hN ≤ c n -1 , (3.16)

for Sp(τh) 6= ∅. In two dimensions this means that n ≈ c h -2.The convergence behavior of Galerkin approximations on uniform meshes

can be determined using the following result from approximation theory: Thereexists an interpolation operator Π : H1

0(Ω) ∩ Hs+1(Ω) → Sp(τh), where 0 ≤s ≤ p, such that

‖v − Πv‖H10 (Ω) ≤ c hs |v|Hs+1(Ω) , ∀v ∈ H1

0(Ω) ∩ Hs+1(Ω) ; (3.17)

see [31, 81]. Combining this with Céa’s Lemma, see Theorem 3.B, yields thefollowing result.

Proposition 3.1 Let u ∈ H10(Ω) ∩ Hs+1(Ω) with 0 ≤ s ≤ p denote the weak

solution to (3.14) and let uh ∈ Sp(τh) denote the Galerkin approximation withrespect to the uniform mesh τh. Then the following a priori estimate holds:

‖u − uh‖H10 (Ω) ≤ c hs |u|Hs+1(Ω) .

In terms of number of degrees of freedom n, the a priori estimate reads:

‖u − uh‖H10 (Ω) ≤ c n - s/N |u|Hs+1(Ω) .

Proof Theorem 3.B applies since its assumptions are satisfied. The estimate interms of h then follows by noting that the best-approximation error is smallerthan the error of the interpolant, ‖u − Πu‖U , which in turn can be estimatedusing (3.17). The estimate in terms of n follows using (3.16). ¤

Proposition 3.1 implies that if the solution is regular enough (as smooth asHp+1(Ω)), then the (asymptotic) convergence is O(hp) as h → 0 or, in termsof n, O(n -p/N) as n → ∞. Such a convergence rate of order p ( -p/N) is called an


optimal convergence rate with respect to h (n). Unfortunately, if the solution isnot smooth enough, then the convergence is suboptimal and can be arbitrarilyslow depending on the level of nonsmoothness.

Let us next consider the convergence of goal functionals of the solution.Assume for example that the goal functional is defined as a weighted integralof the solution, i.e.,

Q(u) =∫

Ωq u ,

where q ∈ L2(Ω). Since the boundary value problem (3.14) corresponds toa self-adjoint operator B : H1

0(Ω) → H -1(Ω), the dual problem (3.9) for z ∈H1

0(Ω) corresponds to the weak form of:

−∆z = q in Ω ,

z = 0 on ∂Ω .

(3.18a)

(3.18b)

The convergence behavior in Q then follows by combining the Babuška–MillerTheorem, see Th. 3.D, with the interpolation estimate (3.17).

Proposition 3.2 Let u ∈ H10(Ω) ∩ Hs+1(Ω) with 0 ≤ s ≤ p denote the weak

solution to (3.14) and let uh ∈ Sp(τh) denote the Galerkin approximation withrespect to the uniform mesh τh. Furthermore, let z ∈ H1

0(Ω) ∩ Ht+1(Ω) with0 ≤ t ≤ p denote the weak solution to (3.18). Then the following a prioriestimate holds:

|Q(u) −Q(uh)| ≤ c hs+t |u|Hs+1(Ω) |z|Ht+1(Ω) .

In terms of number of degrees of freedom n, the a priori estimate reads:

|Q(u) −Q(uh)| ≤ c n - (s+t)/N |u|Hs+1(Ω) |z|Ht+1(Ω) .

Proof The right-hand side of the inequality in Theorem 3.D can be boundedusing Prop. 3.1 and (3.17). ¤

This proposition essentially shows the higher-order convergence for goal func-tionals that can be expected if the dual solution is regular enough. Indeed, theoptimal rate for goal functionals is 2p ( -2p/N) with respect to h (n). If, however,the goal functional is such that the corresponding dual solution is not smoothat all, then the convergence essentially reduces to the same rate as for the norm.

It is well-known that regularity issues are not mere academic issues, butare frequently encountered in practical problems. In general, the regularityof solutions to boundary value problems is negatively affected by nonsmoothdomains (in particular reentrant corners), irregular (boundary or interior) dataand mixed boundary conditions [120, 155].


ve

Figure 3.1: Illustration of newest-vertex bisection. The newest-vertex of an element isindicated by a dot. The shaded triangle is marked for refinement (left). The refinementcauses a hanging vertex in the mesh (middle). Two additional bisections are requiredfor completion (right).

As an illustrative example, suppose that Ω is a convex polygon in two di-mensions, and f and q are in L2(Ω), but not smoother. Then it is known that uand z are in H2(Ω); see for instance [120]. Hence, using linear finite elements onuniform meshes, one can expect optimal convergence of O(h) (O(n -1/2)) in theenergy norm, and O(h2) (O(n -1)) in goal functionals. The use of quadratic (oreven higher order) finite elements makes no difference from an (asymptotic)convergence point of view. It would, of course, matter on optimal adaptivemeshes.

3.2.3 Convergence of optimal discretizations

Let us consider optimal discretizations in the context of adaptive mesh-refinement. To describe such discretizations, we consider meshes that can begenerated from an initial coarse mesh by refining its elements. Various mesh-refinement techniques are described in [166, 229], for example. We shall fo-cus on a mesh-refinement technique which in two dimensions is referred to asnewest-vertex bisection. This bisection technique is becoming very popular sinceit has many theoretically advantageous properties; see [26, 157, 208].

In two dimensions, the bisection technique can be described as follows.Given a triangular mesh, suppose that several elements are marked for refine-ment. An element that is to be refined is bisected by connecting its newest-vertex v to the midpoint of the opposite edge e; see Fig. 3.1. This creates twonew triangles for which their newest-vertex is defined as the midpoint of e. Ofcourse, one has to manually assign newest-vertices to all elements of the ini-tial mesh. To make sure that the resulting mesh is conformal, i.e., there are nohanging vertices, one introduces additional refinements; a process referred toas completion.

To describe approximation properties of spaces based on adaptive meshes,we shall introduce some concepts from nonlinear approximation theory. These


concepts have originally been studied in the context of wavelet methods; see[47, 65, 66]. Only recently have these concepts been used to study optimality ofadaptive finite-element discretizations; see [26, 40, 104, 167, 206, 207]. We shall,however, use the theory only to introduce a definition of optimal discretiza-tions.

Let us denote by τm an admissible mesh in the sense that it can be gener-ated from some initial mesh τ0 by recursively applying the described bisectiontechnique. By m we denote the cardinality of τm (number of elements in τ), i.e.,

m = #τm .

We shall simply call τm an adaptive mesh and τmm a family of adaptivemeshes. The cardinality of τ0 is set as m0 := #τ0. Let us collect all adaptivemeshes τm with at most m ≥ m0 elements in

Tm :=

τm : m0 ≤ m ≤ m

.

With each τm, we associate an adaptive finite-element space Sp(τm). Notethat the number of elements m and corresponding degrees of freedom n =dim Sp(τm) are equivalent in the sense that there is a c > 0 such that

n/c ≤ m ≤ c n .

Moreover, if τm ∈ Tm, we obviously have

n/c ≤ m . (3.19)

An important question at this point is: How well can a function v ∈ H10(Ω)

be approximated using adaptive finite-element spaces? To quantify this, weintroduce the best-adaptive-approximation error

σm(v) := infτm∈Tm

infφ∈Sp(τm)

‖v − φ‖H10 (Ω) . (3.20)

Note that this is the minimal best-approximation error (cf. (3.6)) that can beachieved on an adaptive mesh with at most m elements. The approximabil-ity of v is then characterized by the (asymptotic) convergence rate on the bestadaptive mesh with respect to m:

σm(v) ≤ c m - s/N , (3.21)

where s > 0. The functions v that have a common convergence rate are col-lected in the approximation class

As(Ω) :=

v ∈ H1

0(Ω) : |v|As(Ω) < ∞

,


where the seminorm | · |As(Ω) is defined as the smallest constant c for

which (3.21) holds, i.e.,9

|v|As(Ω) := sup

m≥m0

m s/N σm(v) .

This seminorm definition is particularly useful to present an a priori error esti-mate for the best approximation on the best adaptive mesh. If we denote thisbest adaptive approximation, the φ which minimizes the inf’s in (3.20), by vm,then

‖v − vm‖H10 (Ω) ≤ m - s/N |v|

As(Ω) ≤ c n - s/N |v|As(Ω) . (3.22)

where we invoked (3.19) to introduce n, the number of degrees of freedom ofthe associated best adaptive space.

What can we say about the functions that belong to As(Ω)? In other words,

which functions can be approximated on adaptive meshes with a convergencerate of - s/N? Of course, on uniform meshes, it follows from Prop. 3.1 that atleast H1

0(Ω) ∩ Hs+1(Ω) is a subset of As(Ω) for s ≤ p (values of s larger than

p are not of interest as it concerns trivial functions). However, there are alsoless regular functions in A

s(Ω). These can be described using Besov smooth-ness spaces; see for instance [27, 65] and [26, Sec. 9]. Of particular interestare the functions that can be approximated by the optimal rate s = p. Inthis regard, we mention the important recent result that there are functions inthe optimal class A

p(Ω) that have barely more regularity than H10(Ω), i.e., in

H10(Ω) ∩ H1+ǫ(Ω) for ǫ arbitrarily small; see [106]. Such irregular functions

would have an extremely poor rate of convergence on uniform meshes, how-ever, they can be approximated optimally on adaptive meshes.

Although it is very unlikely that we are able to actually find the best adap-tive mesh and its best adaptive approximation (in reasonable, polynomial,time), we can at least expect our computable adaptive approximations to havea common rate of convergence. This motivates the following definition.

Definition 3.3 (Optimal Norm-Adaptive Discretization) Let u ∈ Ap(Ω) ⊂

H10(Ω) denote the weak solution of (3.14). A family τmm of adaptive meshes

is said to generate optimal norm-adaptive discretizations of (3.14) if

‖u − um‖H10 (Ω) ≤ c n -p/N |u|

Ap(Ω) ,

where um ∈ Sp(τm) is the Galerkin approximation of u with respect to τm. ¤

9We note that the usual definition of As(Ω) concerns a convergence behavior of m - s instead

of m - s/N. We adhere to our definition to keep the similarity with uniform meshes; see Sec. 3.2.2.Other equivalent seminorm definitions used in literature can be found in [207, Sec. 5].


This definition should be compared with the a priori result on uniform meshes;see Prop. 3.1. The main difference is of course that on optimal adaptive mesheswe always expect optimal convergence, even for irregular functions (in A

p(Ω)).To motivate a similar definition for adaptive discretizations that are optimal

with respect to a goal quantity of interest, we note from (3.12) that we now haveto consider the best adaptive mesh for the product of two best-approximationerrors:

σm(u, z) := infτm∈Tm

(inf

φ∈Sp(τm)‖u − φ‖H1

0 (Ω) infψ∈Sp(τm)

‖z − ψ‖H10 (Ω)

). (3.23)

This can be bounded as follows.

Proposition 3.4 If u ∈ As(Ω) and z ∈ A

t(Ω) with s, t > 0, then the followinga priori bound holds:

σm(u, z) ≤ c m - (s+t)/N |u|As(Ω) |z|At(Ω) .

Proof One might be tempted to separate the inf of the product in (3.23) intothe product of two best-adaptive-approximation errors, σm(u) σm(z). How-ever, this does not provide an upper bound on σm(u, z). We do have an upperbound if the best-adaptive-approximation errors are considered with half of theelements, i.e.,

σm(u, z) ≤ σm/2(u) σm/2(z) . (3.24)

To show this, let τm/2(u) and τm/2(z) denote the best adaptive meshes cor-responding to σm/2(u) and σm/2(z), respectively. We then consider the com-mon refinement of these meshes. This is the mesh τcom

m that is the combinatorialunion of both meshes, which can be thought of as refining several elementsin one mesh to include the refined elements of the other. It is obvious that#τcom

m ≤ #τm/2(u) + #τm/2(z) ≤ m. It then holds that

σm(u, z) ≤ infφ∈Sp(τcom

m )‖u − φ‖H1

0 (Ω) infψ∈Sp(τcom

m )‖z − ψ‖H1

0 (Ω) .

Note that Sp(τm/2(u)) ⊂ Sp(τcomm ) and similarly for τm/2(z), hence

σm(u, z) ≤ infφ∈Sp(τm/2(u))

‖u − φ‖H10 (Ω) inf

ψ∈Sp(τm/2(z))‖z − ψ‖H1

0 (Ω)

= σm/2(u) σm/2(z) ,

which proves (3.24). Finally, invoking (3.22), we obtain the proposition withc = 2(s+t)/N. ¤


In view of (3.12) (and (3.19)), this proposition has the following implication. Foreach m, there exists an adaptive mesh τm ∈ Tm such that the error in Q by theGalerkin approximation based on Sp(τm) is O(n - (s+t)/N) as n → ∞. This is themotivation for the following definition.

Definition 3.5 (Optimal Goal-Adaptive Discretization) Let u ∈ Ap(Ω) ⊂

H10(Ω) and z ∈ A

p(Ω) ⊂ H10(Ω) denote the weak solutions of (3.14) and (3.18),

respectively. A family τmm of adaptive meshes is said to generate optimalgoal-adaptive discretizations of (3.14) if

|Q(u) −Q(um)| ≤ c n -2 p/N |u|Ap(Ω) |z|Ap(Ω) ,

where um ∈ Sp(τm) is the Galerkin approximation of u with respect to τm. ¤

We stress that the presented definitions concern optimality of discretiza-tions. Optimal discretizations are necessary to have optimal computational com-plexity, which concerns the number of arithmetic operations required to obtainthe approximations. Complexity, however, is outside the scope of this work;we refer to the recent results in [104, 143, 167, 206, 207].

Having discussed what can be expected on adaptive meshes, let us considernext how one can actually construct such adaptively-refined meshes.

3.2.4 Adaptive discretization procedure

Adaptive discretization methods typically consist of the loop

Solve → Estimate → Mark → Refine .

For a goal-oriented adaptive finite-element procedure controlling the dis-cretization error in the goal functional Q, this can be detailed as follows:

0. Initialize i := 0 and start with a coarse mesh τi := τ0;

1. Solve the approximate primal problem on τi;

2. Estimate the error in Q, based on an approximate dual solution;

3. Mark elements in τi for refinement, based on the error estimate;

4. Refine τi to obtain τi+1; increment i and go back to 1.

In practice, one uses a stopping criterion after step 2, for example, the error es-timate is within a predefined tolerance. For procedures that control the error inmultiple goal functionals, we refer to [84, 127]. We note that a norm-orientedadaptive procedure differs only in step 2 where the error in the norm is esti-mated without resorting to a dual solution.


Having discussed step 4 in Sec. 3.2.3, let us comment on steps 2 and 3 inthe adaptive procedure. For these steps, there are various strategies availablein literature; see [22, 112, 182, 191] and, in particular, the overview in [204,Sec. 4]. The error estimate for Q can be a poor or good estimate; it can evenbe a guaranteed upper bound of the error. The main purpose of the estimate isto motivate the extraction of refinement indicators. Indeed, often the estimateleads to a sum of elemental contributions, which can be identified as refinementindicators. One then proceeds by marking the elements corresponding to thelargest (in absolute value) indicators for refinement. The rationale behind thisis that control of element contributions provides control of the error.

One of the most elementary approaches to obtaining refinement indicatorsis via so-called dual-weighted residuals also known as Type I a posteriori errorbounds; see [21, 22] and [134, 209], respectively. Since we shall employ this ap-proach in our work, we outline it for the model boundary value problem (3.14).The starting point is the error representation formula (3.13). Let u denote theweak solution of (3.14) and uh ∈ Sp(τ) its finite-element approximation on agiven mesh τ. Then (3.13) is specified by

Q(u) −Q(uh) = R(uh; z − ψ) =∫

Ω

(f (z − ψ) −∇uh · ∇(z − ψ)

),

for any ψ ∈ Sp(τ), where z is the weak solution of the dual problem (3.18). Ofcourse, the exact dual solution z is unknown, however, one can easily computea Galerkin approximation to it, say z ∈ V, which then provides an estimate ofthe error:

Q(u) −Q(uh) = R(uh; z − ψ) + R(uh; z − z) ≈ R(uh; z − ψ) .

Note that one has a useless error estimate if the approximate dual space V is asubspace of or equal to the primal space Sp(τ). In that case, R(uh; z − ψ) = 0on account of Galerkin orthogonality. Therefore, in practice, the dual problemis either solved using a larger space V ⊃⊃ Sp(τ), or it is solved on a dedicateddual-problem space such that V * Sp(τ) * V. For such choices of the dualspace, moreover, the dual error, z − z, is relatively small so that R(uh; z − z) canindeed be neglected. A common choice for V, which we shall also employ, is ahigher-order primal space, i.e., V = Sp+1(τ); see [22] for other options.

To obtain element refinement indicators, one rewrites the residual as a sumof local residual contributions as follows. First, we perform an element-wiseintegration by parts:

R(uh; z − ψ) = ∑K∈τ

( ∫

K

(f + ∆uh

)(z − ψ) −

∫

∂K∂nuh (z − ψ)

). (3.25)


Let Γint denote the set of interior element edges (faces in three dimensions). Wethen introduce for each element K ∈ τ the element interior residual rh

K and foreach interior edge e ∈ Γint the flux residual rh

e defined as

rhK := f + ∆uh in K ,

rhe := 1

2 [[∂nuh]] on e ,

where [[∂nuh]] := ∂nuh|K+ + ∂nuh|K− denotes the jump in the normal derivative(flux), K+ and K− being the elements on both sides of e. We summarize thefinal result in the following proposition.

Proposition 3.6 Let u ∈ H10(Ω) denote the weak solution to (3.14) and let

uh ∈ Sp(τ) denote its Galerkin approximation. Furthermore, let z ∈ H10(Ω)

denote the weak solution to (3.18) and let z ∈ V * Sp(τ) denote its Galerkinapproximation. Then the following error representation holds:

Q(u) −Q(uh) = R(uh; z − ψ) + R(uh; z − z) ,

for any ψ ∈ Sp(τ), and

R(uh; z − ψ) = ∑K∈τ

ηK ,

where the ηK ∈ R are computable dual-weighted residual contributions:

ηK :=∫

Krh

K (z − ψ) − ∑e⊂∂K∩Γint

∫

erh

e (z − ψ) ∀K ∈ τ . (3.26)

Proof We only need to show that (3.25) equals ∑K ηK. The K integral termsfollow by definition of rh

K. To show the equality of the edge terms, we collectthese terms edge-wise and then redistribute them again to the elements. Indoing so, we note that z − ψ = 0 on ∂Ω since z, ψ ∈ Sp(τ). Hence,

∑K∈τ

−∫

∂K∂nuh (z − ψ) = ∑

e∈Γint

−∫

e[[∂nuh]] (z − ψ) .

This last sum can again be written as a sum over elements of all element edgeintegrals, introducing the factor 1/2:

∑K∈τ

∑e∈∂K∩Γint

−∫

e

12 [[∂nuh]] (z − ψ) .

This finishes the proof. ¤


The practical meaning of Prop. 3.6 is that if z is accurate enough, then the errorin Q can be bounded by computable contributions:

|Q(u) −Q(uh)| / ∑K∈τ

|ηK| . (3.27)

Hence, this motivates to refine those elements for which |ηK| is large, i.e., onecan use |ηK| as the element refinement indicator.

We are still left with specifying ψ in ηK; see (3.26). Note that if |ηK| is to be asatisfactory indicator, it has to enforce the double rate of convergence expectedfor optimal goal-adaptive discretizations. This implies that ψ ∈ Sp(τ) has to bea good approximation of z; cf. the reasoning in [68]. In practice, one chooses ψas a local projection of z, such as the nodal interpolant.

Let us discuss some of the options for step 3, i.e., the marking strategies.It is clear that a fraction of elements need to be marked for refinement, mostlikely those with the largest indicators. Many strategies are described in [22,Sec. 5] and [112, Sec. 7], and in a more rigorous and generic setting in [169]. Thefollowing list provides a short overview:

• Fixed-fraction marking strategy. A fixed fraction λ ∈ (0, 1] of the elementsis marked corresponding to the largest indicators. This is the most basicstrategy.

• Maximum marking strategy. All elements K′ ∈ τ are marked for whichtheir indicator is at least a fraction λ ∈ [0, 1] of the maximal indicator, i.e.,

|ηK′ | ≥ λ maxK∈τ

|ηK| .

This is one of the first proposed strategies; see [11, Sec. 6]. The fixed-fraction marking strategy and this strategy are very popular owing totheir straightforwardness.

• Modified equidistribution marking strategy. All elements K′ ∈ τ aremarked for which their indicator is at least a fraction λ ∈ [0, 1] of theaverage indicator, i.e.,

|ηK′ | ≥ λ ∑K∈τ

|ηK|/#τ .

This strategy originates from the (heuristic) equidistribution principle: amesh with a given number of elements has the minimal error if the indi-cators are equidistributed among the elements; see [169].

• Dörfler-type marking strategy. A (close to) minimal set τ′ ⊂ τ of elementsis marked for which their sum is at least a fraction λ ∈ (0, 1] of the total


sum, i.e.,

∑K′∈τ′

|ηK′ | ≥ λ ∑K∈τ

|ηK| .

Introduced by Dörfler in [71], this strategy appears to be advantageous intheoretical studies of the convergence of adaptive methods; see [40, 169].

We shall employ the Dörfler-type marking strategy in our work.We remark that convergence of the above described goal-oriented adaptive

procedure is unproven, let alone optimal convergence (even for the consideredsimple boundary value problem (3.14)). That is, it is uncertain whether theobtained goal quantities Q(uh) converge to the exact value Q(u) as the meshis adaptively refined. Rigorous convergence analysis of adaptive methods isa recent topic in literature. Indeed, the first noteworthy convergence analysisappeared in 1996, see [71], when convergence was proved of a basic norm-oriented adaptive method for the Poisson equation in two dimensions. Sincethen, this result has been refined and extended, mainly to linear model prob-lems; see [38, 72, 139, 160, 168, 169], for example. Optimality of norm-orientedadaptive methods have only been established since 2004, see [26, 40, 206, 207],although it has been observed in many numerical computations decades ear-lier. The first noteworthy convergence (and optimality) analysis of a goal-oriented adaptive procedure, different from the approach in this section, hasonly just appeared; see [167]. Further investigations related to the analysis ofgoal-oriented strategies can be found in [52, 163, 174].

3.3 Generalization to nonlinear problems

In this section, we extend the analysis pertaining to abstract linear problems inSec. 3.1 to abstract nonlinear problems. Again, we let U denote a Banach spaceand V a reflexive Banach space. The variational form of the nonlinear problemthat we consider is:

Find u ∈ U :

N(u; v) = L(v) ∀v ∈ V .(3.28)

where N : U × V → R and L ∈ V∗ denote a semilinear and continuous lin-ear form, respectively. We assume the existence of (possibly nonunique) so-lutions u ∈ U. Moreover, to present the analysis, we shall consider so-callednonsingular solutions. At such solutions the derivative of N is well-behaved.We now also consider goal functionals Q : U → R that may be nonlinear. Be-fore we continue, however, we briefly review some elements of differentiationin Banach spaces.

3.3 Generalization to nonlinear problems 43

3.3.1 Differential calculus in Banach spaces

Let F : U → W∗ be a nonlinear map, where W is also a reflexive Banach space.We shall mainly be concerned with W = V or W = R. The following resultscan be found in many books on nonlinear analysis; see [236], for example.

We introduce two fundamental notions of directional differentiability in Ba-nach spaces. The Gâteaux derivative of F at u ∈ U in the direction δu ∈ U isdefined as the limit

F′(u)(δu) := limt→0

F(u + t δu) − F(u)

t, (3.29)

if it exists in W∗. Unfortunately, the map F′(u) : U → W∗ may not be linear.This is the reason to introduce the following definition.

Definition 3.7 (Gâteaux Differentiability) The nonlinear map F : U → W∗ issaid to be Gâteaux differentiable at u ∈ U if the Gâteaux derivative (3.29) exists forall δu ∈ U and the map δu 7→ F′(u)(δu) : U → W∗ is linear and continuous. ¤

A somewhat stronger notion of differentiability, which we require, is the fol-lowing.

Definition 3.8 (Fréchet Differentiability) The nonlinear map F : U → W∗ issaid to be Fréchet differentiable at u ∈ U if it is Gâteaux differentiable at u and

F′(u)(δu) = F(u + δu) − F(u) + o(‖δu‖U) , δu → 0 ,

for all δu ∈ U. In this case F′(u)(δu) is also called the Fréchet derivative of F at uin the direction δu. ¤

Fréchet differentiability has the following nice property: If F is Fréchet differ-entiable at u, then F is continuous at u. Furthermore, note that Fréchet differ-entiability implies Gâteaux differentiability but not vice versa. However, if themap u 7→ F′(u) is continuous at u, then the reverse argument holds. From nowon, if we consider derivatives, we mean Fréchet derivatives, unless specifiedotherwise.

Higher derivatives of order n ≥ 1 can be defined inductively and are de-noted F(n)(u)(δu1) · · · (δun). The classical Taylor’s Theorem has the followinggeneralization.

Theorem 3.E (Generalized Taylor’s Theorem) Assume that F′, F′′, . . . , F(n) ex-ist in a sufficiently large neighborhood of u. Then,

F(u + δu) = F(u) +n

∑k=1

1k!

F(k)(u)(δu)k + rn ,


where rn = o(‖δu‖n

U

)as δu → 0. Moreover, if F(n+1) exists and is continuous,

then rn = Rn = O(‖δu‖n+1

U

)where

Rn :=1n!

∫ 1

0(1 − t)n F(n+1)(u + t δu)(δu)n+1 dt .

As an important special case, n = 0, we obtain the Fundamental Theorem ofCalculus:

Corollary 3.9 (Fundamental Theorem of Calculus) Assume that F′ exists andis continuous on an open convex subset of U that includes u0 and u1. Then,

F(u1) − F(u0) = Fs(u1, u0)(u1 − u0) ,

where the secant form through u0 and u1 is defined as

Fs(u1, u0) :=∫ 1

0F′(u0 + t (u1 − u0)

)dt . (3.30)

The secant form is also referred to as a mean-value linearization.

3.3.2 Nonsingular solutions

We can now introduce the concept of nonsingular solutions of the nonlinearproblem (3.28). The advantage of such solutions is that many a priori anda posteriori error estimates can easily be derived assuming the approximationis close enough; see [229, p. 47], [33] and [113, p. 297]. First, we write (3.28) inoperator form. Let N : U → V∗ be defined as the map that satisfies

⟨N(u), v

⟩V,V∗ = N(u; v) ∀u ∈ U , ∀v ∈ V .

Then, (3.28) is equivalent to finding u ∈ U such that

N(u) = L in V∗ . (3.31)

Definition 3.10 (Nonsingular Solution) A solution u of (3.28), or equivalentlyof (3.31), is said to be nonsingular or regular, if N is differentiable at u, and more-over, N′(u) : U → V∗ is a bijection. ¤

Nonsingular solutions exclude bifurcation points and limit points. Notice thatthe given definition poses conditions on N. To obtain equivalent conditions onN, we note that the derivative of N corresponds to the derivative of N in thefollowing way:

N′(u; v)(δu) := limt→0

N(u + t δu; v) −N(u; v)

t=

⟨N′(u)(δu), v

⟩V∗ ,V


Hence, (Fréchet) differentiability of N at u implies that

N′(u; v)(δu) = N(u + δu; v) −N(u; v) + o(‖δu‖U

)‖v‖V , (3.32a)

as δu → 0, for all δu ∈ U; see Def. 3.8. Furthermore, bijectivity of N′(u) isequivalent to the bilinear form (δu, v) 7→ N′(u; v)(δu) : U × V → R beingcontinuous and satisfying the inf-sup conditions (3.2); see [81, p. 85] for thisequivalence. Hence, at nonsingular solutions u, there exist cN′ , cN′ > 0 suchthat

|N′(u, v)(δu)| ≤ cN′ ‖δu‖U ‖v‖V ∀δu ∈ U , ∀v ∈ V , (3.32b)

infδu∈U\0

supv∈V\0

N′(u, v)(δu)

‖δu‖U‖v‖V

≥ cN′ > 0 . (3.32c)

3.3.3 Galerkin approximations

Let us consider conforming Galerkin approximation of (3.28). For subspacesUh ⊂ U and Vh ⊂ V, we define the Galerkin discretization as

Find uh ∈ Uh :

N(uh; vh) = L(vh) ∀vh ∈ Vh .(3.33)

Note that Galerkin orthogonality is now given by

N(u; vh) −N(uh; vh) = 0 ∀vh ∈ Vh , (3.34a)

which, for a nonsingular solution u, can be combined with (3.32a) to obtain

N′(u; vh)(u − uh) = o(‖u − uh‖U

)‖vh‖V ∀vh ∈ Vh , (3.34b)

as uh → u.Nonlinear counterparts to Cea’s Lemma (Theorem 3.B), i.e., convergence

of Galerkin approximations to nonlinear problems, can usually be derived forrestricted classes of nonlinear problems. For example, for problems involvingstrongly monotone nonlinearities see [45, p. 322] or [238]. For free-boundaryproblems, such results are far from trivial; see for example [79, 194]. In case offluid–structure interaction, only few results are known and these consider onlygeometrically linearized models; see [73, 152]. In the following theorem, wepresent an abstract convergence estimate for approximations that are close tononsingular solutions under the assumption that the discrete inf-sup conditionholds:

infδuh∈Uh\0

supvh∈Vh\0

N′(u, vh)(δuh)

‖δuh‖U‖vh‖V

≥ chN′ > 0 . (3.35)


The theorem furthermore presents a nonlinear extension of the error–residualequivalence, see Th. 3.C, which can be found in [228–230] and [1, p. 225]; seealso [138].

Theorem 3.F (Error Estimates near Nonsingular Solutions) Assume that u isa nonsingular solution to problem (3.28) and that the approximate prob-lem (3.33) has a solution uh ∈ Uh. Then, as uh → u, the following a priorierror estimate holds if the discrete inf-sup condition (3.35) is satisfied:

‖u − uh‖U ≤(1 + cN′/ch

N′)

infφ∈Uh

‖u − φ‖U + o(‖u − uh‖U

), (3.36a)

Furthermore, for any approximation uh ∈ U, the following error-residualequivalence holds:

cN′‖u − uh‖U + o(‖u − uh‖U

)

≤ ‖R(uh; ·)‖V∗ ≤ cN′‖u − uh‖U + o(‖u − uh‖U

), (3.36b)

where the residual v 7→ R(uh; v) : V → R is defined as

R(uh; v) := L(v) −N(uh; v) . (3.37)

Proof The proof of (3.36a) follows closely the proof of Th. 3.B. Let φ ∈ Uh.First, apply the triangle inequality and the discrete inf-sup condition (3.35):

‖u − uh‖U ≤ ‖u − φ‖U + ‖uh − φ‖U

≤ ‖u − φ‖U +1

chN′

supvh∈Vh

N′(u; vh)(uh − φ)

‖vh‖V

= ‖u − φ‖U +1

chN′

supvh∈Vh

N′(u; vh)(u − φ) −N′(u; vh)(u − uh)

‖vh‖V

.

Next, use continuity (3.32b) on the first N′-term and Galerkin orthogonal-ity (3.34b) on the other N′-term:

‖u − uh‖U ≤ ‖u − φ‖U +1

chN′

(cN′ ‖u − φ‖U + o

(‖u − uh‖U

)).

Then, (3.36a) follows by taking the infimum of all φ in Uh. The proof of (3.36b)follows closely the proof of Th. 3.C. Let v ∈ V. First use (3.32a) to rewrite theresidual in terms of N′:

R(uh; v) = N(u; v) −N(uh; v) = N′(u; v)(u − uh) + o(‖u − uh‖U

)‖v‖V


Then by definition of the dual norm:

‖R(uh; ·)‖V∗ = supv∈V

N′(u; v)(u − uh)

‖v‖V

+ o(‖u − uh‖U

).

The upper bound follows using continuity (3.32b) and the lower bound followsfrom the inf-sup condition (3.32c). ¤

Note that, up to higher-order terms, the results in this theorem are essentiallythe same as for linear problems; see Th. 3.B and 3.C. That is, Galerkin approx-imations are quasi-optimal and the residual is equivalent to the error in norm.We stress, however, that the inequalities in Th. 3.F only make sense if uh is closeenough to u. One can quantify the required closeness of approximations andspecify the higher-order terms if one introduces additional assumptions on thenonlinearity, such as Lipschitz continuity; see [33, 229] for more details.

3.3.4 Duality, linearized adjoints and errors in goal quantities

Using duality one can obtain estimates for the error in (possibly nonlinear) goalfunctionals Q of the solution. For nonlinear problems, the associated dual prob-lems are based on linearized adjoints. In the following, we consider goal func-tionals Q that are continuously differentiable on U. Furthermore, to be able toestablish well-posedness of the dual problems, we assume from now on that Uis a reflexive Banach space.

In view of the conditions associated with nonsingular solutions u, it is atthis point most logical to base the dual problem on the adjoint of N′(u), whichis referred to as the linearized adjoint at u. Let uh ∈ Uh denote the solution ofthe approximate problem (3.33). We note that the mean-value linearization (orsecant) of Q, given by

Qs(u, uh) :=∫ 1

0Q′(uh + t (u − uh)

)dt : U → R ,

(cf. definition in (3.30)) is well-defined for continuously differentiable Q. Thedual problem based on the linearized adjoint at u is given by:

Find z ∈ V :

N′(u; z)(δu) = Qs(u, uh)(δu) ∀δu ∈ U .(3.38)

If we denote by N′(u)∗ : V → U∗ the adjoint of N′(u), i.e.,

〈δu, N′(u)∗v〉U,U∗ = 〈N′(u) δu, v〉V∗ ,V = N′(u; v)(δu) ∀δu ∈ U , ∀v ∈ V ,


then this problem is equivalent to finding z ∈ V such that:

N′(u)∗z = Qs(u, uh) in U∗ .

We note that the dual problem is well-posed since at nonsingular solutions thebilinear form (δu, v) 7→ N′(u; v)(δu) as well as its adjoint, satisfy the inf-supconditions; see Prop. A.9.

The dual problem provides the key to errors in goal quantities. Indeed,

Q(u) −Q(uh) = Qs(u, uh)(u − uh) = N′(u; z)(u − uh) , (3.39)

where we used Corollary 3.9 in the first step. From this result we can easilyderive the following a priori estimate, which is an extension of Th. 3.D to thenonlinear case.

Theorem 3.G (Quantity Convergence near Nonsingular Solutions) Let theassumptions of Theorem 3.F hold and let z denote the solution of the dualproblem (3.38). Then, as uh → u, the following a priori estimate holds:

|Q(u) −Q(uh)| ≤(

cN′ ‖u − uh‖U + o(‖u − uh‖U

))inf

ψ∈Vh‖z − ψ‖V

+ o(‖u − uh‖U

)‖z‖V .

Proof To obtain the proof, we use differentiability of N, see (3.32a), two times,which allows us to invoke Galerkin orthogonality in the form (3.34a). We startfrom (3.39), use (3.32a) and subsequently (3.34a):

Q(u) −Q(uh) = N(u; z) −N(uh; z) + o(‖u − uh‖U

)‖z‖V

= N(u; z − ψ) −N(uh; z − ψ) + o(‖u − uh‖U

)‖z‖V ,

for any ψ ∈ Vh. Using again (3.32a) gives:

Q(u) −Q(uh) = N′(u; z − ψ)(u − uh) + o(‖u − uh‖U

) (‖z‖V + ‖z − ψ‖V

).

The proof is obtained by using continuity (3.32b) and taking the infimum of allψ in Vh. ¤

This theorem shows that up to higher-order terms, the convergence in goalquantities is similar as for linear problems; see Th. 3.D. Hence, also in the non-linear case, we can expect higher rates of convergence for goal quantities thanfor the norm.

For the purpose of goal-oriented error estimation and adaptivity, it is im-portant to relate the goal error to the residual. In the linear case, we gave anexact error representation formula in Sec. 3.1.3; see (3.13). Remarkably, such an


exact formula is also possible in the nonlinear case; see [22, Sec. 2], [204, Sec. 6]or [209, Sec. 6]. To present this formula, we shall from now an assume that N iscontinuously differentiable in an open convex subset of U that includes uh andu. The mean-value linearization (or secant form) of N, given by

Ns(u, uh; v)

:=∫ 1

0N′(uh + t (u − uh); v) dt ∀δu ∈ U ,

is then well-defined. We next consider the dual problem based on this mean-value linearization:

Find z ∈ V :

Ns(u, uh; z)(δu) = Qs(u, uh

)(δu) ∀δu ∈ U .

(3.40)

Assuming this problem has a solution z, one obtains from Corollary 3.9 that

Q(u) −Q(uh) = Qs(u, uh)(u − uh)

= Ns(u, uh; z)(u − uh) = N(u; z) −N(uh; z) ,

and by definition of the residual, see (3.37), we obtain the following exact errorrepresentation formula:

Q(u) −Q(uh) = R(uh; z) . (3.41)

However, it is to be remarked that the mean-value linearizations Ns and Qs

in (3.40) depend on the solution u. Although appealing from a theoretical per-spective, this means that (3.40) can not be used in practice to compute dualsolutions. For an investigation of theoretical aspects of (3.40) in case of a non-linear scalar conservation law, we refer to [126, p. 68].

A straightforward approximation to the “exact” dual problem can be ob-tained by replacing u in (3.40) by uh.10 Since Ns(uh, uh; z) = N′(uh; z) andQs(uh, uh) = Q′(uh), one then essentially obtains the dual problem based onthe linearized-adjoint at uh:

Find z ∈ V :

N′(uh; z)(δu) = Q′(uh)(δu) ∀δu ∈ U .(3.42)

Indeed, mainly this dual problem is used to compute goal-oriented error es-timates and drive goal-oriented adaptivity. Although the exact representa-tion (3.41) does not hold, it does hold up to higher-order terms. We summarizethis result in the following theorem.

10If one has access to a better approximation to u than uh, say uh, then one can also replace uin (3.40) by this better uh. Although such a dual problem is not very useful for error estimation, itcan be used to study the effect of nonlinearity on error estimates; see [22, p. 13] for more details.


Theorem 3.H (Linearized-Adjoint-Based Error Representation) Assume thatu is a solution to problem (3.28) and that the approximate problem (3.33) has asolution uh ∈ Uh. Furthermore, assume that z is a solution of the dual prob-lem (3.42). Then the following error representation holds:

Q(u) −Q(uh) = R(uh; z − ψ) + r ,

for any ψ ∈ Vh with remainder r = o(‖u − uh‖U). If N and Q are twicecontinuous differentiable, then the remainder is given by r := RQ − RN =

O(‖u − uh‖2U), where

RQ :=∫ 1

0(1 − t) Q′′(uh + t (u − uh)

)(u − uh)2 dt ,

RN :=∫ 1

0(1 − t) N′′(uh + t (u − uh); z

)(u − uh)2 dt .

Proof The proof uses the Generalized Taylor’s Theorem, see Th. 3.E. Indeed,applying this theorem with n = 1 to Q, we have

Q(u) −Q(uh) = Q′(uh)(u − uh) + rQ ,

where rQ = o(‖u − uh‖U) or rQ = RQ if Q is twice continuously differentiable.Invoking the dual problem (3.42) and subsequently applying the GeneralizedTaylor’s Theorem to N, we obtain

Q(u) −Q(uh) = N′(uh; z)(u − uh) + rQ = N(u; z) −N(uh; z) − rN + rQ ,

where rN = o(‖u − uh‖U) or rN = RN if N is twice continuously differentiable.The proof is obtained by noting that N(u; z) = L(z) and N(uh; ψ) = L(ψ) forany ψ ∈ Vh; see (3.28) and (3.33), respectively. ¤

The result in this theorem forms the basis for many goal-oriented error esti-mates and adaptive discretization procedures. Indeed, the adaptive procedurepresented in Sec. 3.2.4 is easily extended to nonlinear problems by neglect-ing higher-order terms and introducing computable refinement indicators asin Prop. 3.6:

|Q(u) −Q(uh)| ≈ |R(uh; z − ψ)| / ∑K

|ηK| .

Such goal-oriented adaptive procedures for nonlinear problems perform verywell in practice; see the many computational examples in the mentionedoverviews [22, 132, 191, 204, 209] and the cited references therein. Unfortu-nately, it is, at this time, still uncertain whether the known goal-oriented adap-tive strategies applied to nonlinear problems converge at all. Currently, conver-gence results are only known for norm-oriented adaptive methods applied tosome nonlinear problems; we refer the interested reader to [37, 39, 43, 67, 226].

3.4 Nontrivial nonlinearity in free-boundary problems 51

3.4 Nontrivial nonlinearity in free-boundary prob-

lems

The abstract framework for nonlinear problems in Sec. 3.3 is very powerful.Indeed, the main conclusion is that one can generate goal-oriented adaptivediscretizations employing the abstract result in Theorem 3.H. That is, onedetermines an approximate solution of the dual (linearized-adjoint) problemand subsequently computes duality-based refinement indicators. The majordifficulty in applying this strategy to free-boundary problems such as fluid–structure interaction pertains to the determination of a suitable dual problem:How do we linearize the operators in these problems?

Let us see why the nonlinearity is nontrivial for free-boundary problemsby recalling the weak formulation of the Bernoulli free-boundary problem (asimilar reasoning holds for fluid–structure interaction), see (2.2) in Sec. 2.1.2:Find Ω ∈ O and u ∈ H1

h(Ω) such that

N(Ω, u; v) = 0 ∀v ∈ H10,ΓD

(Ω) . (3.43)

Recall that O is the set of bounded open Lipschitz domains in RN. We assume

that we are interested in a goal functional of the solution, Q(Ω, u). It shoulddirectly be noticed that this weak form and the goal Q do not fit the abstractframework we considered in Sec. 3.3. We have the following two complications.

1. How to linearize the domain dependence Ω 7→ N(Ω, u; v) and Ω 7→Q(Ω, u)?

2. How to deal with the fact that the u- and v-space are defined on the apriori unknown domain Ω?

A resolution of these issues should also deal with the following. Suppose thatΩh and uh ∈ H1

h(Ωh) form an approximate solution to (3.43). In the abstractframework, we have continuously considered errors. However, what is a suit-able notion of error between Ω and Ωh? Furthermore, how should the errorbetween u and uh be defined? Their difference, u − uh, is meaningless, since weare comparing functions on different domains; see the illustration in Fig. 3.2.

In the sequel, we present two approaches that deal with these complica-tions. Both approaches rely heavily on concepts from shape differential calcu-lus, which is the topic of the next chapter. Before we delve into these matters,however, we briefly outline each approach.

3.4.1 Domain-map linearization approach

The first approach is referred to as the domain-map linearization approach. Thebasic idea is that the free-boundary problem is transformed to a fixed reference


Γ

Ω

Γh

Ωh

Figure 3.2: Comparing functions on different domains. The solution u ∈ H1h(Ω) lives

on Ω and the approximation uh ∈ H1h(Ωh) lives on Ωh. The dotted lines are isocontours

of u and uh.

domain, say Ω0. This is accomplished by means of a suitable transformationmap

TΩ : Ω0 → Ω .

Instead of Ω, the unknown then becomes the map TΩ. If we denote the trans-formed functions by u0 := u TΩ and v0 := v TΩ, the original problem canthen be equivalently formulated on Ω0 as: Find T ∈ T(Ω0) and u0 ∈ U(Ω0)such that

N0(T, u0; v0

)= 0 ∀v0 ∈ V(Ω0) , (3.44)

where T(Ω0), U(Ω0) and V(Ω0) are suitable spaces defined on the referencedomain and N0 is essentially defined by

N0(T, u0; v0

):= N

(T(Ω0), u0 T -1; v0 T -1) .

Applying the same transformation to Q, we arrive in the standard frameworkof Sec. 3.3. Hence, we can simply linearize with respect to the domain map T.

In the next chapter, we show how domains can suitably be parametrizedby transformations. We apply the domain-map linearization technique to theBernoulli free-boundary problem in Chapter 5. In Chapter 7, we apply this tech-nique to the model fluid–structure interaction problem introduced in Sec. 2.2.

3.4.2 Shape-linearization approach

The second approach is referred to as the shape-linearization approach. Since themaps Ω 7→ N(Ω, u; v) and Ω 7→ Q(Ω, u) for fixed u and v can be viewed asso-called shape functionals, we can employ the techniques of shape differentialcalculus to linearize this dependence. In this procedure, one has to view u andv as functions defined outside their domain by a suitable extension.

Such a shape linearization can, however, only be performed under appro-priate regularity requirement on the involved integrands. These requirements

3.4 Nontrivial nonlinearity in free-boundary problems 53

will be considered in detail in the next chapter. We apply the shape lineariza-ton approach to the Bernoulli free-boundary problem in Chapter 6 and to themodel fluid–structure interaction problem in Chapter 8.

Chapter 4

Shape Differential Calculus

Practical mechanics is the subject that comprises all the manual arts, fromwhich the subject of mechanics as a whole has adopted its name. But sincethose who practice an art do not generally work with a high degree of ex-actness, the whole subject of mechanics is distinguished from geometry bythe attribution of exactness to geometry and of anything less than exactnessto mechanics. Yet the errors do not come from the art but from those whopractice the art. Anyone who works with less exactness is a more imperfectmechanic, and if anyone could work with the greatest exactness, he wouldbe the most perfect mechanic of all. For the description of straight lines andcircles, which is the foundation of geometry, appertains to mechanics.

Sir Isaac Newton (1642–1727), Trinity College, Cambridge, 8 May 1686Philosophiæ Naturalis Principia Mathematica1

Dans le présent mémoire, j’ai principalement en vue l’étude de la loi suivantlaquelle varient les diverses quantités qui interviennent dans la détermina-tion des fonctions biharmoniques lorsqu’on fait varier la forme du domainequi les engendre.2

Jacques Salomon Hadamard (1865–1963), 1907

In boundary-value problems, the domain where the unknown variable is de-fined, is given and therefore fixed. Free-boundary problems differ from normalboundary-value problems in that both the unknown variable and its underly-ing domain are part of the solution. An appropriate differential calculus thatdeals with varying domains is shape differential calculus.

4.1 Introduction

In this chapter, we study functionals that depend on (the shape of) domains.Let us first define such functionals in a more precise manner. We follow [59,

1Mathematical Principles of Natural Philosophy, Will. & Jno. Innys, MDCCXXVI, [172, p. 318].2These are the first sentences of a famous publication in 1907 of Hadamard [123] which marked

the beginning of shape differential calculus.

56 4. Shape Differential Calculus

p. 288]. Let D ⊆ RN denote a sufficiently-large hold-all domain. We denote

by O an admissible family of subsets of D. For our purposes, O generally con-sists of bounded open Lipschitz domains in D, which have a piecewise smoothboundary.

Definition 4.1 (Shape Functional) A functional J : O → R is said to be a shapefunctional if it maps a family O of domains into R, such that for any homeomor-phism T of D with T(Ω) = Ω it holds that J(Ω) = J(T(Ω)), for all Ω ∈ O. ¤

An elementary example is the volume integral given by J(Ω) =∫

ΩdΩ. This

definition can easily be generalized as follows.

Definition 4.2 (Shape Function) A function y : O → W is said to be a shapefunction if it maps a family O of domains into some vector space W, such thatfor any homeomorphism T of D with T(Ω) = Ω it holds that y(Ω) = y(T(Ω)),for all Ω ∈ O. ¤

A simple example of a shape function is the restriction to Ω of a globally definedfunction, e.g., for φ ∈ L2(D) this would be y(Ω) = φ|Ω. A more typical shapefunction corresponds to the solution of a boundary value problem posed on Ω.

Note that the set of admissible domains O does not have a linear (vectorspace) structure (summation and scalar multiplication are ill-defined), makingit impractical to work with. For example, the notion of a difference quotient

(J(Ω) − J(Ω \ ∆Ω)

)/ meas(∆Ω)

and its limit for meas(∆Ω) → 0 leading to the so-called topological derivativesare very involved and unnecessarily complicated for our purposes, see e.g. [24,42, 175, 201].

Since we are dealing with domains that do not change topologically, it ispossible to describe a differential calculus along parametrized families of trans-formations of a given reference domain. Indeed, such a family allows a natu-ral notion of directional derivatives of shape functionals and shape functions.This differential calculus is referred to as shape diffential calculus and the deriva-tives are known as shape derivatives. A classical application field of shape calcu-lus is shape optimization. Accordingly, most of the shape calculus theory canbe found in mathematically-oriented studies of shape optimization; we referto [59, 130, 170, 185, 202] and references therein.

This chapter proceeds as follows. In Sec. 4.2 we introduce variousparametrized families of transformations. Then, in Sec. 4.3, we consider deriva-tives of shape functionals. Although derivatives of shape functions bear sim-ilarities to the linearization approaches of free-boundary problems, they arenot essential in the ensuing chapters. Nevertheless, the interested reader canfind a brief introduction to derivatives of shape functions in the appendix; seeSec. A.2.

4.2 Parametrized families of domain transformations 57

Ωt

Γt

D

Figure 4.1: A one-parameter family of domains Ωt ⊂ D

4.2 Parametrized families of domain transforma-

tions

Consider a bounded open Lipschitz domain Ω ⊂ D ⊆ RN in the hold-all do-

main D. A family of variable domains can be constructed by considering theimages of a continuously varying one-parameter family of transformations of Ω.Let t denote the parameter, which can be thought of as artificial time. We willassume that for each t ∈ [0, t ] the transformation Tt is a diffeomorphism map-ping Ω := Ω0 onto Ωt := Tt(Ω). Furthermore, we assume that Ωt ⊂ D for allt ∈ [0, t ]; see Fig. 4.1.

There are various methods to construct the family of transformations. Con-cise overviews can be found in [59, Ch. 7] and [202, Ch. 2]. We will focusmainly on the so-called Lagrangian methods, excluding for example level setideas [197, 198]. First, we consider the so-called velocity method. This is themost general method, in that other methods correspond to specific velocityfields. Then we discuss perturbations of the identity and an important subclassthereof referred to as scalar perturbations.

4.2.1 The velocity method and general transformations

The velocity (or speed) method is a general method for constructing transfor-mations of Ω, or any bijection of D for that matter. We refer to [56, 59, 202] foroverviews of the method. The main characteristic of the velocity method is itsability to transform continuously to any diffeomorphism of Ω; a property orig-inating from its connection with continuum mechanics. The velocity-methodbasics can be explained best by considering the autonomous case first.

Given an (artificial) velocity field v : D → RN, the transformation associ-

ated with v is defined for each X ∈ D and t ∈ [0, t ] as

Tt(X) = x(t, X), (4.1)


Ω

v

Figure 4.2: The velocity method

where x(t, X) is the solution of the following initial value problem, consistingof a (vector) differential equation with initial conditions:

∂x

∂t(t, X) = v

(x(t, X)

), ∀t ∈ [0, t ] ,

x(0, X) = X .

Essentially, each point x(t, X) = Tt(X) moves along the trajectory t 7→ x(t, X)with current velocity v

(x(t, X)

)and starting point X. It is also said that the

transformations (4.1) correspond to the flow of v . Note that Tt transforms Ω

into Ωt = Tt(Ω). See Fig. 4.2 for a graphical illustration of the velocity method.

The previous construction is readily extended to the more general case ofnonautonomous velocity fields v : [0, t ] × D → R

N. In this case, we associatewith the nonautonomous v the solution x(t, X) of

∂x

∂t(t, X) = v

(t, x(t, X)

), ∀t ∈ [0, t ] ,

x(0, X) = X ,

(4.2a)

(4.2b)

for each X ∈ D and, subsequently, define the transformation as

Tt(X) = x(t, X) . (4.2c)

The initial value problem (4.2a)–(4.2b) has a unique solution under regularityrequirements on the velocity field v , specifically, continuity in time and Lip-schitz continuity in space, according to the generalized Picard-Lindelöf Theo-rem, see e.g. [236, p. 78] or [4, p. 215]. Here, we define v to be an admissiblevelocity field if

v ∈ C([0, t ]; V(D)

), (4.3)


where V(D) is the space of Lipschitz-continuous vector fields which are tangentto the boundary ∂D, i.e.,

V(D) :=

v ∈ C0,1(D; RN)

∣∣v is tangent to ∂D

. (4.4)

In particular, this implies that v · n = 0 on ∂D except at possible singular pointsof ∂D where the normal n is undefined. For these admissible velocity fields, itcan be shown that the associated family of transformations are bijections of D.3

This important result is outlined in the following theorem.

Theorem 4.A (Transformation by Velocity Field) If v ∈ C([0, t ]; V(D)

)is an

admissible velocity field, then there exists a unique family of bijective transfor-mations Tt : D → D, t ∈ [0, t ] which corresponds to the flow of v accordingto (4.2a)–(4.2c). The trajectories t 7→ Tt(X) are continuously differentiable, andthe transformations as well as their time-derivatives and the inverse transfor-mations T -1

t : D → D are Lipschitz fields, that is,

(t, X) 7→ Tt(X) ∈ C1([0, t ]; C0,1(D; RN)

), (4.5a)

(t, x) 7→ T -1t (x) ∈ C

([0, t ]; C0,1(D; R

N))

. (4.5b)

Proof See [59, p. 315] or [56]. ¤

This theorem implies that the family Tt generates the family of domainsΩt, where

Ωt = Tt(Ω) :=

x ∈ RN∣∣ x = Tt(X), ∀X ∈ Ω

,

for which Tt maps interior (resp., boundary) points of Ω onto interior (resp.,boundary) points of Ωt.

Although D is possibly a bounded domain, this is not necessary for the theo-rem, that is, it also includes the unconstrained case for which D can be replacedwith R

N, see [59, p. 300]. Note that the space of admissible velocity fields in thiscase is C

([0, t ]; C0,1(R

N; RN)

). Hence V(D) = V(R

N) := C0,1(RN; R

N) withoutthe tangency requirement.

The velocity method with nonautonomous velocity fields is truly a gen-eral way of constructing transformations. In fact, we have the following resultwhich shows that the velocity and transformation viewpoint are equivalent.

Proposition 4.3 If Tt : D → D, t ∈ [0, t ] is a given family of bijective trans-formations of D satisfying (4.5a) and (4.5b), and for which T0 = I, then thenonautonomous velocity field v : [0, t ] × D → R

N defined as

v(t, x) :=∂Tt

∂t

(T -1

t (x))

(4.6)

3Hence, the associated family is actually a one-parameter group of transformations of D.


Ω

θ

Figure 4.3: Perturbation of the identity

is an admissible velocity field, i.e., v ∈ C([0, t ]; V(D)

), for which the corre-

sponding transformations constructed from this v (according to (4.2a)–(4.2c))coincide with the given Tt : D → D, t ∈ [0, t ].

Proof See [59, p. 315] or [56]. ¤

In view of the stated equivalence, we will call Tt : D → D, t ∈ [0, t ] anadmissible family of transformations if it is bijective and satisfies (4.5a) and (4.5b)on [0, t ].

4.2.2 Perturbations of the identity

An important family of transformations can be constructed by considering per-turbations of the identity. Let us denote the identity map by Id : R

N → RN and

let θ : RN → R

N denote a perturbation vector field. Then we define a perturbedtransformation as

Tθ := Id + θ , (4.7)

i.e., Tθ(X) = X + θ(X) for all X ∈ Ω. This transformation leads to the perturbeddomain

Ωθ := Tθ(Ω) = Ω + θ =

x ∈ RN∣∣ x = X + θ(X), ∀X ∈ Ω

. (4.8)

A corresponding one-parameter family of perturbations for t ≥ 0 is given byTtθ = Id + t θ, which generates the perturbed domains Ωtθ = Ttθ(Ω); seeFig. 4.3 for a graphical illustration of identity perturbations. It is obvious thatin general the perturbations may not be too large or else intersections can oc-cur, yielding a noninvertible transformation. Fortunately, for small enoughLipschitz-continuous perturbation, we have the following.


Theorem 4.B (Transformation by Perturbation of Identity) If the perturba-tion field θ is a Lipschitz vector field on R

N, i.e.,

θ ∈ C0,1(RN; R

N) , (4.9)

then there exists a t > 0 such that the corresponding family of transforma-tions Ttθ : R

N → RN, t ∈ [0, t ] is admissible. That is, Ttθ is bijective on R

N

and satisfies conditions (4.5a) and (4.5b) on [0, t ] with D = RN.

Proof See [59, p. 303] or [56, 69]. ¤

On account of this result, we will call θ admissible if it satisfies (4.9).Since perturbations of the identity generate a family of bijective transfor-

mations, Prop. 4.3, applied to the unconstrained case (D = RN), implies that

there is a corresponding nonautonomous velocity field. This is outlined in thefollowing.

Corollary 4.4 If θ ∈ C0,1(RN; R

N) is an admissible perturbation field, thenthere exists a t > 0 such that the nonautonomous velocity field vθ : [0, t ] ×R

N → RN defined as

vθ(t, x) := θ(T -1

tθ (x))

(4.10)

is admissible, i.e., vθ ∈ C([0, t ]; C0,1(R

N, RN)

), and vθ coincides with the veloc-

ity field associated to the family of identity perturbations Tt = Ttθ.

Proof Noting that the family Tt = Ttθ is admissible for θ ∈ C0,1(RN; R

N),and T0 = Id, we can use Proposition 4.3 with Tt replaced by Ttθ in (4.6). ¤

Perturbations of the identity may seem inferior to the velocity method.However, there are definite advantages of using perturbations of the identity.Indeed, the transformations are simply constructed algebraically, see (4.7). Thisshould be compared to the velocity method where the transformations are dif-ferentially defined. Moreover, since the perturbation field coincides with theassociated velocity field (4.10) at t = 0, both methods can often be consideredequal for t ց 0. In particular, this will be shown for the shape derivative inSection 4.3.

4.2.3 Scalar perturbations

An important class of perturbations of the identity involve perturbations inspecific directions only. Although the direction is fixed, the magnitude of theperturbation is then prescribed by a scalar field at the boundary. There arebasically two scalar perturbation methods. The first method involves pertur-bations in the normal direction (or a smoothed variant thereof), referred to


Ω

n

Figure 4.4: Scalar perturbation in the normal direction

as boundary normal-perturbations. Such perturbations have been proposed byHadamard in 1907 [123]. It is used by several authors; see, for example, theworks of Pironneau [185, 186]. The other method involves perturbations alonga fixed coordinate, say xN. We refer to these as cartesian perturbations. Theseperturbations are used extensively in the books of Haslinger, Neittaanmäki andMäkinen [129, 130] and are very popular in free-surface flow problems; see [55]for example.

To explain scalar perturbations, consider the boundary Γ = ∂Ω of a suffi-ciently smooth domain Ω and the following perturbation Γ (= ∂Ω) of Γ alongthe fixed direction m:

Γ := Γ + m =

x ∈ RN∣∣ x = X + (X) m(X) , ∀X ∈ Γ

,

where : Γ → R is a scalar perturbation field defined at the boundary; seeFig. 4.4. In case of boundary-normal perturbations, we have m = n. For carte-sian perturbations along the coordinate xN, we have m = eN := (0, . . . , 0, 1).

One can show that a scalar perturbation corresponds to a specific pertur-bation of the identity. Let ˜ and m denote extensions4 from Γ to R

N of andm, respectively. Then we can define the transformation T : R

N → RN as the

identity perturbation

T := Id + ˜m . (4.11)

A family of transformations can then be given by Tt which, accordingly,generates the family of perturbed domains Ωt. Moreover, if ˜m is a Lipschitzvector field on R

N, then we can apply Theorem 4.B to θ = ˜m and infer thatthere exists a t > 0 such that the family of transformations Tt is admissibleon [0, t ]. Hence, it follows that Tt maps Γ onto Γt and Ω onto Ωt, independent

4Whenever we mention an extension, we have in mind a sufficiently smooth, linear and contin-uous extension.


Table 4.1: Various methods for the parametrization of domains.

Velocity method Perturbation of identity Scalar perturbation

Main variable v ∈ C(0, t; V(D)

)θ ∈ C0,1(R

N; RN) ∈ C0,1(Γ; R)

Transformation ∂tTt = v(t, Tt) Tθ = Id + θ T = Id + ˜m

Domain Ωt = Tt(Ω) Ωθ = Ω + θ Ω = Ω + ˜m

Free boundary Γt = Tt(Γ) Γθ = Γ + θ Γ = Γ + m

of the extension used. In view of this results, we call an admissible scalarperturbation if ∈ C0,1(Γ).

A disadvantage of boundary normal-perturbations is that when Ω is of classCk,l , the normal n is an element of Ck−1,l(Γ; R

N). Therefore, generally, theseperturbations yield less regular domains Ωt of class Ck−1,l . Indeed, normalperturbations preserve only C∞ domains. This can be remedied by introduc-ing a smoothed normal field, see for example [214], [120, p. 40] or [59, p. 294].Cartesian perturbations have the advantage that one can explicitly constructthe inverse map T -1

; we refer to [202, p. 46].

4.2.4 Overview of parametrizations

To summarize the above exposition on different families of transformations ofdomains, we have gathered in Table 4.1 relevant aspects of the different meth-ods. We stress that scalar perturbations are specific perturbations of the identitywhich in turn are a special case of the velocity method.

4.2.5 Shape continuity based on parametrized domains

Let us consider a shape functional J : O → R as introduced in Sec. 4.1. If wehave a metric on the set of admissible domains O, then we can define shapecontinuity of J using an ǫ-δ definition. Indeed, an appropriate but complicatedcomplete metric is the so-called Courant metric [59, p. 68]. On the other hand,the parametrized families of domains discussed in Section 4.2 naturally pointto the following notion of directional shape continuity. We follow the defini-tion as given in [59, p. 326]. In the definition, v , θ and denote an admissiblevelocity field, admissible perturbation and admissible scalar perturbation, andthe corresponding family of domains are denoted as in Table 4.1.

Definition 4.5 (Shape Continuity) The shape functional J : O → R is said to


be shape continuous at Ω ∈ O in the direction v if

limt→0

J(Ωt) = J(Ω) .

Similarly, J : O → R is said to be shape continuous at Ω ∈ O in the direction θ(or ) if

limt→0

J(Ωtθ) = J(Ω)(

or limt→0

J(Ωt) = J(Ω))

. ¤

We note that this definition can easily be extended to shape continuity of shapefunctions y : O → W.

4.3 Shape derivatives of shape functionals

The parametrized families of domains presented in the previous section are thestarting point to define derivatives of shape functionals. That is, they act asone-dimensional paths along which limits of difference quotients can be takenwhich yield shape derivatives. In this section, we study these derivatives andrelate them to the classical Gâteaux and Fréchet derivatives. Furthermore, weconsider the two most elementary shape functionals which involve a domainintegral and boundary integral. Most of the theory presented in this section canbe found in [60], [59, Ch. 8] and [202, Ch. 2]. We refer to these works for furtherdetails.

4.3.1 Velocity method and shape derivative

We start with the most generic parametrization of domains, viz., as gener-ated by the velocity method. Let v denote an admissible velocity field, i.e.,v ∈ C

([0, t ]; V(D)

)in accordance with (4.3), and let Tt denote the associ-

ated transformation (according to (4.2a)–(4.2c)), which generates the domainΩt = Tt(Ω).

Consider a shape functional J : O → R. The Eulerian semiderivative of J at Ω

in the direction v is defined as the limit

J′(Ω)(v) := limtց0

J(Ωt) − J(Ω)

t, (4.12)

if it exists.5 It is possible that the Eulerian semiderivative exists only for smoothenough v . For that reason, let Θ(D) be a (smooth) subspace of V(D). Further-more, it is known that if v 7→ J′(Ω)(v) is a continuous map on C

([0, t ]; Θ(D)

),

5The prefix “semi” refers to the one-sided limit, t ց 0.

4.3 Shape derivatives of shape functionals 65

then the Eulerian semiderivative only depends on the initial velocity fieldv(0) := v(0, ·); see [59, p. 343], i.e.,6

J′(Ω)(v) = J′(Ω)(v(0)) . (4.13)

We now state the following key definition.

Definition 4.6 (Shape Differentiability) The shape functional J : O → R issaid to be shape differentiable at Ω ∈ O with respect to Θ(D) if the Euleriansemiderivative (4.12) exists in all directions v ∈ C

([0, t ], Θ(D)

)such that (4.13)

holds and the map v(0) 7→ J′(Ω)(v(0)

)is linear and continuous on Θ(D). In

this case J′(Ω)(v) is called the shape derivative of J at Ω in the direction v ¤

Note that shape differentiability implies that J′(Ω) is a linear and continuousfunctional in Θ(D)∗, referred to as the shape gradient at Ω. Because of (4.13), weshall from now on assume v to be autonomous, unless specified otherwise.

To elucidate the structure of a shape derivative, suppose that the velocityv is such that the associated transformation preserves the domain, i.e., Ωt =Tt(Ω) = Ω for all t ≥ 0. For example, this holds if v = 0 at the boundary Γ.Then, for any shape functional J, we have that J(Ωt) = J(Ω) and, consequently,J′(Ω)(v) = 0. Therefore, it is expected that J′(Ω)(v) is nonzero only if v isnonzero at the boundary Γ. This is made precise in the following theorem.

Theorem 4.C (Hadamard–Zolésio Structure Theorem) If the shape functionalJ is shape differentiable at Ω with respect to Θ(D), then its shape gradientJ′(Ω) is supported (as a distribution) on Γ. Furthermore, if Γ is sufficientlysmooth (dependent on Θ(D)),7 then there exists a scalar Γ-distribution j′(Γ)such that

J′(Ω)(v) =⟨

j′(Γ), γ(v) · n⟩

Γ, (4.14a)

with γ(·) the trace operator on Γ.

Proof See [56] or [59, p. 349]. ¤

Indeed, the first part of the theorem states that the shape derivative only de-pends on v at Γ. Moreover, if Γ is smooth enough, then it depends only onthe normal component v · n. In particular, if j′(Γ) ∈ L1(Γ) then (4.14a) can bewritten as

J′(Ω)(v) =∫

Γj′(Γ) v · n . (4.14b)

6A Eulerian semiderivative which depends only on the initial velocity v(0) is called a Hadamardsemiderivative in [59].

7Assume Γ of class Ck+1 if Θ(D) ⊂ Ck(D, RN).


Although Hadamard [123] derived (4.14b) in 1907 for boundary normal-perturbations of a C∞-domain, (4.14b) is generally known as the Hadamard for-mula.

4.3.2 Perturbations of the identity and Gâteaux derivative

A definition of shape differentiability for perturbations of the identity can obvi-ously be obtained by applying Definition 4.6 to their associated velocity fields(see Proposition 4.4). However, because each domain perturbation can be iden-tified with an element θ of the Banach space C0,1(R

N, RN), we can also resort to

standard differentiation in Banach spaces. Many authors use this approach asthe starting point for defining shape derivatives; see [136, 176] for example.

Let Θ := Θ(RN) denote a (smooth) subspace of C0,1(R

N, RN). We consider

an open ǫ-ball Bǫ in Θ containing sufficiently small perturbations such that

∀θ ∈ Bǫ ⊂ Θ , Tθ = Id + θ : RN → R

N is bijective .

We know from Th. 4.B that there exists such an ǫ > 0. Then, in particular, Tθ isa bijection from Ω onto Ωθ = Tθ(Ω). Now, paired with the shape functional J,we consider the functional JΩ : Bǫ ⊂ Θ → R defined as

JΩ(θ) := J(Ωθ) . (4.15)

This brings us in the standard Banach space setting; see Sec. 3.3.1. Hence, wecan introduce the Gâteaux derivative of JΩ at θ ∈ Bǫ ⊂ Θ in the direction δθ ∈ Θ

defined as

J′Ω(θ)(δθ) := limt→0

JΩ(θ + t δθ) − JΩ(θ)

t. (4.16)

Analogous to Definition 3.7, JΩ is said to be Gâteaux differentiable at θ ∈ Bǫ

if the Gâteaux derivative (4.16) exists in all directions δθ ∈ Θ, and the mapδθ 7→ J′Ω(θ)(δθ) : Θ → R is linear and continuous.

Let us consider next the connection between the Gâteaux derivative of JΩ

and the shape derivative of J introduced in Sec. 4.3.1. In view of the rela-tion (4.15), the Gâteaux derivative (4.16) can be expressed in terms of J as

J′Ω(θ)(δθ) = limt→0

J(Ωθ+t δθ) − J(Ωθ)

t.

The right-hand side would equal a Eulerian derivative, if we can find the veloc-ity field which generates the flow from Ωθ to Ωθ+tδθ .8 But this follows essen-tially from Corollary 4.4 applied to Ωθ . We collect this result in the followingtheorem, as well as a full equivalence result.

8This velocity field would have to exist for t in a small neighborhood of 0. Hence, the limit is aEulerian derivative instead of merely a Eulerian semiderivative.


Theorem 4.D (Equivalent Shape Functional Derivatives) If JΩ has a Gâteauxderivative at θ in the direction δθ ∈ Θ, then J has a Eulerian derivative at Ωθ inthe velocity direction

vθδθ(t, x) := δθ T -1

θ+tδθ(x) .

Furthermore, if JΩ is Gâteaux differentiable at θ, then J is shape differentiableat Ωθ , and vice versa. The derivatives are related as follows:

J′Ω(θ)(δθ) = J′(Ωθ)(δθ T -1θ ) , ∀δθ ∈ Θ ,

J′(Ωθ)(v) = J′Ω(θ)(v Tθ) , ∀v ∈ Θ .

Proof See [60] or [59, p. 343]. ¤

We note that in Banach spaces, there also exists the stronger notion of Fréchetdifferentiability; see Sec. 3.3.1. For the sake of completeness, let us state thecorresponding definition for JΩ. That is, JΩ is said to be Fréchet differentiableat θ ∈ Bǫ if it is Gâteaux differentiable at θ and

J′Ω(θ)(δθ) = JΩ(θ + δθ) − JΩ(θ) + o(‖δθ‖Θ

), δθ → 0 ,

for all δθ ∈ Θ. Recall that Fréchet differentiability implies Gâteaux differentia-bility but not vice versa. However, if θ 7→ J′Ω(θ) is continuous at θ, then thereverse holds.

It is illustrative to consider the structure of the shape derivative accord-ing to Th. 4.C, in case of the boundary normal-perturbations considered inSec. 4.2.3. Such a perturbation can be written as a perturbation of the identityas: δθ = δ n, where δ is the scalar boundary perturbation field. We assumethat J is shape differentiable at Ω = Ω0 and Γ is sufficiently smooth so that theHadamard formula (4.14b) holds; see Th. 4.C. Then, in view of the equivalencein Th. 4.D and invoking (4.14b), the shape derivative at Ω due to δ is given by

J′Ω(0)(δ n) = J′(Ω)(δ n) =∫

Γj′(Γ) δ . (4.17)

This is a very simple form of the shape derivative. In fact, Pironneau’s defi-nition of Γ-differentiability of a shape functional J in [185, p. 82], coincides pre-cisely with Fréchet differentiability of JΩ and the existence of a j′(Γ) ∈ L1(Γ)such that (4.17) holds.

4.3.3 Shape functional involving domain integral

Let us consider the shape derivative of an elementary example, viz., a shapefunctional involving the domain integral of a global function φ : R

N → R


defined as

J(Ω) =∫

Ωφ . (4.18)

We assume Ω is an open bounded Lipschitz domain. We denote by v ∈C0,1(R

N, RN) an admissible velocity field and by Tt the associated transforma-

tion generating the domain Ωt = Tt(Ω).9

The shape derivative of J can be obtained by differentiating t 7→ J(Ωt) at t =0. Note that J(Ωt) can be written as an integral over Ω as follows:

J(Ωt) =∫

Ωt

φ =∫

Ωφ Tt |Jt| , (4.19)

where Jt := det DTt is the Jacobian of Tt, with D(·) := ∂(·)/∂(x1, . . . , xN) theJacobian matrix. To differentiate φ Tt, we need the following result.

Lemma 4.7 If φ ∈ W1,1(RN), then the transformed functions φ Tt and φ T -1

t

are in W1,1(RN), and the maps t 7→ φ Tt and t 7→ φ T -1

t are differentiable att = 0 in L1(R

N) and given by

∂

∂t(φ Tt)

∣∣t=0 = ∇φ · v ,

∂

∂t(φ T -1

t )∣∣t=0= −∇φ · v .

Proof See [202, p. 65 and 72] or [120, p. 19]. ¤

Recall that Tt and v are in C0,1(RN; R

N). Then, by Rademacher’s theorem,the components of the Jacobian matrices DTt and Dv are in L∞(R

N); see [86,p. 91].10 Moreover, the Jacobian Jt is in L∞(R

N) and Jt > 0 for admissible trans-formations. For the differentiation of Jt we have the following standard result.

Lemma 4.8 The map t 7→ Jt is differentiable at t = 0 in L∞(RN) and given by

∂

∂tJt

∣∣t=0 = div v .

Proof See [59, p. 352] or any book on continuum mechanics, e.g. [114]. ¤

With these results, we can easily differentiate t 7→ J(Ωt). We summarize theresult in the following theorem.

9For the sake of clarity, we consider the situation where D = RN (unconstrained setting).

10The components of DT -1t are also in L∞(R

N) in view of DT -1t = D(T -1

t ) Tt and T -1t ∈

C0,1(RN; R

N).


Theorem 4.E (Shape Derivative of Domain Integral) For φ ∈ W1,1(RN), the

shape functional in (4.18) is shape differentiable at Ω with respect toC0,1(R

N; RN). Its shape derivative is given by

J′(Ω)(v) =∫

Γφ v · n .

Proof Using Lemma 4.7 and 4.8, we obtain

J′(Ω)(v) =∫

Ω

(∇φ · v + φ div v

)=

∫

Ωdiv(φ v

).

The proof follows by the divergence theorem. ¤

In view of the structure theorem, see Th. 4.C, it is clear that the scalar distri-bution j′(Γ) associated with the shape gradient of J is equal to γ(φ) ∈ L1(Γ).Furthermore, using the characteristic function χΩ : R

N → 0, 1 defined as

χΩ(x) =

1 if x ∈ Ω ,

0 if x /∈ Ω ,

we can also write the shape derivative as

J′(Ω)(v) =∫

RNχΩ div

(φ v

)=

⟨−∇χΩ φ , v

⟩,

where we used the fact that φ(x) → 0 for x → ∞. It follows that the shape gra-dient J′(Ω) can be identified with the distribution −∇χΩ φ ∈ C0,1(R

N; RN)∗.

Let us illustrate the meaning of Theorem 4.E by considering the volumeintegral, φ = 1, for a spherical domain in R

3. If the sphere is inflated, i.e., v =δ n on Γ for some constant δ, then this proposition implies the familiar resultthat the volume of the sphere increases (up to first order) by the perturbationtimes the surface area, δ

∫Γ

dΓ.As a straightforward corollary to Theorem 4.E, we obtain the well-known

Reynolds’ transport theorem.

Corollary 4.9 (Reynolds’ Transport Theorem) For φ(t) ∈ W1,1(RN) and

φ′(t) ∈ L1(RN) for t ∈ [0, t ], the function Φ : [0, t ] → R defined as

Φ(t) :=∫

Ωt

φ(t) ,

has a semiderivative at t = 0 given by

Φ′(0) =∫

Ωφ′(0) +

∫

Γφ(0) v · n .


4.3.4 Shape functional involving boundary integral

Another elementary shape functional is the boundary integral of a global func-tion ψ : R

N → R defined as

J(Ω) =∫

Γψ . (4.20)

To derive the shape derivative of J we require some concepts from tangen-tial calculus. A synopsis of these concepts can be found in the appendix; seeSec. A.1. In the following, let v denote a sufficiently smooth admissible (au-tonomous) velocity field and let Tt denote the associated smooth transforma-tion generating the domain Ωt = Tt(Ω) with boundary Γt = ∂Ωt.

The shape derivative of J can be obtained as follows. We first rewrite J(Ωt)as a boundary integral at Γ:

J(Ωt) =∫

Γt

ψ =∫

Γψ Tt ωt , (4.21)

where we have introduced the surface density

ωt := Jt

∣∣DT -Tt n

∣∣ : Γ → R ,

which is also referred to as the tangential Jacobian [69]. To differentiate ψ Tt, weapply Lemma 4.7 to ψ. Furthermore, we note that at Γ, a gradient can be decom-posed in a tangential gradient (or surface gradient) and a normal component, i.e.,∇(·) = ∇Γ(·) + ∂n(·) n; see Def. A.1 in Sec. A.1. Hence, we obtain

∂

∂t(ψ Tt)

∣∣t=0 =

(∇Γψ +

∂ψ

∂nn)· v , on Γ .

Next, for the derivative of ωt, we have the following.

Lemma 4.10 If v ∈ C1(RN; R

N), then the map t 7→ ωt is differentiable at t = 0in L∞(Γ) and given by

∂

∂tωt

∣∣t=0 = divΓ v .

Proof See [202, p. 80] or [59, p. 353]. ¤

In this lemma, we introduced the tangential divergence (or surface divergence),which is defined as

divΓ(·) = div(·)∣∣Γ− D(·)n · n =: div(·)

∣∣Γ− ∂n(·) · n , on Γ ;

see Def. A.2 in Sec. A.1. The shape derivative of J readily follows from theseresults. This is summarized in the following theorem.


Theorem 4.F (Shape Derivative of Boundary Integral) For ψ ∈ W2,1(RN),

the shape functional in (4.20) is shape differentiable at Ω with respect toC1(R

N; RN). Its shape derivative is given by

J′(Ω)(v) =∫

Γ

(∂ψ

∂nv · n + ∇Γψ · v + ψ divΓ v

). (4.22)

We note that this result also holds under milder conditions of the perturbationfield. For Lipschitz perturbations which are an extension of a Lipschitz bound-ary perturbation in C0,1(Γ; R

N), (4.22) remains valid.In accordance with the structure theorem, see Th. 4.C, the shape gradi-

ent of J is supported on Γ. However, as Ω is only assumed to be Lipschitz,its boundary Γ is generally not smooth enough so that the Hadamard for-mula (4.14b) holds. Indeed, if Γ is assumed to be C1,1, we can invoke the tan-gential Green’s identity:

∫

Γ

(∇Γψ · v + ψ divΓ v

)=

∫

Γκ ψ v · n ,

where κ := divΓ n ∈ L∞(Γ) coincides with the additive curvature (sum of N− 1curvatures) of Γ; see (A.1) in Sec. A.1. Accordingly, (4.22) can be simplified.

Corollary 4.11 If the boundary Γ is C1,1, then the shape derivative in (4.22)simpifies to

J′(Ω)(v) =∫

Γ

(∂ψ

∂n+ κ ψ

)v · n . (4.23)

Hence, j′(Γ) in (4.14b) can be identified with ∂nψ + κ ψ ∈ L1(Γ). Notice the twocontributions to the shape derivative of a boundary integral. The first contri-bution, ∂nψ v · n, corresponds to changes in ψ normal to the boundary due tonormal boundary perturbations. The second, κ ψ v · n, corresponds to a geo-metric effect: if a curved boundary is perturbed, its length changes.

Part II

Scalar

Free-Boundary Problems

Chapter 5

Domain-Map Linearization Approach to


If error is corrected whenever it is recognized as such, the path of error isthe path of truth.

Hans Reichenbach (1891–1953), 1951The Rise of Scientific Philosophy1

In this chapter and the following chapter, we consider goal-oriented error esti-mation and adaptivity for free-boundary problems. This chapter considers, inparticular, the domain-map linearization approach to derive the dual problem.2

5.1 Introduction

Free-boundary problems are seldom discretized in an optimal manner. Sincethe free boundary is of prime interest, often heuristic mesh refinement is per-formed in the vicinity of the free boundary; see [100, 213]. However, the ac-curacy of goal quantities depends on both the accuracy of the approximatesolution and the accuracy of the domain approximation. In general, this de-pendence is non-obvious and heuristic approaches lead to inefficient approx-imations of the goal quantity. Goal-oriented adaptive discretizations can of-fer a significant efficiency improvement. Although we pointed out in Sec. 3.4that free-boundary problems elude the standard goal-oriented–error estimationframework, we briefly presented two approaches that deal with the complica-tions in free-boundary problems.

In this chapter, we consider the so-called domain-map linearization approachto derive the linearized adjoint and obtain an appropriate dual problem. Thisapproach was briefly outlined in Sec. 3.4.1. To illustrate the approach, we con-sider the Bernoulli free-boundary problem introduced in Sec. 2.1. By meansof perturbations of the identity maps, see Sec 4.2.2, the free-boundary problem

1University of California Press, 13th print, 1968, p. 326.2This chapter is based on [223].

76 5. Domain-Map Linearization Approach to Free-Boundary Problems

can be transformed into an equivalent problem on the fixed domain. The vari-ational formulation of the transformed problem is in canonical form, althoughit contains intricate terms involving the domain map. The linearized adjointis obtained by linearizing the transformed problem with respect to the domainmap.

We show that the dual solution obtained by the domain-map linearizationapproach is essentially independent of the selected reference domain, in thatthe dual solutions corresponding to two distinct reference domains are relatedby the obvious map between the reference domains. Furthermore, we give aninterpretation of the dual problem by showing that it corresponds to a Poissonproblem with a nonlocal Robin-type boundary condition.

In the following chapter, we consider the shape-linearization approach toobtain the dual problem, as explained briefly in Sec. 3.4.2. We note that lin-earization approaches are also used for Newton-type iterative solution algo-rithms. For Bernoulli-type free-boundary problems, however, only the shape-linearization approach seems to have been investigated; see [29, 100, 141].

The contents of this chapter are arranged as follows. Sec. 5.2 recalls theBernoulli free-boundary model problem from Sec. 2.1 and specifies some rel-evant goal functionals for this problem. In Sec. 5.3 we consider the domain-map linearization approach, and apply the canonical framework to the free-boundary problem. Sec. 5.4 presents an analysis of the associated dual prob-lem. Numerical experiments are presented in Sec. 5.5. Finally, Sec. 5.6 containsconcluding remarks. We present a comparison of the domain-map linearizationapproach and the shape-linearization approach in Chapter 6.

5.2 Problem statement

In this section, we briefly recall the Bernoulli free-boundary problem. Further-more, we parametrize the unknown domain using perturbations of the identity.Accordingly, we derive a new weak formulation in terms of the perturbationfield. Finally, several relevant goal functionals are presented.

5.2.1 Bernoulli free-boundary problem

Let the set O consist of bounded open Lipschitz domains Ω for which theboundary ∂Ω of Ω consists of two complementary parts, viz., a fixed part ΓD

and a variable part, Γ, the free boundary; see Fig. 2.1. The Bernoulli free-boundary problem consist in seeking a domain Ω ∈ O and a function u : Ω →

5.2 Problem statement 77

R such that

−∆u = f in Ω , (5.1a)

∂nu = g on Γ , (5.1b)

u = h∣∣Γ

= 1 on Γ , (5.1c)

u = h∣∣ΓD

on ΓD , (5.1d)

where we assume f ∈ C0(RN), g ∈ C1(R

N) together with a lower bound g ≥g0 > 0, and h ∈ C1(R

N). Note that, in accordance with (5.1c), h|Γ = 1 isrequired for all admissible free boundaries.

5.2.2 Parametrization of the unknown domain

We construct variable domains as transformations of a reference domain Ω0 ∈O by perturbations of the identity map Id : R

N → RN; see Sec. 4.2.2. The bound-

ary ∂Ω0 = Γ0 ∪ ΓD consists of the fixed parts ΓD and Γ0, where Γ0 correspondsto the free boundary in the reference configuration.

Let us denote by ΘLip := ΘLip(Ω0) the space of Lipschitz perturbation-vector fields which vanish at ΓD , i.e.,

ΘLip(Ω0) :=

θ ∈ C0,1(Ω0; RN)

∣∣ θ = 0 on ΓD

.

To each θ ∈ ΘLip we associate a transformation map Tθ := Id + θ on Ω0. Thistransformation leads to the perturbed domain Ωθ and the corresponding freeboundary Γθ :

Ωθ := Tθ(Ω0) =

x ∈ RN∣∣ x = Tθ(x0), ∀x0 ∈ Ω0

,

Γθ := Tθ(Γ0) =

x ∈ RN∣∣ x = Tθ(x0), ∀x0 ∈ Γ0

;

see Fig. 5.1. Note that the free boundary is fixed at possible intersections withthe fixed part of the boundary. For Lipschitz domains and Lipschitz perturba-tion fields, the transformation Tθ is invertible and both Tθ and T -1

θ are Lipschitzcontinuous, provided that θ is not too large; see Sec 4.2.2.

Obviously, many perturbation fields in ΘLip vanish at the free boundary Γ0and, accordingly, do not yield perturbed domains. Furthermore, a particularperturbed domain has nonunique parametrizations in ΘLip, i.e., there exist dis-tinct perturbation fields that give the same domain. To have a unique associ-ation between the domains and their parametrization, we need to consider asubspace Θ ⊂ ΘLip of suitable perturbation fields. Essentially, these perturba-tion fields are Lipschitz continuous extensions of boundary perturbations de-fined on the free boundary Γ0; see Sec 4.2.3 on scalar boundary perturbations.We assume the boundary perturbations to be in in C0,1(Γ; R

N).


ΩθΩ0

ΓθΓ0Tθθ(x0)

x0

x = Tθ(x0)

Figure 5.1: Illustration of the transformation Tθ , mapping the reference domain Ω0 ontoΩθ .

5.2.3 Weak form of the free-boundary problem

For an admissible θ ∈ Θ and its corresponding domain Ωθ , recall that

H10,γ(Ωθ) :=

v ∈ H1(Ωθ) : v = 0 on γ

.

Furthermore, to deal with nonzero traces, we defined H1h(Ωθ) := h|Ωθ

+H1

0,∂Ωθ(Ωθ). A weak formulation of (5.1) based on perturbations of the iden-

tity parametrizations is given by:

Find θ ∈ Θ and u ∈ H1h(Ωθ) :

N((θ, u); v

)= 0 ∀v ∈ H1

0,ΓD(Ωθ) ,

(5.2)

where

N((θ, u); v

):= A

(θ; u, v

)− F

(θ; v

)− G

(θ; v

)(5.3)

and the semi-linear forms are defined as

A(θ; u, v

):=

∫

Ωθ

∇u · ∇v , F(θ; v

):=

∫

Ωθ

f v , G(θ; v

):=

∫

Γθ

g v .

Note that the variational statement (5.2) is still noncanonical in the sense that uand v reside in function spaces that depend on the solution component θ. Wewill return to this issue in Sec. 5.3.

5.2.4 Goal functionals and approximation errors

Given a solution (θ, u) of (5.2), we shall be particularly interested in specificgoal functionals, i.e., quantities of interest Q(θ, u) ∈ R. An example is theweighted average of u defined by

Qave(θ; u) :=∫

Ωθ

qave u

5.3 Goal-oriented error estimation by domain-map linearization 79

where the weight qave ∈ H1(RN) is a given function. Another example of rele-

vance in free-surface flows, is the weighted elevation of the free-boundary:

Qelev(θ) :=∫

Γ0

qelev αθ .

Here, the weight qelev ∈ L2(Γ0) is given and the elevation αθ := α(Ωθ) : Γ0 → R

is a scalar function which associates to a specific domain Ωθ the vertical devia-tion of the free boundary with respect to the rest position, Γ0.

Let θh ∈ Θ and uh ∈ H1h(Ωθh) be approximations, obtained by applying

for example the Galerkin method to (5.2) with suitable finite-dimensional sub-spaces. For later reference, we notice that the approximation uh thus satisfiesthe Dirichlet boundary condition, uh = 1, on the approximate free boundaryΓθh . The corresponding approximate value of the goal functional is Q(θh, uh).To enable goal-adaptive discretizations, it is of key importance to derive a dual-based estimate of the goal error

EQ := Q(θ, u) −Q(θh, uh) ;

see Chapter 3. In the next section we show how we can apply the theory fromChapter 3 to our free-boundary problem.

5.3 Goal-oriented error estimation by domain-map

linearization

The weak formulation (5.2) is not yet in canonical form, since the test and trialspaces depend on θ. To cast (5.2) into canonical form, we introduce a domainmap which provides an isomorphism between the θ-dependent domain and afixed reference domain, and apply this map to remove the θ-dependence of thetest and trial spaces from the variational formulation. In Sec. 5.3.1 we considerthe transformation to the most obvious reference domain, Ω0. In Sec. 5.3.2 weconsider the transformation to the approximate domain, Ωθh , which yields amore natural dual formulation. Finally, it is shown in Sec. 5.3.3 that the dualproblems corresponding to the two transformations are equivalent.

5.3.1 Domain-map linearization at reference domain

Recall from Sec. 5.2.2 the transformation Tθ = Id + θ from the reference domainΩ0 to Ωθ . For all admissible θ ∈ Θ, Tθ constitutes a C0,1-diffeomorphism, andthe function transportation map

H1(Ω0) ∋ v0 7→ v0 T -1θ ∈ H1(Ωθ)


is a linear bijection; see [120, p. 21] or [59, p. 406]. In essence, this transportationof domain-dependent functions allows a reformulation of the free-boundaryproblem on a fixed domain. As ΓD is invariant under Tθ , we have the equalityof spaces

H10,ΓD

(Ωθ) =

v = v0 T -1θ : v0 ∈ H1

0,ΓD(Ω0)

. (5.4)

Transformed free-boundary problem Let us introduce the semilinear formN0 :

(Θ × H1(Ω0)

)× H1(Ω0) → R defined as:

N0((θ, w0); v0

):= N

((θ, w0 T -1

θ ); v0 T -1θ

)∀v0, w0 ∈ H1(Ω0) . (5.5)

This is essentially the transformed form of N taking functions on Ω0. Further-more, if we denote by

u0 := u Tθ ∈ H1h(Ω0) (5.6)

the solution of (5.2) transformed to Ω0, then by using (5.4), we can easily verifythat the solution (θ, u0) satisfies

N0((θ, u0); v0

)= 0 ∀v0 ∈ H1

0,ΓD(Ω0) .

To specify this abstract variational statement, let us denote by

DTθ := ∂Tθ(x1, . . . , xN)/∂(x1, . . . , xN) and Jθ := det DTθ ,

the Jacobian matrix and the Jacobian determinant, respectively, of the transfor-mation map Tθ . Furthermore, let

ωθ := Jθ |DT -Tθ n| ,

denote the tangential Jacobian on Γ0, of use in transforming surface integrals.We summarize the specification in the following proposition.

Proposition 5.1 The transformed free-boundary problem solution (θ, u0) ∈Θ × H1

h(Ω0) satisfies

∫

Ω0

(Aθ∇u0) · ∇v0 −∫

Ω0

fθ v0 −∫

Γ0

gθ v0 = 0 ∀v0 ∈ H10,ΓD

(Ω0) , (5.7)

where

Aθ := Jθ DT -1θ DT -T

θ , fθ := Jθ ( f Tθ) , gθ := ωθ (g Tθ) .


Proof Basically the proof follows by transforming the integrals to Ω0. Thesetransformations are essentially (4.19) and (4.21) in Sec 4.3. Consider any v ∈H1

0,ΓD(Ωθ). To transform A(θ; u, v) in (5.3), we use (4.19) and the identity

(∇w) Tθ = DT -Tθ ∇(w Tθ) ∀w ∈ H1(Ωθ) ,

to obtain

A((θ, u); v

)=

∫

Ω0

(DT -T

θ ∇(u Tθ))·(

DT -Tθ ∇(v Tθ)

)Jθ .

Replacing u Tθ with u0 in accordance with (5.6), and setting v Tθ =: v0 ∈H1

0,ΓD(Ω0), we obtain the first term in (5.7). Similarly, the other two terms fol-

low from (4.19) and (4.21). ¤

The goal functional Q can be expressed in terms of u0 as

Q(θ, u) = Q(θ, u0 T -1θ ) =: Q0(θ, u0) .

Note that Q0 is defined on Θ × H1(Ω0). For the weighted average functional,we obtain, in particular,

Qave0 (θ; u0) =

∫

Ω0

qaveθ u0 ,

where

qaveθ := Jθ (qave Tθ) .

As the other goal functional, the weighted elevation functional, is independentof u, we simply have Qelev

0 = Qelev.

Dual-based error representation Because N0 and Q0 act on fixed spaces, wecan essentially follow the standard error-estimation framework; see Sec. 3.3.First, we denote by

uh0 := uh Tθh ∈ H1

h(Ω0) (5.8)

the approximation uh transported to Ω0. Accordingly, we define the dual prob-lem by linearizing N0 and Q0 about (θh, uh

0):

Find z0 ∈ H10,ΓD

(Ω0) :

N′0((θh, uh

0); z0)(δθ, δu0) = Q′

0(θh, uh

0)(δθ, δu0)

∀(δθ, δu0) ∈ ΘΓ0 × H10,∂Ω(Ω0) ,

(5.9)


We refrain here from a precise specification of the derivatives in (5.9). InSec. 5.3.3 it will be shown that (5.9) can be equivalently expressed on the ap-proximate domain Ωθh , and this equivalent formulation will be considered inmore detail in Sec. 5.4. Proceeding under the assumption that there exists aunique dual solution z0 to (5.9), this z0 is indeed appropriate for linking theerror in the goal with the residual of the primal problem (5.2),

R((θh, uh); ·

):= −N

((θh, uh); ·

). (5.10)

This is expressed by the following theorem.

Theorem 5.A (Error Representation Based on z0) Given any approximation(θh, uh) ∈ Θ × H1

h(Ωθh) of the solution (θ, u) ∈ Θ × H1h(Ωθ) of the free-

boundary problem (5.2), let z0 ∈ H10,ΓD

(Ω0) be the solution of dual prob-lem (5.9). It holds that

EQ := Q(θ, u) −Q(θh, uh) = R((θh, uh); z0 T -1

θh

)+ r , (5.11)

with r = o(‖eθ‖Θ, ‖eu

0‖H10,ΓD

(Ω)

)and the errors are defined as eθ := θ − θh and

eu0 := u Tθ − uh Tθh .

The proof is delayed until the end of this section.This error representation formula for free-boundary problems is the ana-

logue of the canonical formula in Theorem 3.H. It shows how the dual so-lution z0 in the reference domain is employed in the residual evaluation forobtaining the error estimate. That is, before evaluation in the residual, z0 istransported back to the approximate domain Ωθh .

Theorem 5.A also provides an interpretation of the error terms in the higher-order remainder r. With respect to the exact u ∈ H1(Ωθ) and approximate uh ∈H1(Ωθh), which reside on different domains, the remainder forms a higher-order term in their difference on the reference domain, that is, eu

0 ∈ H10(Ω0).

Moreover, trivially, r is a higher-order term in the error eθ = θ − θh ∈ Θ.Let us now give a proof of Theorem 5.A. An essential element of the proof

is provided by Taylor series formulae of the functionals Q and N:

Lemma 5.2 The following Taylor series formulae hold:

Q(θ, u) = Q(θh, uh) +Q′0(θh, uh

0)(eθ , eu

0 ) + rQ0 , (5.12a)

N((θ, u); z0 T -1

θ

)= N

((θh, uh); z0 T -1

θh

)+ N′

0((θh, uh

0); z0)(eθ , eu

0 ) + rN0

(5.12b)

for any z0 ∈ H10,ΓD

(Ω0), with remainders rQ0 , rN0 = o(‖eθ‖Θ, ‖eu

0‖H10,ΓD

(Ω)

).


It is to be noted that these formulae relate the values of the functionals on differ-ent domains and for different functions by a linear functional on the referencedomain (upto higher-order terms).

Proof By the definitions of N0, u0 and uh0 in (5.5), (5.6) and (5.8), respectively,

we have the identity

N((θ, u); z0 T -1

θ

)−N

((θh, uh); z0 T -1

θh

)= N0

((θ, u0); z0

)−N0

((θh, uh

0); z0)

The first 2 entries of N0((·, ·); z0) are elements of the fixed spaces Θ and H1

h(Ω0).Therefore, we can apply a standard Taylor-series formula, see Theorem 3.E,to the right-hand side, yielding (5.12b). Eq. (5.12a) can be established analo-gously. ¤

Proof (of Theorem 5.A) Consider the goal error EQ = Q(θ, u)−Q(θh, uh). Us-ing (5.12a), and subsequently invoking the dual problem (5.9), we obtain

EQ = Q′0(θh, uh

0)(eθ , eu0 ) + rQ0 = N′

0((θh, uh

0); z0)(eθ , eu

0 ) + rQ0 .

Next, applying (5.12b), it follows that

EQ = N((θ, u); z0 T -1

θ

)−N

((θh, uh); z0 T -1

θh

)+ rQ0 − rN0 .

Notice that N((θ, u); z0 T -1

θ

)= 0 in accordance with our primal problem (5.2).

Finally, we obtain the proof by substituting the residual R = −N accordingto (5.10). ¤

5.3.2 Domain-map linearization at approximate domain

A more natural dual formulation is obtained by transforming the free-boundary problem to the approximate domain corresponding to θh. For conve-nience of notation, we introduce the notation

Ω := Ωθh and Γ := Γθh .

We now require a bijective transformation which maps Ω onto admissible do-mains Ωθ . We denote this map by

Tθ : Ω → Ωθ .

It is convenient (but not necessary) to define Tθ via the transformation T(·) in-troduced in Sec. 5.2.2:

Tθ := Tθ T -1θh = Id + (θ − θh) T -1

θh ∀θ ∈ Θ ; (5.13)


Ωθ

Ω

Ω0

Γθ

Γ

Γ0

Tθ

T -1θh

Tθ

Figure 5.2: Defining the map Tθ : Ω → Ωθ via the reference domain, i.e., Tθ : ΩT -1

θh−→Ω0

Tθ−→ Ωθ .

see Fig. 5.2 for a graphical illustration. Note that Tθ constitutes a perturbationof the identity with perturbation vector field (θ − θh) T -1

θh . The correspondingfunction transportation map leads to the equality of spaces

H10,ΓD

(Ωθ) =

v = v T -1θ : v ∈ H1

0,ΓD(Ω)

. (5.14)

Transformed free-boundary problem Proceeding as in Sec. 5.3.1, let us nowintroduce the transformed functional N :

(Θ × H1(Ω)

)× H1(Ω) → R:

N((θ, w); v

):= N

((θ, w T -1

θ ); v T -1θ

)∀v, w ∈ H1(Ω) . (5.15)

Next, let us denote the u-solution of (5.2) transformed to Ω by

u := u Tθ ∈ H1h(Ω) . (5.16)

By invoking (5.14), it follows that

N((θ, u); v

)= 0 ∀v ∈ H1

0,ΓD(Ω) .

The precise specification of this abstract variational statement can be de-rived by applying Proposition 5.1 to this situation with the necessary modifi-cations. First, define the Jacobian and tangential Jacobian associated with Tby

Jθ := det DTθ and ωθ := Jθ |DT -Tθ n| . (5.17)


Proposition 5.3 The transformed free-boundary problem solution (θ, u) ∈ Θ×H1

h(Ω) satisfies∫

Ω(Aθ∇u) · ∇v −

∫

Ωfθ v −

∫

Ωgθ v = 0 ∀v ∈ H1

0,ΓD(Ω) ,

where 3


θ , fθ := Jθ ( f Tθ) , gθ := ωθ (g Tθ) .

The corresponding transformation of Q is given by

Q(θ, u) := Q(θ, u T -1θ ) = Q(θ, u) .

Dual-based error representation In this case, contrary to linearization at Ω0,it is not neccesary to transport the approximation uh, as it is already defined onΩ. Hence, we can immediately proceed to the definition of the dual problem at(θh, uh):

Find z ∈ H10,ΓD

(Ω) :

N′((θh, uh); z)(δθ, δu) = Q′(θh, uh

)(δθ, δu)

∀(δθ, δu) ∈ Θ × H10,∂Ω(Ω) .

(5.18)

We provide a specification of the functionals in (5.18) in Sec. 5.4.1. Continuingunder the assumption that (5.18) has a unique solution z, we provide the errorrepresentation formula based on z:

Theorem 5.B (Error Representation Based on z) Given any approximation(θh, uh) ∈ Θ × H1

h(Ω) of the solution (θ, u) ∈ Θ × H1h(Ωθ) of the free-boundary

problem (5.2), let z ∈ H10,ΓD

(Ω) be the solution of dual problem (5.18). It holdsthat

EQ := Q(θ, u) −Q(θh, uh) = R((θh, uh); z

)+ r , (5.19)

with r = o(‖eθ‖Θ, ‖eu‖H1

0,ΓD(Ω)

)and where eθ := θ − θh and

eu := u Tθ − uh .

Note that the remainder now forms a higher-order term in the difference on theapproximate domain, that is, eu ∈ H1

0(Ω).

Proof Proceeds analogously as the proof of Theorem 5.A. ¤

3To avoid the proliferation of “ ˆ” symbols, we allow ambiguous notations here. The preciseconnotation of Aθ , fθ or gθ will be clear from the context.


5.3.3 Equivalence of dual problems

The essential difference between mapping to Ω0 and Ω occurs in the corre-sponding dual problems (5.9) and (5.18). The corresponding dual solutions z0on Ω0 and z on Ω are however equivalent in the folowing sense:

Proposition 5.4 Given the transformation Tθ according to (5.13), the solution z0of dual problem (5.9) transported to the approximate domain Ω is equal to thesolution z of dual problem (5.18), that is,

z0 T -1θh = z ∈ H1

0,ΓD(Ω) .

Note that this implies that the residuals and the remainders in the error rep-resentations corresponding to Ω0 and Ω, (5.11) and (5.19), coincide. In fact, itdoes not matter which domain is taken as a reference: the dual solutions corre-sponding to two distinct reference domains are related by the map between thedomains.

Proof (of Proposition 5.4) The proof is obtained by showing that z0 T -10,θh sat-

isfies dual problem (5.18). Consider v0 ∈ H0,ΓD(Ω0) and w0 ∈ H1

h(Ω0). By thedefinitions of N0 and N, (5.5) and (5.15), we have the key identity

N0((θ, w0); v0

)= N

((θ, w0 T -1

θ ); v0 T -1θ

)

= N((θ, w0 T -1

θ Tθ); v0 T -1θ Tθ

)

= N((θ, w0 T -1

θh ); v0 T -1θh

),

where we used (5.13) in the last step. Taking the derivative at the approxima-tion (θh, uh

0) yields the relation between N′0 and N′:

N′0((θh, uh

0); v0)(δθ, δu0)

= limt→0

1t

(N0

((θh + t δθ, uh

0 + t δu0); v0)−N0

((θh, uh

0); v0))

= limt→0

1t

(N

((θh + t δθ, uh + t δu0 T -1

θh ); v0 T -1θh

)− N

((θh, uh); v0 T -1

θh

))

= N′((θh, uh); v0 T -1θh

)(δθ, δu0 T -1

θh ) .

Notice that we used uh0 = uh Tθh in the second step, see (5.8). Similarly, we

have

Q′0(θh, uh

0)(δθ, δu0) = Q′(θh, uh

)(δθ, δu0 T -1

θh ) .

Hence, substituting the above identities in the Ω0-dual problem (5.9), it followsthat z0 satisfies

N′((θ, uh); z0 T -1θh )

)(δθ, δu0 T -1

θh ) = Q′(θ, uh)(δθ, δu0 T -1

θh ) ,

5.4 Analysis of the dual problem 87

for all (δθ, δu0) ∈ Θ × H10,∂Ω0

(Ω0). Finally, recall that the function transporta-

tion map, δu0 7→ δu0 T -1θh is a linear bijection, cf. (5.4), implying the equality of

spaces

H10,∂Ω

(Ω) =

δu = δu0 T -1θh : δu0 ∈ H1

0,∂Ω0(Ω0)

.

Hence, we have

N′((θ, uh); z0 T -1θh )

)(δθ, δu) = Q′(θ, uh

)(δθ, δu) ,

for all (δθ, δu) ∈ Θ × H10,∂Ω

(Ω), which concludes the proof. ¤

5.4 Analysis of the dual problem

In this section, we analyze the Ω-dual problem (5.18). First, we specify thederivatives in (5.18). Then, we interpret the dual problem by extracting thecorresponding partial differential equation and boundary conditions.

Recall the Ω-dual problem (5.18):

Find z ∈ H10,ΓD

(Ω) :

N′((θh, uh); z)(δθ, δu) = Q′(θh, uh

)(δθ, δu)

∀(δθ, δu) ∈ Θ × H10,∂Ω(Ω) .

(5.20)

The semilinear form N is given by

N((θ, w); v

)=

∫

Ω(Aθ∇w) · ∇v −

∫

Ωfθ v −

∫

Γgθ v

= A(θ; w, v

)− F

(θ; v

)− G

(θ; v

),

where, for convenience, we have introduced transformed functionals of A, F

and G:

A(θ; w, v

)= A

(θ; w T -1

θ , v T -1θ

),

F(θ; v

)= F

(θ; v T -1

θ

),

G(θ; v

)= G

(θ; v T -1

θ

),

for all v, w ∈ H1(Ω). We consider the dual problem for a goal functional con-sisting of the sum of the average and elevation functional. When transformedto Ω, the goal functional is given by

Q(θ, u) = Qave(θ; u) + Qelev(θ) =∫

Ωqave

θ u +∫

Γ0

qelev αθ ,

with

qaveθ := Jθ (qave

θ Tθ) . (5.21)


5.4.1 Specification of the dual problem

The variational statement (5.20) can be logically separated into two equationscorresponding to δu and δθ. Since only A and Qave depend on u and, moreover,the dependence is linear, the δu-equation is simply

A(θh; δu, z

)= Qave(θh; δu

)∀δu ∈ H1

0,∂Ω(Ω) .

Furthermore, in view of Tθh = Id, we have A(θh; ·, ·

)= A

(θh; ·, ·

)and

Qave(θh; ·)

= Qave(θh; ·). Hence, the above expression corresponds to

∫

Ω∇δu · ∇z =

∫

Ωqave δu ∀δu ∈ H1

0,∂Ω(Ω) . (5.22a)

The δθ-equation, on the other hand, is given by

A′(θh; uh, z)(δθ) − F′(θh; z

)(δθ) − G′(θh; z

)(δθ)

= Qave′(θh; uh)(δθ) + Qelev′(θh)(δθ) ∀δθ ∈ Θ .

For a specification of this equation, we require the derivatives of Aθ , fθ , gθ ,qave

θ and αθ . Let us first state some elementary derivatives. Generally, suchderivatives are given for a linearization at θ = 0, that is, at the unperturbedconfiguration; see [59, 202] and Chapter 4, for example. However, linearizationsabout nonzero θ can simply be obtained by translation. In particular, note thatTθ can be written as a perturbation of the identity starting from θh:

Tθh+t δθ = Id + t (δθ T -1θh ) = Id + t δθ ,

where

δθ := δθ T -1θh ∈ Θ :=

δθ = δθ T -1

θh , ∀δθ ∈ Θ

;

see (5.13). A proof of the following lemmata then follows from standard resultsin Chapter 4 or [59, 202].

Lemma 5.5 For Tθ , J and ω defined in (5.13) and (5.17), we have⟨∂θDTθh , δθ

⟩= Dδθ ,

⟨∂θ Jθh , δθ

⟩= div δθ ,

⟨∂θDT -1

θh , δθ⟩

= −Dδθ ,⟨∂θωθh , δθ

⟩= divΓ δθ .

for all δθ ∈ Θ.

Recall that the tangential divergence in Lemma 5.5 is defined as

divΓ(·) := div(·)∣∣Γ− ∂n(·) · n ;

see Sec. 4.3.4 or Def. A.2 in Sec. A.1.


Lemma 5.6 Let φ ∈ H1(RN). Then the map θ 7→ φ Tθ is differentiable at

θh ∈ Θ in L2(Ω). The derivative is given by⟨∂θ(φ Tθ)

∣∣θh , δθ

⟩= ∇φ · δθ .

for all δθ ∈ Θ.

Using these results, we can easily derive the derivatives of Aθ , fθ , gθ and qaveθ

from their definitions in Prop. 5.3 and (5.21):⟨∂θ Aθh , δθ

⟩= (div δθ) I − Dδθ − DδθT ,

⟨∂θ fθh , δθ

⟩= div( f δθ) ,

⟨∂θ gθh , δθ

⟩= g divΓ δθ + ∇g · δθ ,

⟨∂θqave

θh , δθ⟩

= div(qave δθ) ,

with I the identity matrix. The derivative of αθ required for the linearization ofQelev is a bit more involved. Therefore, it is derived in Appendix A.4.2 for thetwo-dimensional case. Its final result is the linearization

Qelev′(θh)(δθ) =∫

Γqelev δθ · n ,

where since qelev is only defined on Γ0, it should be interpreted with the aid ofa projection along the xN-axis, that is

qelev(x1, . . . , xN) = qelev(x1, . . . , xN−1, xΓ0N )

with xΓ0N being the xN-coordinate of Γ0.

The above results lead to the following specification of the δθ-equation:

Proposition 5.7 Given an approximation θh ∈ Θ with corresponding do-main Ω = Ωθh and an approximation uh ∈ H1

h(Ω), the δθ-equation in dualproblem (5.20) is given by

∫

Ω

([div δθ I − Dδθ − DδθT

]∇uh

)· ∇z

−∫

Ωdiv( f δθ) z −

∫

Γ

(g divΓ δθ + ∇g · δθ

)z

=∫

Ωdiv(qave δθ) uh +

∫

Γqelev δθ · n ∀δθ ∈ Θ .

(5.22b)

For a given approximate domain Ω and approximation uh the complete dualproblem for z ∈ H1

0,ΓD(Ω) is specified by (5.22a) and (5.22b). Note that the dual

problem is independent of the particular parametrization in ΘLip that gives Ω.The dual problem is however dependent on the extension into Ω of the pertur-bations δθ ∈ Θ, cf. the final remark in Sec. 5.2.2.


5.4.2 Interpretation of the dual problem

At this point, we are ready to interpret the dual problem. A priori we know thatthe dual solution z is in H1

0,ΓD(Ω). Hence, z satisfies the boundary condition

z = 0 on ΓD .

To extract the partial differential equation in Ω and the boundary condition onΓ, we assume that z ∈ H1

0,ΓD(Ω) ∩ H2(Ω) and, furthermore, that Γ is smooth

enough, for example, Γ is C1,1. By integration by parts and standard variationalarguments, the δu-equation (5.22a) yields a Poisson equation driven by our in-terest in the average goal:

−∆z = qave in Ω .

The δθ-equation in principle specifies a boundary condition on Γ, which com-pletes the boundary value problem for z. However, it does not generally corre-spond to an ordinary local boundary condition. In particular, the δθ-equationenforces a boundary condition involving a non-local operator associated withthe residual. This is evidenced by the following theorem, whose proof we delayuntil the end of this section:

Theorem 5.C (Dual Boundary Condition) If Γ is C1,1 and z ∈ H10,ΓD

(Ω) ∩H2(Ω), then the δθ-equation (5.22b) can be written as

R((θh, uh);∇z · δθ

)−

∫

Γ

(g ∂nz +

(f + ∂ng + κ g

)z + qave + qelev

)δθ · n = 0 ,

for all δθ ∈ Θ, where κ := divΓ n coincides with the additive curvature (sum ofN − 1 curvatures) of Γ.

To establish that the above condition indeed corresponds to a non-localboundary condition, we recall from the final remark in Sec. 5.2.2 that Θ con-sists of perturbation fields that are extensions of functions on Γ and that yieldunique perturbed domains. For a C1,1 free boundary, this implies that δθ · n 6= 0for all δθ ∈ Θ \ 0 and, moreover, δθ1 · n 6= δθ2 · n for distinct δθ1, δθ2 ∈ Θ.Accordingly, we can identify the residual term with a local free-boundary termby means of the L2(Γ) Riesz representant rh(z):

∫

Γrh(z) δθ · n = R

((θh, uh);∇z · δθ

)∀δθ ∈ Θ .

Note that rh(z) is dependent on the particular extension into Ω of perturbationsδθ ∈ Θ. With the L2(Γ) identification, we can summarize the dual problem for


z as:

−∆z = qave in Ω ,

z = 0 on ΓD ,

rh(z) − g ∂nz −(

f + ∂ng + κ g)

z = qave + qelev on Γ .

At the solution (θ, u) the residual vanishes and accordingly the nonlocal bound-ary term rh(z) vanishes too. The boundary condition on Γ then reduces to anordinary Robin boundary condition, and its dependency on the particular ex-tension into Ω of the perturbations δθ ∈ Θ disappears.

Similar Robin problems are also encountered in the shape-linearizedBernoulli free-boundary problem, cf. [100, 141], and in its shape-linearizedadjoint which is considered in the following chapter. A standard sufficiencycondition for wellposedness of the dual problem at the solution (for whichrh(z) = 0) is ( f + ∂ng)/g + κ ≥ 0 on Γ. Such conditions on the data alsoappear in [78, 79].

Proof (of Theorem 5.C.) We will rewrite the terms in (5.22b) one after another.To rewrite the first term, we need the gradient of an innerproduct. That is, let ξand η denote two H1 vector functions. Then ∇(ξ · η) = DξT η + DηT ξ. We canthen verify

∫

Ω

([− Dδθ − DδθT

]∇uh

)· ∇z

=∫

Ω

(δθ · ∇(∇uh · ∇z) −∇uh · ∇(∇z · δθ) −∇z · ∇(∇uh · δθ)

)

=∫

Ω

(δθ · ∇(∇uh · ∇z) −∇uh · ∇(∇z · δθ) + ∆z (∇uh · δθ)

)

−∫

Γ∂nuh ∂nz δθ · n ,

where in the last step, we performed an integration by parts on the third term.Furthermore, we invoked δθ = 0 on ΓD and the fact that uh is constant (=1) onΓ, so that ∇uh = ∂nuh n on Γ. Substituting the above result in the first term of(5.22b) gives

∫

Ω

([div δθ I − Dδθ − DδθT ]∇uh

)· ∇z

=∫

Ω

(div

(δθ (∇uh · ∇z)

)−∇uh · ∇(∇z · δθ) + ∆z (∇uh · δθ)

)

−∫

Γ∂nuh ∂nz δθ · n

=∫

Ω

(−∇uh · ∇(∇z · δθ) + ∆z (∇uh · δθ)

), (5.23a)


where in the last step we used the divergence theorem on the first term andinvoked the same arguments as before on δθ and uh to cancel the Γ-term. Next,we continue with the terms involving f , qave and qelev in (5.22b). By integrationby parts, we simply obtain

−∫

Ωdiv( f δθ) z =

∫

Ωf ∇z · δθ −

∫

Γf z δθ · n , (5.23b)

∫

Ωdiv(qave δθ) uh +

∫

Γqelev δθ · n = −

∫

Ωqave (∇uh · δθ)

+∫

Γ

(qave + qelev)

δθ · n . (5.23c)

Finally, we take up the term involving g in (5.22b). For this, we require addi-tional tangential calculus; see for instance [59, 60]. At Γ, a gradient splits up ina tangential gradient and a normal component: ∇(·) = ∇Γ(·) + ∂n(·) n. Hence

∇g · δθ = ∇Γg · δθ + ∂ng δθ · n .

We can combine the tangential divergence and tangential gradient and apply atangential Green’s identity as follows:∫

Γ

(g divΓ δθ + ∇Γg · δθ

)z =

∫

ΓdivΓ (g δθ) z =

∫

Γκ g z δθ · n −

∫

Γg δθ · ∇Γz .

It then follows that the term involving g in (5.22b) can be written as

−∫

Γ

(g divΓ δθ + ∇g · δθ

)z =

∫

Γ

(g∇z · δθ −

((∂ng + κ g) z + g ∂nz

)δθ · n

).

(5.23d)

We finish by gathering the contributions in (5.23a)–(5.23d). Basically, we candistinguish three different groups: domain contributions involving ∇uh · δθand ∇z · δθ, and free-boundary contributions involving δθ · n. The first groupcancels since −∆z = qave. The second group adds up to the residual termR

((θh, uh);∇z · δθ

). The last group forms the free-boundary integral as stated

in the proposition. ¤

5.5 Numerical experiments

In this section, we present numerical experiments. First, to exemplify essentialattibutes, we consider in Sec. 5.5.1 the free-boundary problem in one dimen-sion. Similar one-dimensional free-boundary problems have been consideredin [51, 100, 221] and [140, p. 37]. One-dimensional free-boundary problems areattractive for a number of reasons. The first being that the free boundary has

5.5 Numerical experiments 93

no geometry, i.e., it is merely a point. Also, it is rather effortless to obtain exactexpressions for dual solutions. Therefore, error estimates are inexact only dueto nonlinearity.

In Sec. 5.5.2 we take up the free-boundary problem in two dimensions. Ap-proximations to the free boundary problem are obtained by using linear finiteelements. Here, we focus on the effectivity of the error estimate on uniformmeshes. In addition, we show an example of goal-oriented adaptive mesh re-finement.

5.5.1 One-dimensional application

In the one-dimensional setting, we characterize the variable domain as Ωϑ =(0, ϑ) ⊂ R. The Dirichlet boundary and free boundary correspond to singlepoints, ΓD = 0 and Γϑ = ϑ, respectively. The semilinear form N (= − R)and the goal functionals are given by

N((ϑ, u); v

)=

∫

Ωϑ

(ux vx − f v

)dx − g(ϑ) v(ϑ) ,

Qave(ϑ; u) =∫

Ωϑ

qave u dx ,

Qelev(ϑ) = qelev ϑ ,

where (·)x = d(·)/dx and qelev ∈ R. To a free boundary approximation ϑh> 0,

we associate a domain transformation from Ω = Ωϑh to Ωϑ by the linear map

x = Tϑ(x) = ϑϑh x = x + ϑ−ϑh

ϑh x .

Let, furthermore, uh ∈ H1h(Ω) be given. It can be verified that the Ω-dual

problem (5.18) reduces in this setting to: Find z ∈ H10,ΓD

(Ω):∫

Ωδux zx dx =

∫

Ωqave δu dx ,

− δϑϑh

∫

Ω

(uh

x zx + ( f x)x z)

dx − gx(ϑh) z(ϑh) δϑ = δϑϑh

∫

Ω(qave x)x uh dx

+ qelev δϑ ,

for all (δϑ, δu) ∈ R × H10,∂Ω

(Ω). The dual problem translates into the boundaryvalue problem:

−zxx(x) = qave(x) ∀x ∈ Ω ,

z(0) = 0 ,

R((ϑh, uh); zx x/ϑh

)−

(g zx + ( f + gx) z

)(ϑh) = qave(ϑh) + qelev .


Table 5.1: Specification of the data for the one-dimensional example.

f (x) g(x) qave(x) qelev ϑ u(x) Qave(ϑ; u) Qelev(ϑ)

− 12 x − 1 1 1 2 1

4 x2 23 2

−1

0

1

ϑ = 2ϑh

uuh

−1

0

1zave

zelev

Figure 5.3: Exact solution (ϑ, u) and approximation (ϑh, uh) [left]. Dual solutions zave

and zelev corresponding to goal functional Qave and Qelev, respectively [right].

Typical error estimate In the following numerical example, we consider thedata and goal functionals as indicated in Table 5.1. Table 5.1 also contains thecorresponding exact solution. Consider the following approximation of the so-lution and the corresponding goal values:

(ϑh, uh(x)

)=

( 32 , 2

3 x)

, Qave(ϑh; uh)

= 34 , Qelev(

ϑh)

= 32 .

Figure 5.3 [left] shows a graphical illustration of the exact and approximatesolutions. Furthermore, Figure 5.3 [right] shows the dual solutions for Qave

and Qelev:

zave(x) = 4586 x − 1

2 x2 , zelev(x) = − 2443 x .

The corresponding dual-based error estimate, EstQ := R((ϑh, uh); z

), and the

true goal-error, EQ, are as follows:

EstQave = 15344 , EstQelev = 39

86 ,

EQave = − 112 , EQelev = 1

2 .

Note that the difference in the error estimate and the true error is caused bylinearization. The only source of nonlinearity is the domain dependence andone can easily verify that the estimates are exact if ϑh = ϑ.

Convergence of error estimates In this example, the data is again specifiedas in Table 5.1. To investigate the convergence of the dual-based error estimate,we consider the following ∆ϑ-family of approximate solutions:

(ϑh, uh

)=

(ϑ − ∆ϑ , u Tϑ

). (5.24)


−1 0 1 ∆ϑ

−0.5

0

0.5EQave

EstQave

100

10-2

10-4

10-6

10-3 10-2 10-1∥∥(eϑ, eu)

∥∥

|EQ

ave−

EstQ

ave|

slope 2

∆ϑ>0∆ϑ<0

Figure 5.4: True goal error EQave and dual-based error estimate EstQave for the ∆ϑ-familyof approximations (ϑh, uh) given in (5.24) [left]. Convergence of the error in the errorestimate with respect to the norm

∥∥(eϑ, eu)∥∥ [right].

This family converges to the exact solution as ∆ϑ → 0. Note that for each∆ϑ, uh is simply a scaling of u along the x-axis. This also implies that eu =u Tϑ − uh = 0. Hence, from the perspective of the error representation (seeTheorem 5.B), the only relevant error is eϑ = ∆ϑ.

For the goal functional Qave, Figure 5.4 [left] plots the true value EQave

and the dual-based estimate EstQave with respect to ∆ϑ. It can be seen thatthe estimate approaches the exact error as ∆ϑ → 0. Moreover, the slopesof the two curves are identical at ∆ϑ = 0. To further elucidate the conver-gence behavior, Figure 5.4 [right] presents a log-log plot of the error in the esti-mate |EQave − EstQave | versus the norm of the error:

∥∥(eϑ, eu)∥∥2

= |ϑ − ϑh|2 +∣∣u Tϑ − uh

∣∣2H1(Ω)

= |∆ϑ|2 .

Both figures confirm that the estimate converges as O(‖(eϑ, eu)‖2)

.

5.5.2 Two-dimensional application

Next, we turn to the two-dimensional case. We denote coordinates by (x, y) ∈R

2. In the following examples, we compute approximations (θh, uh) of (5.2)based on piecewise-linear finite elements on triangles. Accordingly, the ap-proximate free-boundary is a piecewise-linear curve composed of the edges ofadjacent elements. The nonlinear problem is solved using a fixed point iterationsimilar to the explicit Neumann scheme in [100], where we allow the verticesof the free-boundary to move only vertically. Hence θh

1 = 0 and θh2 = αθh on Γ0.

The dual problem (5.22) is solved on the same mesh as the approximationbut with quadratic shape functions. That is, z is piecewise quadratic and van-


1

0.75

0.5

0.25

0

Figure 5.5: Parabolic-free-boundary test problem. The exact domain and solution (con-tour plot) [left], and the approximate domain and solution corresponding to the coarsestmesh [right].

ishes on ΓD and the test functions δu are piecewise-quadratic shape functionsthat are zero on ∂Ω. Furthermore, the test functions δθ in (5.22b) are suitableextensions of vertical-perturbation fields δϑ on Γ:

δθ2(x, y) = y−yb(x)

yΓ(x)−yb(x)δϑ2

(x, yΓ(x)

)∀(x, y) ∈ Ω ,

where yb represents the bottom of the domain and yΓ = yΓ0 + αθh describes theposition of the approximate free boundary. Moreover, for δϑ2 we use piecewise-quadratic shape functions which vanish on ∂Γ.

Effectivity for the parabolic-free-boundary testcase First, we investigate theeffectivity of the dual-based error estimate under uniform mesh refinement.We consider a test problem with a geometric lay-out and solution depicted inFig. 5.5. We have yb = 0 and yΓ0 = 1 and Ωθ = (0, 2) × (0, 1 + αθ). The data f , g, h of the problem is manufactured to yield the parabolic free-boundaryelevation and solution

αθ(x) = 12 x (2 − x) ,

u(x, y) = y1+αθ(x)

+ αθ(x) y1+αθ(x)

(1 − y

1+αθ(x)

).

Our interest is the average goal functional with qave = 1. For the exact solution,we have Qave(θ; u) = 67/45 = 1.4888 . . . . An illustration of the coarsest meshapproximation is also visible in Fig. 5.5. For this approximation, we find thevalue Qave(θh; uh) = 1.1573 . . . .

In Fig. 5.6, we depict the approximate dual solution z for the coarsestmesh, and for a very fine mesh. The convergence of the corresponding es-timates, EstQave = R

((θh, uh); z

), on uniformly refined meshes is reported in


-2

-1.5

-1

-0.5

0

Figure 5.6: Parabolic-free-boundary test problem. The approximate dual solution (con-tour plot) associated with a very fine mesh [left] and the coarsest mesh [right].

Table 5.2. Note that the effectivity index EstQave /EQave approaches 1, whichclearly demonstrates the consistency of the error estimate.

Goal-oriented adaptivity for free-surface flow over a bump To investigatethe applicability of the dual-based error estimate to drive adaptive mesh refine-ment, we consider a domain with a reentrant corner at the bottom; see Fig. 5.7[top]. We take yΓ0 = 1 and Ωθ = (0, 4) × (yb, 1 + αθ) and f = 0, g = 1.

Table 5.2: Convergence of the goal-oriented error estimate EstQave under uniform meshrefinement.

Elements Dofs Qave(θh; uh) EQave EstQave Effectivity

8 8 1.1573 0.33163 0.22131 0.66716 15 1.3145 0.17440 0.13852 0.79432 23 1.3694 0.11947 0.09994 0.83664 45 1.4284 0.06045 0.05499 0.910

128 77 1.4555 0.03339 0.03055 0.915256 153 1.4715 0.01740 0.01676 0.963512 281 1.4803 0.00860 0.00808 0.940

1,024 561 1.4843 0.00458 0.00450 0.9842,048 1,073 1.4867 0.00217 0.00205 0.9474,096 2,145 1.4877 0.00117 0.00115 0.9918,192 4,193 1.4883 0.00054 0.00051 0.949

16,384 8,385 1.4886 0.00029 0.00029 0.993

∞ ∞ 1.4888 0


1

0.75

0.5

0.25

0

-0.6

-0.4

-0.2

0

Figure 5.7: Free-surface flow over a bump. The exact domain and solution (contourplot) [top], and the approximate domain and dual solution corresponding to the coarsestmesh [bottom]. We have indicated the free-boundary elevation point of interest (at x0 =2 +

√2).

Moreover, h is 0 at the bottom and increases linearly to 1 along the sides of thedomain. Our interest is the elevation of the free-boundary at the specific pointx0 = 2 +

√2; see Fig. 5.7. This interest corresponds to the elevation goal func-

tional Qelev with qelev a Dirac measure at x0. The linearization of this functionalis elaborated in Appendix A.4.2. Fig. 5.7 [bottom] displays the correspondingcoarsest-mesh dual solution.

To drive the adaptivity, element refinement indicators are extracted fromthe error estimate formula, as usual (see Sec. 3.2.4). (In particular, we integrateby parts element-wise and assign weighted interior and edge residuals to theassociated elements to obtain element contributions. The absolute values ofthe element contributions are then identified as the element indicators.) Basedon these indicators, we mark a set of elements for refinement. This set is theminimal set for which the sum is a fraction of the total sum of indicators (a so-called Dörfler-type marking). We take this fraction as 0.4. The marked elementsare refined using newest-vertex bisection; see Sec. 3.2.4).

In Fig. 5.8, we plot the convergence of the error estimate versus the totalnumber of degrees of freedom, which is denoted by n. A plot of the “true” er-ror is also displayed. This true error has been obtained by computing the goal

5.6 Concluding remarks 99

10-2

10-3

10-4

10-5101 102 103 104

n

|E| (uniform)|Est| (uniform)|E| (adaptive)|Est| (adaptive)slope -3/4

slope -1

Figure 5.8: Convergence of the “true” error E = EQelev and error estimate Est = EstQelev

under uniform and adaptive mesh refinement versus the total number of degrees offreedom n.

on a uniformly refined mesh with 245,760 elements and n = 123,585 resultingin the reference value Qelev(θ) ≈ 0.02271. The results indicate that the accuracyof this reference value is surpassed on adaptively refined meshes for n > 1,000.This explains the drop in the true error for the adaptive case for n > 1,000. Fur-thermore, the plots reveal an asymptotic convergence rate of O(n-1) for adaptiverefinements. This is expected for optimal refinements and should be comparedwith the suboptimal convergence rate of approximately O(n-3/4) for uniformrefinements. Figure 5.9 shows several adaptively refined meshes. Apart fromthe refinement at the reentrant corner, the refinement at the free boundary andparticularly near the elevation point of interest is noteworthy.

5.6 Concluding remarks

We showed that free-boundary problems elude the standard goal-oriented–error estimation framework on account of the fact that their typical variationalform is non-canonical. To obtain an appropriate dual problem (linearized ad-joint), we presented the domain-map linearization approach. In this approachthe free-boundary problem is transformed into an equivalent problem on afixed reference domain which has a canonical variational form. The dual prob-lem is then obtained by linearization with respect to the domain map. Weshowed that the solution of the dual problem is essentially independent of theselected reference domain: dual solutions corresponding to distinct referencedomains are related by the obvious map between the reference domains.

For a Bernoulli-type free-boundary problem, we showed that the dual prob-lem corresponds to a Poisson problem with a nonlocal Robin-type boundarycondition. The nonlocal term depends on the particular extension of boundary


Figure 5.9: Adaptively refined meshes, controlling the error in the free-boundary ele-vation at x0 = 2 +

√2, obtained after 10, 18 and 29 iterations with 120, 793 and 5,447

elements, respectively.

perturbations into the domain but, being of residual type, the nonlocal termvanishes at the exact free-boundary solution. The effectivity of the dual-basederror estimate and its usefulness in goal-oriented adaptive mesh-refinementwas demonstrated by numerical experiments in one and two dimensions.

The presented approach admits several extensions. For example, we con-sidered constant Dirichlet data at the free boundary which means that the datais invariant under domain transformations. Nonconstant Dirichlet data can beincluded by means of a free-boundary Lagrange multiplier in the variationalformulation. Such a formulation moreover allows nonconforming trial func-tions that violate the Dirichlet data.

The domain-map linearization approach bears similarities to the classi-cal material derivative in shape optimization in view of the comparison offunctions in a reference domain; see Sec. A.2. An alternative in the shape-optimization field is the so-called shape derivative. The shape-linearizationapproach can also be used to obtain a suitable dual problem for goal-orientederror estimation of free-boundary problems. This is the subject of the followingchapter. In this chapter, we moreover present a comparison of the two differentapproaches.

Chapter 6

Shape-Linearization Approach to


Two powerful internal driving forces have strongly influenced the directionof theoretical research yet which usually go unmentioned in serious scien-tific writings–for fear, no doubt, that these influences may seem to havedrifted too far from the strict rules of proper scientific procedure. The firstof these is beauty, or elegance. [The second:] the irresistible allure of whatare frequently termed ‘miracles’.

Roger Penrose, 2005The Road to Reality: A Complete Guide to the Laws of the Universe1

In this chapter, we consider goal-oriented error estimation and adaptivity forfree-boundary problems using the shape-linearization approach to derive thedual problem.2

6.1 Introduction

In the previous chapter, we considered goal-oriented error estimation andadaptivity for free-boundary problems using the domain-map linearization ap-proach. We explained that free-boundary problems elude the standard goal-oriented–error-estimation framework because their typical variational form isnoncanonical. In pursuit of a canonical form, we introduced the domain-maplinearization approach at a reference domain which in essence reformulates thefree-boundary problem to a fixed reference domain. Accordingly, the dual (lin-earized adjoint) problem is obtained by linearizing the transformed problemwith respect to the domain map. This approach is straightforward. However,the dual problem contains nonstandard and nonlocal interior and boundaryterms, which is inconvenient from an implementation point of view. Moreover,there is some arbitrariness in the dual problem due to the heuristic extension

1Vintage, 20052This chapter is based on [224]. The one-dimensional results have also appeared in [221].

102 6. Shape-Linearization Approach to Free-Boundary Problems

of boundary perturbations into the domain. A similar arbitrariness appears inshape optimization in the so-called material derivative approach; see Sec. A.2.An elegant alternative in shape optimization is the shape derivative whosevariational formulation consists only of standard interior and boundary terms.Finding an analogous linearization approach for free-boundary problems hasbeen the main motivation of this chapter.

This chapter considers the so-called shape-linearization approach to derive thelinearized adjoint and obtain an appropriate dual problem. This approach wasbriefly outlined in Sec. 3.4.2. To illustrate the approach, we reconsider theBernoulli free-boundary problem of the previous chapter. We show that theassociated very weak form and goal functional of interest can be formulated asa function of the unknown domain. This shape dependence can be linearizedusing techniques from shape differential calculus, which we discussed in Chap-ter 4.

The shape-linearization approach does not pursue a canonical formulationand therefore requires a slight deviation from the usual procedure to goal-oriented error-estimation. To obtain a suitable dual problem, we reason asfollows. The very weak form of the free-boundary problem and the goal func-tional of interest admit an appropriate shape linearization. This linearizationyields a linear (adjoint) equation. In the usual procedure, this linear equationdirectly defines the dual problem; see [22, 112, 191, 209]. In our case, how-ever, the linear equation provides only a specification of the dual solution, butit does not suitably define the dual problem. Instead, we extract from the linearequation an appropriate dual problem by means of a consistent reformulation.

The dual problem can be found by straightforward variational arguments ifthe linearization takes place at a domain with a smooth free-boundary. The dualproblem then corresponds to a standard Poisson problem with a Robin bound-ary condition that involves the curvature. For nonsmooth free-boundaries,however, the construction of an appropriate dual problem requires that weconsider specific domain perturbations. This dual problem is a generalizationof the smooth case that admits singular curvatures at singular points of theboundary.

It is noteworthy that many authors have presented linearizations of free-boundary problems in an effort to arrive at Newton-based iteration algorithms.Most of these address the free-boundary problem in a discretized setting; seefor instance [64, 159, 213]. Our linearization is, however, in the continuoussetting where one requires intricate shape differential calculus. It is thereforemore related to Newton-based iteration algorithms in continuous settings aspresented in [29, 51, 100], for example. In particular, we mention the works ofKärkkäinen and Tiihonen [140–142] who appropriately apply the techniques ofshape differential calculus, in a formal sense though.


The contents of this chapter are arranged as follows: Sec. 6.2 briefly recallsthe Bernoulli free-boundary problem. In Sec. 6.3, we apply shape linearizationat smooth free boundaries to the very weak form of the free-boundary prob-lem. Furthermore, we present the dual problem suitable for goal-oriented errorestimation. Sec. 6.4 considers shape linearization at nonsmooth free bound-aries. In Sec. 6.5, we present numerical experiments and compare the shape-linearization approach with the domain-map linearization approach of Chap-ter 5. Finally, Sec. 6.6 contains concluding remarks.


We briefly recall the Bernoulli free-boundary problem and the correspondingweak formulation introduced in Sec. 2.1. We present relevant goal function-als. In addition, we introduce a very weak form of the free-boundary problemwhich shall be suitable for shape-linearization.

6.2.1 Bernoulli free-boundary problem

Let D ⊂ RN denote a sufficiently-large hold-all domain and let O denote the

set of bounded open Lipschitz domains Ω ⊂ D with boundary ∂Ω consistingof a fixed part ΓD and a variable part Γ, the free boundary; see Fig. 2.1. TheBernoulli free-boundary problem consists in seeking a domain Ω ∈ O and ascalar function u defined on Ω such that:

−∆u = f in Ω , (6.1a)

∂nu = g on Γ , (6.1b)

u = h∣∣Γ

= 1 on Γ , (6.1c)

u = h∣∣ΓD

on ΓD . (6.1d)

where we assume f ∈ C0(D), g ∈ C1(D) together with a lower bound g ≥g0 > 0, and h ∈ C1(D). Note that, in accordance with (6.1c), h|Γ = 1 is requiredfor all admissible free boundaries. For Ω ∈ O, recall that

H10,γ(Ω) :=

v ∈ H1(Ω) : v = 0 on γ

,

and the (affine) space incorporating h is defined as H1h(Ω) := h|Ω + H1

0,∂Ω(Ω).

A weak formulation of (6.1) has been given in Sec 2.1, see (2.2), which is re-peated here for convenience:

Find Ω ∈ O and u ∈ H1h(Ω) :

∫

Ω∇u · ∇v =

∫

Ωf v +

∫

Γg v ∀v ∈ H1

0,ΓD(Ω) .

(6.2)


6.2.2 Errors in goal quantities

In the previous chapter, we introduced two relevant goal functionals; viz., aweighted average of u and a weighted elevation of the free-boundary,

Qave(Ω; u) :=∫

Ωqave u and Qelev(Ω) :=

∫

Γ0

qelev α(Ω) ,

respectively, where qave ∈ H1(D) and qelev ∈ L2(Γ0). The elevation α(Ω) :Γ0 → R is a scalar function which associates to a specific domain Ω the verticaldeviation of the free boundary with respect to a horizontal rest position, Γ0.

Given an approximate domain Ω ∈ O and corresponding approximationuh ∈ H1

h(Ω), we aim to derive by shape-linearization principles a dual-basedestimate of the goal error

EQ := Q(Ω, u) −Q(Ω, uh) .

For this, the required dual problem is extracted from the linearization of thefree-boundary problem and the goal functional with respect to (Ω, u). As usual,the linearization of the free-boundary problem yields the linearized adjoint op-erator, and the linearization of the goal functional yields the right-hand side forthe dual problem.

6.2.3 Very weak form of the free-boundary problem

For a succesful linearization of the free-boundary problem, it is of crucial im-portance that we can vary Ω and u independent of each other for fixed testfunctions v. This is possible only if there are no Ω-dependent constraints onthe u- and v-space. Furthermore, we need to view u and v as functions definedon the whole of D by suitably extending them outside Ω. This gives rise to theembedding H1(Ω) ⊂ H1(D) for all Ω ∈ O.

In view of the free-boundary constraint in the space H1h(Ω), the weak form

in (6.2) is not suitable for the linearization. This constraint can be removed intwo ways. The first is by means of a Lagrange multiplier that enforces the con-straint u = h on Γ. This is the approach taken by Kärkkäinen and Tiihonenin [140–142]. The downside of this approach are the additional difficulty anddual interpretation of the Lagrange multiplier, and the inability to perform theshape linearization without unnecessary smoothness assumptions. We there-fore prefer a second approach, which enforces the constraint weakly. This ap-proach does not encounter the mentioned problems.

A variational statement that weakly enforces Dirichlet boundary conditionsis provided by the so-called very weak form. The function space that accomo-


dates test functions for the very weak form is:

H10,ΓD

(∆; D) :=

v ∈ H10,ΓD

(D) : ∆v ∈ L2(D)

.

The very weak form N :(O× H1(D)

)× H1

0,ΓD(∆; D) → R is given by

N((Ω, u); v

):=

∫

Ω−u ∆v −

∫

Ωf v −

∫

Γg v +

⟨∂nv, h

⟩∂Ω

, (6.3)

where the brackets, 〈·, ·〉∂Ω, imply a duality pairing of H−1/2(∂Ω) andH1/2(∂Ω). It can easily be verified that the free-boundary-problem solutionΩ ∈ O and u ∈ H1(Ω) ⊂ H1(D) satisfy

N((Ω, u); v

)= 0 , ∀v ∈ H1

0,ΓD(∆; D) . (6.4)

We are now ready to linearize N((Ω, u); v

)(for fixed v) and Q(Ω, u) (view-

ing it as the map Q : O × H1(D) → R). Note that the dependence on Ω isan unusual nonlinearity. However, for fixed u ∈ H1(D) and v ∈ H1

0,ΓD(∆; D),

the maps Ω 7→ N((Ω, u); v

)and Ω 7→ Q(Ω, u) are shape functionals. Hence,

their linearization can be dealt with by using techniques from shape differentialcalculus discussed in Chapter 4.

6.2.4 Free-boundary perturbations

To linearize the shape functionals at Ω, we need to introduce a suitable familyof free-boundary perturbations. We consider perturbations of the identity mapId : D → D; see Sec. 4.2.2. Let us denote by Θ the space of bounded Lipschitzperturbation-vector fields that are extensions of Lipschitz boundary perturba-tions and that vanish at ΓD , i.e.,

Θ := Θ(Ω) :=

δθ ∈ C0,1(D; RN) : δθ

∣∣Γ∈ C0,1(Γ; R

N) and δθ = 0 on ΓD

.

For δθ ∈ Θ, we define the perturbed transformation map as Tδθ := Id + δθ, andthe associated one-parameter family of domains and free boundaries is definedas

Ωt := Tt δθ(Ω) =

x ∈ RN∣∣ x = Tt δθ(X), ∀X ∈ Ω

,

Γt := Tt δθ(Γ) =

x ∈ RN∣∣ x = Tt δθ(X), ∀X ∈ Γ

;

see Fig. 6.1.3 Note that Ω0 = Ω. For small t, both Tt δθ and T -1t δθ are Lipschitz

continuous and Ωt ∈ O; see Sec. 4.2.2.3Note that the notation deviates at this point from the notation in Chapter 4, where Ωt and Γt

were denoted as the family of domains generated by the velocity method.


Ω

Γ

Γtt δθ

Figure 6.1: A perturbation of Γ by t δθ generates the domain Ωt with free boundary Γt.

Let J : O → R denote a shape functional that is shape differentiable at Ω

with respect to Θ. We denote the shape derivative of J at Ω in the direction δθ ∈Θ by

J′(Ω)(δθ) = limt→0

J(Ωt) − J(Ω)

t.

Note that the shape derivative coincides with the Gâteaux derivative of the mapθ 7→ J(Tθ(Ω)) at θ = 0 in the direction δθ; see Theorem 4.D. If this map is fur-thermore Fréchet differentiable at Ω, then the following Taylor-series identityholds:

J(Ω1) = J(Ω) + J′(Ω)(δθ) + o(‖δθ‖Θ

). (6.5)

Before we turn to the linearization of the free-boundary problem, we needto introduce the following particular case of the shape derivative of a domainintegral

∫Ω

φ ; see Theorem 4.E. It is clear from this theorem that the shapederivative at Ω vanishes if φ = 0 at Γ. However, this holds also for less regularφ than assumed in the theorem, in particular, for φ corresponding to the prod-uct of two functions of which one vanishes at Γ. We collect this result in thefollowing proposition.

Proposition 6.1 Let φ = φ1 φ2 where φ1 ∈ L2(D) and φ2 ∈ H1(D) with φ2 = 0on Γ. Then the shape functional J =

∫Ω

φ is shape differentiable at Ω ∈ O withrespect to Θ, and the shape derivative vanishes:

J′(Ω)(δθ) = 0 .

Proof It appears that this is a new result. The proof is rather elaborate and istherefore transferred to the appendix; see Sec. A.4.3. ¤

This innocent looking generalization is very important in differentiating in arigorous manner some of the terms that we will encounter in the next section.

6.3 Goal-oriented error estimation by shape linearization 107

6.3 Goal-oriented error estimation by shape lin-

earization

We now turn our attention to goal-oriented error estimation for the free-boundary problem (6.1). We proceed by linearizing the very weak form and thegoal functional at the approximation (Ω, uh) ∈ O × H1

h(Ω). In this section, weassume that the approximate free boundary Γ is sufficiently smooth, i.e., Γ is aC1,1-boundary. The more general case of Lipschitz continuous free-boundariesis taken up in Sec. 6.4.

6.3.1 Linearization of the free-boundary problem

We can write the very weak form given in (6.3) as

N((Ω, u); v)

)= −B(Ω; u, v) − F(Ω; v) − G(Ω; v) + H(Ω; v) , (6.6)

where the semi-linear forms are defined as

B(Ω; u, v) :=∫

Ωu ∆v , F(Ω; v) :=

∫

Ωf v ,

H(Ω; v) :=∫

Ω

(h ∆v + ∇h · ∇v

), G(Ω; v) :=

∫

Γg v .

Note that we replaced the duality pairing 〈∂nv, h〉∂Ω with two domain integrals;this shall be convenient for the shape derivative.

Next we consider a fixed v ∈ Hdual(Ω; D) ⊂ H10,ΓD

(∆; D), where

Hdual(Ω; D) :=

v ∈ H10,ΓD

(D) : ∆v ∈ L2(D) and ∂nv∣∣Γ∈ L2(Γ)

.

It will shortly become clear that Hdual(Ω; D) is a suitable space for the dualvariable. The linearization of u 7→ N

((Ω, u); v

)at uh ∈ H1

h(Ω) ⊂ H1(D) isstraightforward as only B depends on u, and moreover, this dependence islinear. Denoting this derivative by 〈∂u(·), δu〉, we have

⟨∂uN

((Ω, uh); v

), δu

⟩=

∫

Ω−δu ∆v , (6.7a)

for all δu ∈ H10,ΓD

(D). This implies that the linearized-adjoint operator corre-

sponds to the minus Laplacian in Ω.The shape-linearization of Ω 7→ N

((Ω, uh); v

)at Ω splits up in three contri-

butions. As the first contribution, we consider the combined B- and H-term:

−B(Ω; uh, v) + H(Ω; v) =∫

Ω−(uh − h) ∆v +

∫

Ω∇h · ∇v .


Observe that (uh − h) and ∇h are in H1(D) and vanishes on Γ on account ofuh ∈ H1

h(Ω) and h = 1 on all admissible free boundaries. Hence, by Prop. 6.1,the shape derivatives of these terms are zero. To obtain the shape derivative ofF(Ω; v), we can invoke Theorem 4.E since f v ∈ W1,1(D). For δθ ∈ Θ this gives

F′(Ω; v)(δθ) =∫

Γf v δθ · n .

For the final contribution, G(Ω; v), we note that g ∈ C1(D) and v ∈Hdual(Ω; D). Furthermore, since Γ is C1,1, we can use Corollary 4.11, yielding

G′(Ω; v)(δθ) =∫

Γ

(g ∂nv + (∂ng + κ g) v

)δθ · n .

We summarize these results in the following proposition:

Proposition 6.2 (FBP Shape Derivative: Smooth Free Boundaries) Let Ω ∈ O

with C1,1 free boundary Γ. For any uh ∈ H1h(Ω) ⊂ H1(D) and v ∈ Hdual(Ω; D),

the shape functional Ω 7→ N((Ω, uh); v

)is shape differentiable at Ω with re-

spect to Θ. Its shape derivative is given by

⟨∂ΩN

((Ω, uh); v

), δθ

⟩= −

∫

Γ

(g ∂nv +

(f + ∂ng + κ g

)v)

δθ · n . (6.7b)

From this proposition it is clear that the shape gradient can be identified withthe boundary function

j′(Γ) = −g ∂nv − ( f + ∂ng + κ g) v .

Essentially this implies that the linearized adjoint operator corresponds to aRobin-type boundary condition on Γ.

6.3.2 Linearization of the goal functional

We consider the goal functional consisting of a linear combination of the aver-age and elevation functional:

Q(Ω, u) = Qave(Ω; u) +Qelev(Ω) =∫

Ωqave u +

∫

Γ0

qelev α(Ω) ,

Again, the linearization with respect to u is straightforward:

⟨∂uQ(Ω, uh), δu

⟩=

∫

Ωqave δu , (6.8a)



(D). The shape-linearization of Ω 7→ Qave(Ω; uh) follows

from Theorem. 4.E as qaveuh ∈ W1,1(D) for qave, u ∈ H1(D). Furthermore,the shape-linearization of Ω 7→ Qelev(Ω; uh) was considered in the previouschapter, where we employed a Gâteaux derivative approach for the elevationfunction θ 7→ α(Tθ(Ω)); see Sec. 5.4.1 and Sec. A.4.2. Combining these results,we have for δθ ∈ Θ that

⟨∂ΩQ(Ω, uh), δθ

⟩=

∫

Γ

(qave + qelev)

δθ · n . (6.8b)

Note that we substituted uh = h = 1 on Γ. Furthermore, since qelev is onlydefined on a horizontal rest position Γ0, it should be interpreted with the aid ofa projection along the xN-axis, that is,

qelev(x1, . . . , xN) = qelev(x1, . . . , xN−1, xΓ0N )

with xΓ0N being the xN-coordinate of Γ0. The result in (6.8b) is valid for any

Ω ∈ O, i.e., a C1,1 free boundary is not required.

6.3.3 Dual problem and goal-error estimate

We are now ready to define the appropriate dual problem based on the lin-earized adjoint operator and goal functional linearization. Let the dual solu-tion z be defined as the solution of the following variational problem:

Find z ∈ H10,ΓD

(Ω) :∫

Ω∇δu · ∇z +

∫

Γ

1g ( f + ∂ng + κ g) z δu

=∫

Ωqave δu −

∫

Γ

1g (qave + qelev) δu ∀δu ∈ H1

0,ΓD(Ω) .

(6.9)

It can easily be shown that z is a weak solution of the following Poisson problemwith a Robin-type boundary condition at the approximate free boundary Γ:

−∆z = qave in Ω , (6.10a)

g ∂nz + ( f + ∂ng + κ g) z = −(qelev + qave) on Γ , (6.10b)

z = 0 on ΓD . (6.10c)

Note that on account of coercivity, a unique solution of (6.9) exists if ( f +∂ng)/g + κ ≥ 0. Under the implied requirements of the data, we obtainfrom (6.10a) that ∆z ∈ L2(Ω), and from (6.10b) that ∂nz ∈ L2(Γ). Hence, we


have a unique dual solution z ∈ Hdual(Ω; D). We remark that similar condi-tions on the data are given in [78, 79] in a completely different setting of theBernoulli free-boundary problem, though.

The main result of this section is that this dual problem provides a solu-tion that is consistent with the linearization. We outline this in the followingtheorem, whose proof is delayed until the end of this section.

Theorem 6.A (Dual Consistency: Smooth Free Boundaries) Given an ap-proximation (Ω, uh) ∈ O × H1

h(Ω) with a C1,1 free boundary Γ, of thesolution (Ω, u) ∈ O × H1

h(Ω) of the free-boundary problem (6.1), the solutionz ∈ H1

0,ΓD(Ω) ⊂ H1(D) of dual problem (6.9) satisfies

N′((Ω, uh); z)(δθ, δu) = Q′(Ω, uh

)(δθ, δu) ,

for all (δθ, δu) ∈ Θ × H10,ΓD

(D).

If we compare the shape-linearized dual problem (6.9) with the dual prob-lem obtained by domain-map linearization in Sec. 5.4.2, we notice that the lat-ter contains a nonlocal residual-type boundary term. However, at the free-boundary problem solution (Ω, u), this residual-type term vanishes, and bothdual problems are equivalent.

From the standard goal-oriented–error-estimation framework it is clear thatdual consistency plays a key role in goal-oriented error estimates; see Sec. 3.3.To present this estimate, let eΩ ∈ Θ denote a nontrivial perturbation-vector fieldsuch that Ω = TeΩ

(Ω). Then ‖eΩ‖ essentially measures the domain differencebetween Ω and Ω. Let eu denote the error in u by subtracting uh from it, therebyviewing u and uh as members of H1(D). Since both u = h and uh = h on ΓD ,we have eu := u − uh ∈ H1

0,ΓD(D). The following proposition shows that, up to

high-order terms, the error in our goal is related to the residual at (Ω, uh):

v 7→ R((Ω, uh); v

):=

∫

Ωf v +

∫

Γg v −

∫

Ω∇uh · ∇v .

Proposition 6.3 (Error Representation: Smooth Free Boundaries) Under theconditions of Theorem 6.A, let z ∈ H1

0,ΓD(Ω) be the solution of dual problem

(6.9). It holds that

EQ := Q(Ω, u) −Q(Ω, uh) = R((Ω, uh); z

)+ r , (6.11)

with remainder r = o(‖eΩ‖Θ, ‖eu‖H1

0,ΓD(D)

).


Proof The proof follows closely the proof of Theorem 3.H in Sec. 3.3. Wefirst derive the following Taylor series expressions. Applying (6.5) to Ω 7→Q(Ω, uh), we have

Q(Ω, uh) = Q(Ω, uh) +⟨∂ΩQ(Ω, uh) , eΩ

⟩+ o

(‖eΩ‖Θ

).

For the linearization of Q with respect to both its arguments, this implies theTaylor series formula

Q(Ω, u) = Q(Ω, uh) +Q′(Ω, uh)(eΩ, eu) + rQ , (6.12)

where rQ = o(‖eΩ‖Θ, ‖eu‖H1

0,ΓD(D)

). We have a similar expression for N:

N((Ω, u); v

)= N

((Ω, uh); v

)+ N′((Ω, uh); v

)(eΩ, eu) + rN , (6.13)

for any v ∈ Hdual(Ω; D). Consider the goal error EQ = Q(Ω, u)−Q(Ω, uh). Us-ing the Taylor-series formula (6.12) and the dual-consistency theorem, Th. 6.A,we obtain

EQ = Q′(Ω, uh)(eΩ, eu) + rQ = N′((Ω, uh); z)(eΩ, eu) + rQ

Since the dual z ∈ Hdual(Ω; D), we can invoke (6.13) to obtain

EQ = N((Ω, u); z

)−N

((Ω, uh); z

)+ rQ − rN .

The first term on the right-hand side vanishes on account of consistency of thesolution (Ω, u) with the very weak form; see (6.4). Furthermore, expandingN

((Ω, uh); z

)in accordance with (6.3), it follows that

EQ =∫

Ωuh ∆z +

∫

Ωf z +

∫

Γg z −

⟨∂nz, h

⟩∂Ω

+ r .

Finally, by applying an integration by parts on the first term and using uh = hon ∂Ω, we obtain the proof. ¤

Proof (of Theorem 6.A) The linear equation in Th. 6.A can be logically sepa-rated into two equations corresponding to δu and δθ:

⟨∂uN

((Ω, uh); z

), δu

⟩=


⟩∀δu ∈ H1

0,ΓD(D) , (6.14a)

⟨∂ΩN

((Ω, uh); z

), δθ

⟩=

⟨∂ΩQ(Ω, uh), δθ

⟩∀δθ ∈ Θ . (6.14b)

Firstly, we show that z satisfies (6.14a). The explicit expression of (6.14a) followsfrom (6.7a) and (6.8a) and is given by

∫

Ω−δu ∆z =

∫

Ωqave δu ∀δu ∈ H1

0,ΓD(D) .


Since H10,ΓD

(Ω) is dense in L2(Ω), we essentially need to show that

−∆z = qave a.e. in Ω . (6.15)

This follows from (6.9) by elementary variational arguments: Choosing a δuin (6.9) that vanishes on ∂Ω, i.e., δu ∈ H1

0,∂Ω(Ω), we have

∫

Ω∇δu · ∇z =

∫

Ωqave δu ,

and an integration by parts on the left-hand side followed by a density argu-ment proves (6.15). We next show that z satisfies (6.14b). An explicit expressionof (6.14b) follows from (6.7b) and (6.8b). Hence, we need to show that

−∫

Γ

(g ∂nz +

(f + ∂ng + κ g

)z)

δθ · n =∫

Γ

(qave + qelev)

δθ · n , (6.16)

for all δθ ∈ Θ. For this, we integrate by parts the first integral in (6.9) and usethe fact that z satisfies (6.15) to obtain

+∫

Γ

(∂nz + 1

g ( f + ∂ng + κ g) z)

δu = −∫

Γ

1g (qave + qelev) δu ,


(Ω). For any δθ ∈ Θ, we can form the function g δθ · n whichresides in C0,1(Γ) since g ∈ C1(D) and n ∈ C0,1(Γ, R

N) for a C1,1 free boundaryΓ. Note that we can extend this to a function in H1

0,ΓD(Ω). Hence by setting

δu = −g δθ · n, we obtain (6.16). ¤

6.4 Extension to nonsmooth free boundaries

In numerical computations the approximate free boundary is often piecewisesmooth, i.e., Γ is Lipschitz continuous. In this case, the curvature term κ =divΓ n is singular at singular points of the free boundary and definition (6.9) ofthe dual problem does not apply. In this section, we extend the dual problemto Lipschitz free-boundaries by introducing a generalization of the curvatureterm. Accordingly, we obtain goal-oriented error estimates for any approxima-tion (Ω, uh) ∈ O × H1

h(Ω) of our free-boundary problem. We note that similargeneralizations of curvature terms have been studied in [64, 108, 140, 195].

6.4.1 Shape linearization at nonsmooth free boundaries

The singular contribution associated with κ appears in the shape-derivative ofthe free-boundary problem weak form N. Specifically, it originates from the

6.4 Extension to nonsmooth free boundaries 113

Γ

Figure 6.2: Illustration of a Lipschitz-continuous m-vector field at the free boundary Γ.The part of m that is orthogonal to Γ is equal to the normal n, that is, m · n = 1 a.e. on Γ.

shape linearization of G. Therefore, a natural extension of this term can be ob-tained by extending this linearization to Lipschitz free-boundaries. To derivethis extension, we shall consider particular perturbation-vector fields δθ ∈ Θ.

Recall from the structure theorem, Theorem 4.C, that for sufficiently smoothboundaries, the significant perturbations are nonzero in the normal direction,i.e., δθ · n 6= 0. For Lipschitz domains Ω, a similar role shall be played byperturbations in a smoothed-normal direction m = m(Γ); see Sec. 4.2.3. Thissmoothed normal m is a bounded Lipschitz-continuous vector field which isextendable onto D, i.e., m ∈ C0,1(D, R

N). Furthermore, we normalize m ac-cording to

m · n = 1 a.e. on Γ . (6.17)

An example of m in two dimensions is illustrated in Fig. 6.2. We next definea particular perturbation in the m-direction as δθ = δ m. Here, δ is a scalarLipschitz-continuous function that vanishes on ΓD . The corresponding spacefor perturbations in the m-directions shall be denoted by Θ(m) and is definedas

Θ(m) :=

δθ = δ m, ∀δ ∈ C0,1(D) : δ∣∣Γ∈ C0,1(Γ) and δ = 0 on ΓD

.

Note that these perturbations are admissible in the sense that Θ(m) ⊂ Θ.We now turn to the shape-linearization of Ω 7→ G(Ω; v) for perturbations

δ m. Since the free boundary is nonsmooth, we apply Theorem 4.F and obtain

G′(Ω; v)(δ m) =∫

Γ

(∂n(g v) δ + divΓ(g v δ m)

), (6.18)

where we invoked the normalization (6.17) two times. Comparing this resultwith the shape derivative of G in Sec. 6.3.1, we observe that the curvature con-tribution has been replaced with a tangential divergence term. A suitable spacefor v in (6.18) is provided by the intersection Hdual(Ω; D) ∩ H1

0,ΓD(Ω), where

H10,ΓD

(Ω) :=

v ∈ H10,ΓD

(Ω)∣∣ ∫

x |v|2 dx < ∞ for all singular points x ⊂ Γ 4

.


The space H10,ΓD

(Ω) accounts for the boundedness of the tangential divergenceterm in (6.18). As this is not immediately clear, we show this in Appendix A.3.1.We can now present an extension of Prop. 6.2 which holds for nonsmooth free-boundaries. This result follows easily from the preceding developments.

Proposition 6.4 (FBP Shape Derivative) Let Ω ∈ O. For any uh ∈ H1h(Ω) ⊂

H1(D) and v ∈ Hdual(Ω; D) ∩ H10,ΓD

(Ω), the shape functional Ω 7→N

((Ω, uh); v

)is shape differentiable at Ω with respect to Θ(m). Its shape

derivative is given by

⟨∂ΩN

((Ω, uh); v

), δ m

⟩= −

∫

Γ

((g ∂nv + ( f + ∂ng) v

)δ + divΓ(g v δ m)

).

(6.19)

As a side remark, we note that for smooth (C1,1) free boundaries, the resultsreduce to those of Sec. 6.3.1. In fact, in the smooth case, m = n is Lipschitzcontinuous, and we have H1

0,ΓD(Ω) = H1

0,ΓD(Ω). Furthermore, (6.19) reduces

to (6.7b) since

divΓ(g v δ n) = g v δ divΓ n + ∇Γ(g v δ ) · n = κ g v δ ,

and δ = δθ · n.


Based on the extension of the linearization to Lipschitz domains Ω, we canintroduce an analogous extension of the dual problem in (6.9). The extendeddual problem relies on the smoothed-normal field m introduced previously. Letz be the solution of the following variational problem:

Find z ∈ H10,ΓD

(Ω) :∫

Ω∇δu · ∇z +

∫

Γ

(1g ( f + ∂ng) z δu + divΓ(z δu m)

)

=∫

Ωqave δu −

∫

Γ

1g (qave + qelev) δu ∀δu ∈ H1

0,ΓD(Ω) .

(6.20)

The existence of unique solutions to (6.20) can be established based on a co-ercivity estimate under similar assumptions on the data as in Sec. 6.3.3. Asthe derivation of this estimate is rather involved, we have deferred it to Ap-pendix A.3.2.

4In two dimensions x consists of zero-dimensional singular points xi ∈ Γ, and∫

x |v|2 dx =

∑i |v(xi)|2. In N dimensions x is the (N − 2)-dimensional subset of Γ where κ is singular.

6.4 Extension to nonsmooth free boundaries 115

Analogous to the smooth case in Theorem 6.A, the dual problem in (6.20)provides a solution that is consistent with the linearization of N and Q. Theallowed domain perturbations in the linearization are, however, in the m-direction only. We summarize this dual consistency in the following theorem,whose proof is delayed to the end of this section:

Theorem 6.B (Dual Consistency) Given an approximation (Ω, uh) ∈ O ×H1

h(Ω) of the solution (Ω, u) ∈ O× H1h(Ω) of the free-boundary problem (6.1).

The solution z ∈ H10,ΓD

(Ω) ⊂ H1(D) of dual problem (6.20) satisfies

N′((Ω, uh); z)(δθ, δu) = Q′(Ω, uh

)(δθ, δu) ,

for all (δθ, δu) ∈ Θ(m) × H10,ΓD

(D).

We immediately obtain a goal-oriented error estimate by the same argu-ments as in Sec. 6.3.3. As the allowed domain perturbations are in the m-direction only, we first need to introduce a domain difference between Ω and Ω

along m. For this, let eΩ(m) ∈ Θ(m) such that Ω = TeΩ(m)(Ω).

Proposition 6.5 (Error Representation) Under the conditions of Theorem 6.B,let z ∈ H1

0,ΓD(Ω) be the solution of dual problem (6.9). It holds that

EQ := Q(Ω, u) −Q(Ω, uh) = R((Ω, uh); z

)+ r , (6.21)

with remainder r = o(‖eΩ(m)‖Θ, ‖eu‖H1

0,ΓD(D)

).

Proof The proof of this proposition follows by the same arguments as in theproof of Prop. 6.3. ¤

The dual problem in (6.20) is an extension of the dual problem in (6.9) in thesense that for smooth free boundaries (6.20) reduces to (6.9). This can be verifiedby recalling that in this case m = n and H1

0,ΓD(Ω) = H1

0,ΓD(Ω). Furthermore,

we have

divΓ(z δu m) = z δu divΓ n + ∇Γ(z δu) · n = κ z δu .

Proof (of Theorem 6.B) As in the proof of Theorem 6.A, we consider the u- andΩ-linearized equations separately. The satisfaction of

⟨∂uN

((Ω, uh); z

), δu

⟩=


⟩∀δu ∈ H1

0,ΓD(D) ,

follows from the same variational arguments as in the proof of Theorem 6.A. Asa result we obtain −∆z = qave in Ω, and thus z ∈ Hdual(Ω; D) ∩ H1

0,ΓD(Ω). We


are left with showing that z satisfies the Ω-linearized equation defined by (6.19)and (6.8b):

−∫

Γ

((g ∂nz + ( f + ∂ng)

)z δ + divΓ(g z δ m)

)=

∫

Γ

(qave + qelev)

δ , (6.22)

for all δ ∈ C0,1(D) with δ = 0 on ΓD . To show this, we integrate by parts thefirst integral in (6.20) to obtain

⟨∂nz, δu

⟩Γ+

∫

Γ


)= −

∫

Γ

1g (qave + qelev) δu ,


(Ω). Choosing δu = −g δ ∈ H10,ΓD

(Ω) we obtain (6.22). ¤


To enable a comparison between the shape-linearization approach and thedomain-map linearization of the previous chapter, we consider the same nu-merical experiments as before. We demonstrate that the shape-linearizationapproach provides an elegant alternative to the domain-map linearization ap-proach. First, we consider in Sec. 6.5.1 the Bernoulli free-boundary problemin one-dimension. The shape linearization of this one-dimensional problem isessentially equivalent to the so-called total linearization method used in [51]to obtain a Newton-type solution algorithm. In Sec. 6.5.2, we consider theBernoulli free-boundary problem in two dimensions. We demonstrate the ef-fectivity of the dual-based error estimate on uniformly refined meshes, andpresent an example of goal-oriented adaptive mesh refinement.

6.5.1 One-dimensional application

In the one-dimensional setting, the variable domain is the interval Ω = (0, ϑ) ⊂R. The Dirichlet boundary and free boundary are the points ΓD = 0 andΓ = ϑ, respectively. The approximate domain is given by Ω = (0, ϑh). It canbe verified that the dual problem (6.9) reduces in the one-dimensional settingto: Find z ∈ H1

0,ΓD(Ω) :

∫

Ωδux zx dx +

(1g ( f + gx) z δu

)(ϑh) =

∫

Ωqave δu dx −

(1g (qave + qelev) δu

)(ϑh) ,


(Ω), where (·)x = d(·)/dx and qelev ∈ R.5

5Let us note that in the one-dimensional setting, the use of shape calculus is, of course, notnecessary as one can use the Leibniz integral rule to differentiate under the integral sign.


−1

0

1

ϑ = 2ϑh

uuh

−1

0

1zave

zelev

Figure 6.3: Exact solution (ϑ, u) and approximation (ϑh, uh) [left]. Dual solutions zave

and zelev corresponding to goal functional Qave and Qelev, respectively [right].

Typical error estimate We consider the data and goal functionals, and the cor-responding exact solution as indicated in Table 5.1. To show some typical errorestimation results, consider the following approximation and correspondinggoal values:

(ϑh, uh(x)

)=

( 32 , 2

3 x)

, Qave(ϑh; uh)

= 34 , Qelev(

ϑh)

= 32 .

Figure 6.3 [left] shows a graphical illustration of the exact and approximatesolutions. Furthermore, Figure 6.3 [right] shows the dual solutions for Qave

and Qelev:

zave(x) = 14 x − 1

2 x2 , zelev(x) = − 45 x .

The corresponding dual-based error estimate, EstQ := R((Ω, uh); z

)=∫

Ωf z dx + (g z)(ϑh) −

∫Ω

uhx zx dx , and the true goal-error, EQ, are as follows:

EstQave = 1764 , EstQelev = 13

20 ,

EQave = − 112 , EQelev = 1

2 .

The difference in the error estimate and the true error is caused by linearization.Let us note that both the dual solutions and error estimates are slightly differentfrom the results obtained by means of the domain-map linearization approach.

Convergence of error estimates In the following example, the data is againspecified as in Table 5.1. We investigate the convergence of the dual-based errorestimate for the following ∆ϑ-family of approximate solutions: 6

ϑh = ϑ + ∆ϑ , (6.23a)

uh(x) =

u(x) x ∈ [0, ϑ] ,

u(ϑ) x ∈ (ϑ, ϑ + ∆ϑ] .(6.23b)

6To arrive at a convenient expression for the norm of the error, the family of approximationsin (6.23) is slightly different from that in Sec. 5.5.1.


0 1 ∆ϑ−2

0

2EQave

EstQave

100

10-2

10-4

10-3 10-2 10-1∥∥(eϑ, eu)

∥∥

|EQ

ave−

EstQ

ave|

slope 2

Figure 6.4: True goal error EQave and dual-based error estimate EstQave for the ∆ϑ-familyof approximations (ϑh, uh) given in (6.23) [left]. Convergence of the error in the errorestimate with respect to the norm

∥∥(eϑ, eu)∥∥ [right].

This family converges to the exact solution as ∆ϑ ց 0. Note that uh is simplya constant extension of u on the approximate domain. Hence, if u is conceivedof as a member of H1

0,ΓD(Ω) by constant extension outside [0, ϑ], then eu =

u − uh = 0. From the perspective of the error representation (see Prop. 6.3),only the error eΩ = ∆ϑ is then relevant.

Fig. 6.4 [left] plots the actual error EQave and the dual-based estimate EstQave

versus ∆ϑ for the goal functional Qave. Furthermore, Fig. 6.4 [right] plots theerror in the estimate |EQave − EstQave | versus the norm of the error:

∥∥(eϑ, eu)∥∥2

= |ϑ − ϑh|2 + ‖ux − uhx‖

2L2(Ω) = |∆ϑ|2 .

Both figures illustrate that the convergence of the estimate is indeed second-order, in accordance with the theory.

6.5.2 Two-dimensional application

We now consider the two-dimensional case. We denote coordinates by (x, y) ∈R

2. We compute approximations of (6.2) by means of Galerkin’s method.Hence, the approximate domain Ω ∈ O and corresponding approximate so-lution uh ∈ Vh

h (Ω) satisfy∫

Ω∇uh · ∇v =

∫

Ωf v +

∫

Γg v ,

for all v ∈ Vh0,ΓD

(Ω), where Vhh (Ω) ⊂ H1

h(Ω) and Vh0,ΓD

(Ω) ⊂ H10,ΓD

(Ω) denotestandard finite element spaces consisting of piecewise-linear functions on trian-gles. Accordingly, the approximate free-boundary is a piecewise-linear curve


composed of the edges of the adjacent elements. The nonlinear problem issolved using a fixed point iteration similar to the explicit Neumann schemein [100], where we allow the vertices of the free-boundary to move only verti-cally.

Since our approximate domains have piecewise-linear free-boundaries, wehave to use the dual problem (6.20). This dual problem is discretized on thesame triangular mesh as the primal problem but with piecewise-quadratic func-tions (that vanish on ΓD ). The tangential divergence term in (6.20) is imple-mented by means of identity (A.15); see Appendix A.3 for more details.

Effectivity for the parabolic-free-boundary testcase First, we reconsider theparabolic-free-boundary testcase introduced in Sec. 5.5.2; see Fig. 5.5. The data f , g, h of the free-boundary problem is manufactured to yield the exact do-main Ω = (0, 2) × (0, 1 + αΩ) with

αΩ(x) = 12 x (2 − x) ,

and the corresponding solution

u(x, y) = y1+αΩ(x)

+ αΩ(x) y1+αΩ(x)

(1 − y

1+αΩ(x)

).

Our interest pertains to the average goal with qave = 1, which yieldsQave(Ω; u) = 67/45 = 1.4888 . . . at the solution. In Fig. 6.5, we depictthe approximate dual solution z for the coarsest mesh and a very fine mesh.The convergence of the corresponding dual-based error estimates EstQave =R

((Ω, uh); z

)on uniformly refined meshes is reported in Table 6.1. The effec-

tivity index EstQave /EQave approaches 1 demonstrating the consistency of theerror estimate. The small deviation from 1 is conjecturally caused by weak sin-gularities in the dual solution at the kinks in the approximate free boundary. Toenable a comparison, Table 6.1 also presents the results obtained in Sec. 5.5.2 forthe domain-map linearization approach. On coarse meshes, shape-linearizationyields a more accurate estimate than domain-map linearization. However, theresults of both approaches are essentially identical.

Goal-oriented adaptivity for free-surface flow over a bump To investigatethe applicability of the error estimate to drive adaptive mesh refinement, weregard the problem of free-surface flow over a bump with domain Ω = (0, 4)×(yb, 1 + αΩ); see Fig. 5.7 [top]. The function yb describes the bottom and hastriangular bump at 1 < x < 2. For the data, we take f = 0 and g = 1. Moreover,h is 0 at the bottom and increases linearly to 1 along the lateral boundaries ofthe domain. Our interest is the elevation of the free-boundary at x0 = 2 +

√2.

Fig. 6.6 displays the corresponding coarsest-mesh dual solution. We construct


-2

-1.5

-1

-0.5

0

Figure 6.5: Parabolic-free-boundary test problem. The approximate dual solution (con-tour plot) associated with a very fine mesh [left] and the coarsest mesh [right].

element refinement indicators, apply a Dörfler-type marking and refine usingnewest-vertex bisection as in the previous chapter.

In Fig. 6.7, we plot the error estimate and the “true” error versus the totalnumber of degrees of freedom, which is denoted by n. The drop in the trueerror for the adaptive case for n > 1,000 is caused by the non-negligible errorin the reference value. The adaptive refinement yields an optimal convergence

Table 6.1: Convergence of the dual-based error estimate EstQave under uniform meshrefinement for the shape-linearization aproach and the domain-map linearization ap-proach of Chapter 5.

Shape Domain-mapElements Dofs Qave EQave EstQave Effect. EstQave Effect.

8 8 1.1573 0.33163 0.32160 0.970 0.22131 0.66716 15 1.3145 0.17440 0.16101 0.923 0.13852 0.79432 23 1.3694 0.11947 0.12285 1.028 0.09994 0.83664 45 1.4284 0.06045 0.06044 1.000 0.05499 0.910

128 77 1.4555 0.03339 0.03584 1.073 0.03055 0.915256 153 1.4715 0.01740 0.01810 1.040 0.01676 0.963512 281 1.4803 0.00860 0.00933 1.085 0.00808 0.940

1,024 561 1.4843 0.00458 0.00482 1.054 0.00450 0.9842,048 1,073 1.4867 0.00217 0.00235 1.083 0.00205 0.9474,096 2,145 1.4877 0.00117 0.00123 1.057 0.00115 0.9918,192 4,193 1.4883 0.00054 0.00059 1.081 0.00051 0.949

16,384 8,385 1.4886 0.00029 0.00031 1.057 0.00029 0.993

∞ ∞ 1.4888 0


-0.6

-0.4

-0.2

0

Figure 6.6: Free-surface flow over a bump. The dual solution corresponding to thecoarsest mesh. We have indicated the free-boundary elevation point of interest (atx0 = 2 +

√2).

rate of O(n-1) while a suboptimal convergence rate of O(n-3/4) is obtained foruniform refinement. Figure 6.7 also shows that the convergence behavior ofadaptive refinement with shape linearization and with domain-map lineariza-tion is similar.

Figure 6.8 [left] presents several adaptively refined meshes obtained byshape-linearization. The different refinements at the three bump corners as wellas the local refinement near the elevation point of interest are noteworthy. Forcomparison, Figure 6.8 [right] recalls from Sec. 5.5.2 the sequence of adaptivelyrefined meshes obtained with domain-map linearization. The meshes in Fig-ure 6.8 [left] have been selected in such a manner that they have similar num-bers of elements. Note, however, that owing to our marking strategy the cor-responding iteration number can be distinct. It is to be noted that the domain-map linearization approach yields significantly more refinement near the freeboundary. This is in line with the analysis in Sec. 6.3.3, which conveys that thedifference between the two approaches consists of a residual-type boundaryterm. However, although the refinements are different, both approaches givesimilar and optimal convergence behavior; see Fig. 6.7.


On the basis of the Bernoulli free-boundary problem, we presented a shape lin-earization approach to goal-oriented error estimation for free-boundary prob-lems. We showed that the associated very weak form and goal-functional ofinterest can be formulated as a function of the unknown domain. The domaindependence was linearized using techniques from shape calculus. We extractedfrom the linear (adjoint) equation an appropriate dual problem by means of aconsistent reformulation. This dual problem corresponds to a Poisson problem


10-2

10-3

10-4

10-5101 102 103 104

n

slope -3/4

slope -1

|E| (uniform)|Est| (uniform)|E| (adapt S)|Est| (adapt S)|E| (adapt DM)|Est| (adapt DM)

Figure 6.7: Convergence of the “true” error E = EQelev and error estimate Est = EstQelev

under uniform mesh refinement, adaptive mesh refinement using shape linearization(adapt S) and adaptive mesh refinement using domain-map linearization (adapt DM)versus the total number of degrees of freedom n.

Figure 6.8: Adaptively refined meshes, controlling the error in the free-boundary eleva-tion at x0 = 2 +

√2, obtained with shape-linearization [left] after 10, 18 and 26 iterations

(120, 848 and 5,408 elements, respectively) and with domain-map linearization [right]after 10, 18 and 29 iterations (120, 793 and 5,447 elements, respectively).

with a Robin-type boundary condition involving the curvature. Moreover, wederived a generalization of the dual problem for nonsmooth free boundarieswhich includes a natural extension of the curvature term. To demonstrate theeffectivity of the dual-based error estimate and its usefulness in goal-orientedadaptive mesh-refinement, we presented numerical experiments in one and


two dimensions.The shape-linearization approach provides an attractive alternative to the

domain-map linearization approach in Chapter 5, as it avoids the nonstandardand nonlocal interior and boundary terms of the latter. We showed that theessential difference between the two approaches is that the dual problem inChapter 5 contains a nonlocal residual-type boundary term. At the solutionof the free-boundary problem, this residual-type term vanishes and both dualproblems are equivalent. A comparison of the numerical results obtained byshape linearization with the results obtained by domain-map linearization re-vealed no essential differences.

Various extensions of our model problem can be envisaged. For example,we considered constant Dirichlet data at the free boundary, and conforminguh in the sense that uh = h = 1 on Γ. This is convenient as the shape lin-earization of the associated combined term −B(Ω; uh, v) + H(Ω; v) vanishes;see Sec. 6.3.1. However, nonconstant h|Γ and uh 6= h on Γ can be included ifthe associated terms are shape linearized accordingly. As a result, additionalterms involving the Laplace-Beltrami operator appear in the dual Γ-boundarycondition. However, this requires additional regularity on the dual solution.

Instead of deriving linearized adjoints, the present shape-linearization ap-proach can also be used to obtain Newton-based iteration algorithms for free-boundary problems, cf. [140–142]. The use of shape calculus appears to providea convenient rigorous setting compared to formal asymptotic developments asin [29, 100]. The proposed extension to nonsmooth free boundaries in Sec. 6.4,however, has to our knowledge not been considered before in this context. Thisextensions provides a natural generalization of curvatures to nonsmooth freeboundaries, obviating heuristic curvature reconstruction.

Part III

Steady Incompressible


Chapter 7

Domain-Map Linearization Approach to


A mathematician who wants to occupy himself with physics is confrontedwith the following difficulty.(a) In the physical literature the mathematical apparatus is not always cor-rectly handled. [. . . ] This has the consequence that many mathematiciansrestrict themselves to purely mathematical considerations, or if they dotreat physical subjects, then physicists have difficulty in recognizing thefamiliar physical material.(b) On the other hand, creative physicists have a number of objections tomathematics. “The rigor of mathematics is a luxury. We need concrete an-swers to our problems. The general results of mathematicians are oftenuseless for our purposes, or the formulation is so abstract that we cannotrecognize whether a mathematical result is useful to us or not. Thus wehave to develop our own methods and cannot wait until they are mathe-matically justified. By experience, this takes much too long.”For a student who wants to learn both mathematics and physics, aschizophrenic situation may occur, and he might be caught on the hornsof a dilemma.

Eberhard Zeidler, 1987Nonlinear Functional Analyis and its Applications, IV:

Applications to Mathematical Physics1

In this chapter and the following chapter, we consider goal-oriented error esti-mation and adaptivity for fluid–structure interaction. This chapter considers, inparticular, the domain-map linearization approach to derive the dual problem.2

7.1 Introduction

A recent development in goal-oriented error estimation and adaptive methodsconcerns their extension to muliphysics problems, involving multiple coupled

1Springer-Verlag, 1988 [237, p. viii].2Some of the results in this chapter have appeared in [225].

128 7. Domain-Map Linearization Approach to Fluid–Structure Interaction

problems. In the context of coupled boundary value problems, we mentionthe works by Larson et al. [25, 146–148] and by Estep et al. [36, 82, 83, 85].For the coupling of particle and continuum models, we mention the works byPrudhomme and Bauman et al. [15, 188]. Some authors have also consideredthe extension to fluid–structure interaction. This has shown to be quite difficultowing to their free-boundary character: The derivation of a suitable dual (lin-earized adjoint) problem is highly nontrivial. Indeed, as mentioned in Chap-ter 1, the approaches by Dunne et al. [74–76] and Grätsch and Bathe [119] all by-pass this tedious linearization procedure by ignoring the domain dependenceand resorting to approximations of the linearized operators.

In this chapter, we do not bypass the derivation of the linearized opera-tors, but rigorously derive them using the so-called domain-map linearizationapproach. This approach was briefly outlined in Sec. 3.4.1 and applied inChapter 5 in the context of the Bernoulli free-boundary problem. To illustratethe approach, we consider the fluid–structure-interaction model introduced inSec. 2.2. By means of a suitable domain map (based on perturbations of theidentity, see Sec 4.2.2, but in discrete settings known as the ALE (arbitraryLangrangian–Eulerian) map), the fluid–structure problem can be transformedinto an equivalent problem on a fixed domain. The variational formulation ofthe transformed problem is then in canonical form, although it contains intri-cate terms involving the domain map. One can then linearize the transformedproblem with respect to the domain map.

We show that the domain-map linearized dual problem corresponds to alinearized fluid subproblem coupled to a dual structure displacement subprob-lem. These subproblems have a kinematically straightforward coupling, how-ever, the complementary coupling condition depends in a nonstandard andnonlocal manner on the dual fluid variables. This is similar to the domain-maplinearized dual problem of the Bernoulli free-boundary problem in Chapter 5.

We note that domain-map linearization approaches have also recently beenused to derive Newton-type iterative solution algorithms; see the works of Fer-nández et al. [94, 98] and Bazilevs et al. [17, 19] and also [158].

The contents of this chapter are arranged as follows. Sec. 7.2 recalls thefluid–structure-interaction model from Sec. 2.2. In Sec. 7.3 we consider thedomain-map linearization approach, and apply the canonical framework ofgoal-oriented error estimation oulined in Sec. 3.3. Numerical experiments arepresented in Sec. 7.4. Finally, Sec. 7.5 contains concluding remarks.


In this section, we recall the two-dimensional fluid–structure interaction modelintroduced in Sec. 2.2, viz. an incompressible fluid coupled to a (vector) string


model. To deal with deformable fluid domains, we consider perturbations ofthe identity map, which in discrete settings are commonly known as ALE maps;see [19, 70]. We also consider goal quantities and their approximation based onconforming discretizations.

7.2.1 Fluid–structure-interaction model

We consider a fluid–structure system such as graphically depicted in Fig. 2.2.To introduce the model problem, consider an admissible string displacementθ : Γ0 → R

2 that is sufficiently smooth, i.e., θ ∈ W1,∞(Γ0) ∩ H10(Γ0). Since

W1,∞(Γ0) coincides with C0,1(Γ0) for convex Γ0, there exists a Lipschitz exten-sion of θ onto Ω0 which is zero on the other boundaries of Ω0. We shall denotethis extension also by θ and note that it is a member of the space

Θ :=

θ = ExtΓ0 η ∈ C0,1(Ω0; R2) , η ∈ W1,∞(Γ0) : θ = 0 on ∂Ω0 \ Γ0

,

with ExtΓ0 a given (sufficiently smooth) extension operator. Accordingly, weintroduce the perturbation of the identity map (see Sec. 4.2.2)

Tθ = Id + θ : Ω0 → Tθ(Ω0) := Ωθ ,

which associates to each structural displacement, a fluid domain Ωθ withboundary ∂Ωθ consisting of the interface Γθ := Tθ(Γ0) and fixed boundaries Γfx.

Within the domain Ωθ , we consider the following fluid problem for the ve-locity u : Ωθ → R

2 and pressure p : Ωθ → R:

−ν ∆u + u · ∇u + ∇p = f

−div u = 0

in Ωθ ,

u = 0 on Γθ ,

(7.1a)

(7.1b)

where ν is the fluid kinematic viscosity and f ∈ L2(R2) is a given body force.

The fixed boundaries consist of two parts, ΓD and ΓN , on which Dirichlet andNeumann boundary conditions are imposed. For the sake of clarity, we assumethat these conditions are homogeneous.

The string displacement θ satisfies the following (vector) string equation:

−E ∂2s θ = g +

((−ν ∂nu + p n) Tθ

)ωθ in Γ0 , (7.1c)

where the constant E is the string’s Young modulus, ∂s(·) := ∂(·)/∂s is thederivative along Γ0, and g ∈ L2(Γ0)

2. The tangential Jacobian is given byωθ =

∣∣∂sTθ |Γ0

∣∣ = Jθ |DT -Tθ n|, with Jθ = det DTθ the Jacobian determinant;

cf. Sec. 4.3.4.


7.2.2 Weak formulation

We recall the bounded weak formulation that was given in Sec. 2.2.4. To sup-port structure displacements, we introduce

H10(Γ0) :=

η ∈ H1(Γ0) : η = 0 on ∂Γ0

.

The velocity space enforcing homogeneous Dirichlet bounday conditions is de-fined as

H10(Ωθ) :=

v ∈ H1(Ωθ) : v = 0 on Γθ ∪ ΓD

.

The weak formulation of the coupled fluid–structure problem is then given by:


0(Ωθ) × L2(Ωθ) :

S(θ, η) + N(Ωθ , (u, p); (v + ExtΓθ

η, q))

= G(η)

∀(η, v, q) ∈ H10(Γ0) × H1

0(Ωθ) × L2(Ωθ) .

(7.2)

where the fluid semilinear form N consists of the momentum and continuityresidual:

N(Ω, (u, p); (v, q)

):= −Rm(

Ω, (u, p); v)−Rc(Ω, u; q

).

Rm(Ω, (u, p); v

):=

∫

Ω

(f · v − ν ∇u · ∇v − u · ∇u · v + p div v

),

Rc(Ω, u; q)

:=∫

Ωq div u ,

By S and G, we denote the bilinear form associated with the structure operatorand the linear functional associated with external structure forces:

S(θ, η

):=

∫

Γ0

E ∂sθ · ∂sη ,

G(η) :=∫

Γ0

g · η .

Note that in (7.2), the (v, q)-equations correspond to the fluid problem in thevariable domain Ωθ . The η-equation in (7.2) corresponds to the string equa-tion (7.1c) with a weakly-enforced fluid traction. Weak enforcement is accom-plished by means of the extension map ExtΓθ

defined for nonzero θ as the com-position of extension and transformation:

η 7→ ExtΓθη := (ExtΓ0 η) T -1

θ : H10(Γ0) → H1

0,Γfx(Ω0) → H1

0,Γfx(Ωθ) , (7.3)

where H10,Γfx

denotes the space of H1-functions with zero trace on Γfx.


Ω0 Ωθh

Γ0 Γθh

Tθh

Figure 7.1: Conforming finite-element-type discretization

We shall call a displacement θ admissible if it is a member of W1,∞(Γ0) ∩H1

0(Γ0) and if it leads to a Lipschitz fluid domain Ωθ . In the sequel, we as-sume the existence of a (possibly non-unique) admissible displacement θ and acorresponding velocity and pressure (u, p) ∈ H1

0(Ωθ)× L2(Ωθ) that solve (7.2).

7.2.3 Goal functionals and discretization errors

Our interest will be specific (bounded and continuous differentiable) goal func-tionals Q : (θ, u, p) ∈ H1

0(Γ0) × H10(Ωθ) × L2(Ωθ) → R of the solution. For

simplicity, we will restrict the ensuing analysis to functionals

Q(θ, u, p) =∫

Γ0

qdisp · θ +∫

Ωθ

(qvel · u + qpres p

). (7.4)

where qdisp ∈ L2(Γ0) and qvel, qpres ∈ H1(R2). We note that other bounded

functionals are also possible.Numerical approximations of the fluid–structure interaction problem can

be obtained by employing a suitable discretization method. We shall focus onconforming finite-element discretizations that directly employ the weak for-mulation using suitable discrete spaces. Let us assume that the approximatestructure displacement θh is in some approximation space Vh

s ⊂ H10(Γ0). To

each θh one can associate the approximate fluid domain Ωθh = Tθh(Ω0). Typ-ically, the approximate velocity and pressure (uh, ph) are then members of theθh-dependent space

Vhf (θh) :=

(v, q) = (v0, q0) T -1

θh : (v0, q0) ∈ Vhf

⊂ H1

0(Ωθh) × L2(Ωθh) ,

where Vhf := Vh

f (0) is some suitable approximate velocity and pressure space inthe referential domain; see Fig. 7.1. Such an approximation is called conformingin the sense that the resulting velocity and pressure live on the approximatedomain Ωθh . In general, this is achieved by having matching fluid and structuremeshes at the interface.


Given a conforming approximation (θh, uh, ph), our objective is to providea dual-based estimate of the goal-error Q(θ, u, p) −Q(θh, uh, ph). As discussedin Chapter 3, such an estimate is of key importance in goal-adaptive discretiza-tions.

7.3 Goal-oriented error estimation by domain-map

linearization

To enable the application of the standard goal-oriented error estimation frame-work, we shall cast the weak formulation in (7.2) in canonical form by reformu-lating the fluid part on a fixed domain. We can then linearize the transformedproblem with respect to the domain map yielding the dual problem.

Essentially, it does not matter which fixed domain is chosen (cf. the analysisin in Sec. 5.3.3). However, it is most natural to choose the fixed domain closestto the linearization state, viz., the approximate domain corresponding to θh. Forconvenience, let us denote the approximate domain and interface throughoutthe rest of this chapter by


This section basically follows the procedure in Chapter 5 concerning theBernoulli free-boundary problem.

7.3.1 Reformulation to the approximate domain

We transform the fluid problem to the approximate domain by means of themap Tθ : Ω → Ωθ which is defined using the map Tθ as

Tθ := Tθ T -1θh = Id + (θ − θh) T -1

θh ∀θ ∈ Θ ; (7.5)

see also Fig. 5.2. Recall from Sec. 5.2.2 that for admissible θ, Tθ constitutes aC0,1-diffeomorphism. For such diffeomorphisms, it holds that

H10(Ωθ) =

v = v T -1

θ : v ∈ H10(Ω)

,

L2(Ωθ) =

q = q T -1θ : q ∈ L2(Ω)

;

see [120, p. 21] or [59, p. 406].Let us now introduce the transformed semilinear form N : Θ × (H1(Ω) ×

L2(Ω))2 → R defined as

N(θ, (w, r); (v, q)

):= N

(Ωθ , (w, r) T -1

θ ; (v, q) T -1θ

)(7.6)

∀(v, q), (w, r) ∈ H1(Ω) × L2(Ω) .


We can then formulate the fluid–structure interaction problem equivalently onfixed domains as follows.

Proposition 7.1 Let θ denote an admissible structure solution of (7.2) and let(u, p) := (u, p) Tθ ∈ H1

0(Ω) × L2(Ω) denote the corresponding fluid solutionof (7.2) transformed to Ω. Then it holds that

S(θ, η) + N(θ, (u, p); (v + ExtΓ η, q)

)= G(η)

∀(η, v, q) ∈ H10(Γ0) × H1

0(Ω) × L2(Ω) ,(7.7)

where N is defined in (7.6). Explicitly,3

N(θ, (u, p); (v, q)

)= −Rm(

θ, (u, p); v)− Rc(θ; u; q

), (7.8a)

Rm(θ, (u, p); v

)=

∫

Ω

(fθ · v − ν (Aθ ∇u) · ∇v ,

− u · (Bθ ∇)u · v + p (Bθ ∇) · v)

(7.8b)

Rc(θ; u; q)

=∫

Ωq (Bθ ∇) · u , (7.8c)

with


θ , Bθ := Jθ DT -Tθ , fθ := Jθ( f Tθ) , Jθ := det DTθ .

Proof To show that (7.7) holds, recall that ExtΓθη = (ExtΓ0 η) T -1

θ ; see (7.3).Hence this extension transforms to Ω as

(ExtΓθη) Tθ = (ExtΓ0 η) T -1

θ Tθ = (ExtΓ0 η) T -1θh = ExtΓ η ,

where we used (7.5) in the second step. This result combined with (7.6)establishes (7.7). The specification of N given by (7.8b) and (7.8c) can beproven by transforming the integrals to Ω. These transformations are essen-tially (4.19) and (4.21) in Sec 4.3. Consider any v ∈ H1

0,ΓD(Ωθ). To transform∫

Ωθν ∇u · ∇v =

∫Ωθ

ν ∇ui · ∇vi, we use (4.19) and the identity

(∇wi) Tθ = DT -Tθ ∇(wi Tθ) ∀wi ∈ H1(Ωθ) ,

to obtain∫

Ωθ

ν ∇u · ∇v =∫

Ω

(DT -T

θ ∇(ui Tθ))·(

DT -Tθ ∇(vi Tθ)

)Jθ

=∫

Ω

(Aθ ∇(ui Tθ)

)· ∇(vi Tθ) .

3Expression involving matrices are meant to be evaluated as in ordinary matrix-matrix ormatrix-vector products. For example Aθ∇u = (Aθ)i,k (∇u)k,j = (Aθ)i,k ∂xk

uj. Hence, (Aθ∇u) ·∇v = (Aθ)i,k ∂xk

uj ∂xivj = (Aθ ∇uj) · ∇vj. Furthermore, Bθ∇ should be interpreted as the vector-

valued operator (Bθ)i,k ∂xk.


Replacing u Tθ with u and setting v Tθ =: v ∈ H10(Ω), we obtain the Aθ-term

in (7.8b). The other terms in (7.8b) and (7.8c) follow from (4.19) and (4.21) in asimilar manner. ¤

The goal functional can be transformed to Ω in a similar manner as N, i.e.,

Q(θ, u, p) = Q(θ, u T -1θ , p T -1

θ ) =∫

Γ0

qdisp · θ +∫

Ω

(qvel

θ · u + qpresθ p

). (7.9)

with

qvelθ := Jθ (qvel Tθ) and q

presθ := Jθ (qpres Tθ) .

7.3.2 Dual problem and error representation

Having cast our problem in canonical form, we can derive the dual problem bylinearization. Accordingly, we shall linearize the left-hand side of (7.7) and Qwith respect to (θ, u, p) at (θh, uh, ph). Let us denote the dual velocity, pressureand displacement by z, s and ζ, respectively.

First, we consider the linearization with respect to (u, p). It is clear thatthis linearization results in a dual fluid problem. The appearance of the sumv + ExtΓ η as the test function in (7.7) motivates the replacement of this entiresum by the dual velocity z. Hence, we view z at Γ as being equal to the dualdisplacement ζ T -1

θh , i.e., z is a member of

H1ζ,Γ(Ω) :=

v ∈ H1(Ω) : v|Γ = ζ T -1

θh

.

Straigthforward linearization of N with respect to (u, p) gives

∂(u,p)N(θh, (uh, ph); (z, s)

)(δu, δp)

=∫

Ω

(ν ∇δu · ∇z +

(uh · ∇δu + δu · ∇uh

)· z − δp div z

)−

∫

Ωs div δu .

The same linearization of Q gives

∂(u,p)Q(θh, uh, ph

)(δu, δp) =

∫

Ω

(qvel · δu + qpres δp

).

Next, let us consider the linearization of N and Q with respect to θ. Basically,this amounts to linearizing the θ-dependent terms in (7.8b), (7.8c) and (7.9). Wecollect the derivatives of these terms in the following proposition.

Proposition 7.2 Denoting δθ := δθ T -1θh , it holds that

⟨∂θ Aθh , δθ

⟩= (div δθ) I − Dδθ − DδθT ,

⟨∂θq

presθh , δθ

⟩= div(qpres δθ) ,

⟨∂θ Bθh , δθ

⟩= (div δθ) I − DδθT ,

⟨∂θ(qvel

θh )i, δθ⟩

= div(qveli δθ) ,

⟨∂θ fθh , δθ

⟩= div( f δθ) ,


with I the identity matrix.

Proof The terms are similar to those in Sec. 5.4.1. Hence, the proof follows fromLemmata 5.5 and 5.6. ¤

The linearizations lead to the following dual problem.

Definition 7.3 (Domain-Map Linearized Dual FSI Problem) The domain-map linearized dual problem at the approximation domain Ω is definedas:

Find ζ ∈ H10(Γ0) and (z, s) ∈ H1

ζ,Γ(Ω) × L2(Ω) :

S(δθ, ζ) + N′(θh, (uh, ph); (z, s))(δθ, δu, δp) = Q′(θh, uh, ph)(δθ, δu, δp)

∀(δθ, δu, δp) ∈ H10(Γ0) × H1

0(Ω) × L2(Ω) ,

(7.10)

with the δu-, δp- and δθ-equation specified by∫

Ω

(ν ∇δu · ∇z +

(uh · ∇δu + δu · ∇uh

)· z − s div δu

)=

∫

Ωqvel · δu , (7.11a)

−∫

Ωδp div z =

∫

Ωqpres δp , (7.11b)

∫

Γ0

E ∂sδθ · ∂sζ + ∂θN(θh, (uh, ph); (z, s)

)(δθ) = ∂θQ(θh, uh)(δθ) (7.11c)

The derivatives ∂θN and ∂θQ in (7.11c) follow from their definitions in (7.8) and(7.9), and the derivatives in Prop. 7.2. ¤

It should be noted that the dual fluid problem, (7.11a) and (7.11b), correspondsto a linearized fluid problem for (z, s). Furthermore, note that, in (7.10), the dualvelocity satisfies a nonzero boundary condition at the interface Γ, coupling it tothe dual structure displacement ζ. The dual structure equation (7.11c) is es-sentially a string equation coupled to the dual fluid via nonlocal terms in ∂θN.Let us remark that this nonlocal dependence is similar to the domain-map lin-earized dual problem in the context of the Bernoulli free-boundary problem;see Sec. 5.3.

In the following theorem, we show that the dual problem is suitable to pro-vide a goal-oriented error estimate.

Theorem 7.A (Domain-Map Linearized Dual-Based Error Representation)

Given any admissible structure approximation θh and corresponding fluiddomain Ω = Ωθh and fluid approximation (uh, ph) ∈ H1

0(Ω) × L2(Ω) of thefluid–structure solution (θ, u, p) of (7.2), assume the dual problem (7.10) has a


solution (ζ, z, s). Then it holds that

Q(θ, u, p) −Q(θh, uh, ph) = R(θh, uh, ph; ζ, z, s

)+ r ,

where the coupled fluid–structure residual is given by

R(θh, uh, ph; ζ, z, s

)= G(ζ) − S(θh, ζ) −N

(Ω, (uh, ph); (z, s)

).

The remainder r = o(‖eθ‖H1

0 (Γ0), ‖eu‖H1

0 (Ω), ‖ep‖L2(Ω)

)and the errors are de-

fined as

eθ := θ − θh , eu := u Tθ − uh , ep := p Tθ − ph .

This error representation formula is a specification of the abstract formula inTheorem 3.H for our fluid–structure-interaction problem. The goal-quantity er-ror can be estimated by evaluating the fluid–structure residual R with a discretedual solution. In specific discretized settings, one can invoke Galerkin orthog-onality to subtract arbitrary discrete test-functions, as usual; see Chapter 3 formore details.

Note that the theorem shows how fluid variables, that live on distinct do-mains, are compared. Indeed, the remainder forms a higher-order term of er-rors, u Tθ − uh and p Tθ − ph, measured in the approximate domain.

Proof The proof is similar to the proof of Theorem 5.A. An essential element ofthe proof are the following Taylor-series-like formulae:

Q(θ, u, p) = Q(θh, uh, ph) + Q′(θh, uh, ph)(eθ , eu, ep) + rQ , (7.12a)

N(Ωθ , (u, p); (v, q) T -1

θ

)= N

(Ω, (uh, ph); (v, q)

)

+ N′(θh, (uh, ph); (v, q))(eθ , eu, ep) + r

N, (7.12b)

for any (v, q) ∈ H1(Ω) × L2(Ω), with remainders rQ, rN

of the same order as r

as provided in the theorem. Let us show that (7.12b) holds. By definition of N

in (5.15), we have the identity

N(Ωθ , (u, p); (v, q) T -1

θ

)−N

(Ω, (uh, ph); (v, q)

)

= N(θ, (u, p) Tθ ; (v, q)

)− N

(θh, (uh, ph); (v, q)

).

Since N acts on fixed spaces, we can apply a standard Taylor-series formula,see Theorem 3.E, to the right-hand side. This yields (7.12b). Eq. (7.12a) canbe established analogously. Next, consider the goal error EQ := Q(θ, u, p) −Q(θh, uh, ph). Using (7.12a), and subsequently invoking the dual prob-lem (7.10), we obtain

EQ = Q′(θh, uh, ph)(eθ , eu, ep) + rQ

= S(eθ , ζ) + N′(θh, (uh, ph); (z, s))(eθ , eu, ep) + rQ .


Next, applying (7.12b), it follows that

EQ = S(eθ , ζ) + N(Ωθ , (u, p); (z, s) T -1

θ

)−N

(Ω, (uh, ph); (z, s)

)+ rQ − r

N.

The proof follows by noticing that, since z|Γ = ζ, we have

S(θ, ζ) + N(Ωθ , (u, p); (z, s) T -1

θ

)= G(ζ) ,

according to the primal weak formulation (7.2). ¤


In this section, we present numerical experiments. For the sake of simplicity,we consider Stokes flow coupled with the string model.

The fluid subproblem is discretized using standard (p2-p1) Taylor-Hood fi-nite element on triangles. The string equation is discretized using linear finiteelements. These elements coincide with the adjacent fluid-element edges. Wenote that the coupled problem is conforming; see Fig. 7.1. To solve the non-linear coupled problem, we use a standard subiteration (fixed-point iteration)scheme, where at each iteration the fluid mesh is deformed to accommodatethe new displacement. Dual problems are discretized using the same mesh asthe primal problem, but the dual shapefunctions are of one order higher, asdescribed in Sec. 3.2.4.

To drive adaptive mesh refinement, the residual R in Theorem 7.A is local-ized to obtain dual-weighted residual contributions (see Sec. 3.2.4). At the inter-face, we combine the standard structure contribution of a particular structureelement with the standard fluid contribution of the adjacent fluid element intoone fluid–structure residual contribution for that fluid element. We employ aDörfler-type marking strategy and refine fluid elements using newest-vertex bi-section; see Sec. 3.2.4. Structure elements are bisected if the adjacent fluid edgehas been bisected. This preserves conformity of the resulting discretization.

7.4.1 Parabolic interface testcase

First, we consider a manufactured fluid–structure interaction problem. It cor-responds to flow in a channel with a flexible top. The momentum and stringequation have a forcing such that the interface solution is parabolically shaped;see Fig. 7.2 [top-left]. The parameters in this problem are ν = 1 and E = 10.The right boundary of the domain is a Neumann boundary.

Let us briefly describe, how we compute the forcings. We start with the fluidsolution (u0, p0) in a 2 × 1 rectangular channel (parabolic velocity, linear pres-sure). The velocity is then mapped to the domain with the parabolic interface


0.0

-5.0

5.0

0.0

-0.1

-0.2

-0.3

-0.4

Figure 7.2: Parabolic interface testcase. Primal solution [top-left] and dual solution[bottom-left] on a fine mesh. Primal solution [top-right] and dual solution [bottom-right] on the coarsest 8-element mesh. The illustrations show (primal or dual) velocityvectors over the (primal or dual) pressure.

by means of the Piola transform:

u =(

J -1θ DTθ u0

) T -1

θ ,

where Tθ is the map corresponding to a parabolic θ2. The Piola transform hasthe nice property that the velocity remains divergence-free; see [28, 105, 170].The pressure is simply extended onto Ωθ . The so-defined solution (θ, u, p) isthen substituted in the primal problem to yield the forcings.

Our goal quantity of interest is the vertical displacement of the string at apoint

√2/4 ≈ 0.3536 from the left-side. The exact value of this quantity is

(4√

2 − 1)/16. Fig 7.2 [bottom-left] displays the corresponding dual solutionon a fine mesh. The primal and dual approximations on the coarsest meshare also visible in this figure. The convergence of the corresponding estimateEstQdisp and exact error EQdisp on uniformly-refined meshes is reported in Ta-ble 7.1. Note that the effectivity EstQdisp /EQdisp approaches 1. This clearlydemonstrates the consistency of the error estimate.

In Fig. 7.3, we plot the convergence of the exact error and the dual-basederror estimate on uniform and adaptive meshes versus the total number of de-grees of freedom, which is denoted by n. On adaptive meshes we observe anasymptotic convergence rate of O(n-2). This corresponds to optimal behaviorfor Taylor-Hood elements with quadratic velocities, a linear pressure and linear


10-0

10-2

10-4

10-6102 103 104

n

slope -1

slope -2

Error uniformEstimate unif.Error adaptiveEstimate adapt.

Figure 7.3: Parabolic interface testcase. Convergence of the error and error estimateunder uniform and adaptive mesh refinement versus the total number of degrees offreedom n.

structure displacements. Note that for uniform refinements, the rate is subop-timal. Fig. 7.4 displays several adaptively-refined meshes. The refinement nearthe interface as well as the point of interest are noteworthy.

7.4.2 Driven cavity with flexible bottom

Next, we consider a lid-driven cavity problem with a flexible bottom; seeFig. 7.5. This testcase has been studied in [232] and is similar to the test-case

Table 7.1: Convergence of the goal-oriented error estimate EstQdisp under uniform meshrefinement.

Elements Dofs Qdisp EQdisp EstQdisp Effectivity

8 60 0.1843 0.10674 0.10750 1.00716 115 0.2683 0.02280 0.02272 0.99632 187 0.2677 0.02331 0.02396 1.02864 369 0.2842 0.00683 0.00682 0.999

128 657 0.2842 0.00689 0.00708 1.027256 1,309 0.2901 0.00092 0.00093 1.005512 2,461 0.2901 0.00093 0.00098 1.053

1,024 4,917 0.2907 0.00039 0.00040 1.0062,048 9,525 0.2907 0.00039 0.00041 1.0324,096 19,045 0.2910 0.00009 0.00009 1.008

∞ ∞ 0.2911 0


Figure 7.4: Parabolic interface testcase. The initial mesh and adaptively refined meshesafter 8, 16 and 24 iterations, controlling the error in the vertical structure displacementat a point.

ΓN

Ωθ

Γθ

Figure 7.5: The driven cavity with flexible bottom testcase

in [74]. The undeformed domain corresponds to a unit square. The parametersin this problem are ν = 1 and E = 5. We impose a quadratic inflow with max-imum 1 at the top quarter of the left side, and a unit horizontal velocity at thetop of the domain. A homogeneous Neumann condition is imposed at the topquarter of the right side. This is also referred to as a leaky lid. Figure 7.6 showsthe solution on a fine mesh.

Our goal quantity of interest is again the vertical displacement of the stringat the point

√2/4 ≈ 0.3536 from the left-side; see Fig. 7.5. Fig. 7.7 displays the

dual solution corresponding to this goal functional. Note the dual pressure sin-gularity at the connection point of the Neumann and Dirichlet boundary. Fur-thermore, note the weak singularity in the vertical dual velocity at the point ofinterest. In Fig. 7.8, we plot the convergence of the dual-based error estimate onuniform and adaptive meshes. We also plot the error with respect to a referencevalue of 0.230862 obtained on a uniform mesh with 16, 384 elements and 74, 501degrees of freedom. Unfortunately, the reference value is not accurate enoughto fully compare it with the adaptive results; see the flattening behavior or theadaptive error plot. Nevertheless, we observe an optimal convergence rate onadaptive meshes. We show several adaptively-refined meshes in Fig. 7.9.


0.0

-5.0

-10.0

-15.0

-20.0

1.0

0.5

0.0

-0.5

-1.0

0.15

0.10

0.05

0.00

-0.05

-0.10

0.15

Figure 7.6: Driven cavity with flexible bottom testcase. Solution on a fine mesh: veloc-ity vectors [top-left], pressure [top-right], horizontal velocity [bottom-left] and verticalvelocity [bottom-right].

7.4.3 Backward-step with flexible bottom

The last application we consider is the testproblem introduced in [95]. It in-volves a backward step and a flexible bottom; see Fig. 7.10. Similar flexiblechannels without a step have been investigated in [131, 156], for example. Theparameters in our problem are ν = 1 and E = 15. A quadratic inflow withmaximum 3 is imposed at the left inflow boundary. The Neumann boundarycondition ν ∂nu − p n = (12, 0) is imposed at the boundary ΓN . Fig. 7.11 dis-plays the solution on a fine mesh.

Our goal quantity of interest is the integral of the vorticity in the quartercircular domain Ωω indicated in Fig. 7.10:

Qω(u) =∫

Ωω∇× u ,

In Fig. 7.12, we depict the approximate dual solution for the coarsest mesh.Note that the dual solution is local to Ωω. In fact, the dual velocity changesstrongly at the circular boundary of Ωω. The dual pressure displays singular-ities at the edges of the quarter circular boundary. In Fig. 7.13, we plot the


0.0

1.0

2.0

3.0

4.0

5.0

0.2

0.1

0.0

-0.1

-0.2

0.08

0.04

-0.04

-0.08

0.00

Figure 7.7: Driven cavity with flexible bottom testcase. Dual solution on the coarsestmesh for a vertical point-displacement goal functional: dual velocity vectors [top-left],dual pressure [top-right], horizontal dual velocity [bottom-left] and vertical dual veloc-ity [bottom-right].

10-1

10-3

10-7

10-5

102 103 104n

slope -1

slope -2


Figure 7.8: Convergence of the error and error estimate under uniform and adaptivemesh refinement versus the total number of degrees of freedom n.

convergence of the dual-based error estimate on uniform and adaptive meshes.For uniformly refined meshes, we also plot the error with respect to a reference


Figure 7.9: Adaptively refined meshes after 8, 16 and 24 iterations, controlling the errorin the vertical structure displacement at a point.

ΓN

ΩωΓθ

Figure 7.10: The backward step with flexible bottom testcase

value of −1.154 obtained on a uniform mesh with 15, 360 elements and 70, 085degrees of freedom. Owing to the observed convergence rate of O(n−1/2), thereference value is very inaccurate to be compared with the adaptive results. Weobserve an optimal convergence rate of O(n−2) for the dual-based error esti-mates on adaptive meshes. Fig. 7.14 shows several adaptively-refined meshes.Note the refinement at the boundary of the circular domain Ωω.


We obtained a dual problem (linearized adjoint) for fluid–structure interac-tion by means of the domain-map linearization approach. In this approach,the fluid subproblem is transformed onto a fixed domain. The dual problemis then obtained by linearization with respect to the domain transformationmap. We showed that the dual problem corresponds to a linearized fluid sub-problem coupled to a dual structure displacement subproblem. These sub-problems are coupled kinematically in a simple manner: the dual structuredisplacement acts as a Dirichlet boundary condition for the dual fluid veloc-ity. The traction coupling condition, however, involves nonstandard, nonlocal


45.0

30.0

15.0

0.0

-15.0 3.0

1.5

0.0

-1.5

-3.0 1.0

0.5

0.0

-0.5

-1.0

Figure 7.11: Backward step with flexible bottom testcase. Solution on a fine mesh, fromtop to bottom: velocity vectors, pressure, horizontal velocity and vertical velocity.

fluid terms. The effectivity of the dual-based error estimate and its usefulnessin goal-oriented adaptive mesh-refinement was demonstrated by several nu-merical experiments.

The elegant alternative provided by the shape-linearization approach tofree-boundary problems in Chapter 6 motivates us to also investigate this al-ternative for fluid–structure interaction. This is the subject of the followingchapter.


0.0

0.5

1.0

-1.0

-0.5

0.06

0.03

0.00

-0.03

-0.06 0.06

0.03

0.00

-0.03

-0.06

Figure 7.12: Backward step with a flexible bottom. Dual solution on the coarsest meshfor a local vorticity goal functional. From top to bottom: dual velocity vectors, dualpressure, horizontal dual velocity and vertical dual velocity.


10-1

10-2

10-3

10-4

10-5102 103 104

n

slope -1/2slope -2

Error uniformEstimate unif.Estimate adapt.


Figure 7.14: Adaptively refined meshes after 10, 20 and 30 iterations, controlling theerror in the local vorticity goal functional.

Chapter 8

Shape-Linearization Approach to


What is clear and easy to grasp attracts us; complications deter.

David Hilbert (1862–1943)

Creating mathematics is a painful and mysterious experience. Often theobject of the proof is clear, but the route is shrouded in fog, and the math-ematician stumbles through a calculation, terrified that each step might betaking the argument in completely the wrong direction. Additionally thereis the fear that no route exists.

Simon Singh, 1997Fermat’s Last Theorem1

In this chapter, we consider goal-oriented error estimation and adaptivity forfluid–structure interacton using the shape-linearization approach to derive thedual problem.

8.1 Introduction

In the previous chapter, we considered goal-oriented error estimation andadaptivity for fluid–structure interaction using the domain-map linearizationappoach. The resulting dual problem, however, contains a nonstandard, non-local boundary condition. This is similar to the domain-map linearized dualproblem of the Bernoulli free-boundary problem; see Chapter 5. For that prob-lem, we derived an elegant alternative using the shape-linearization approachin Chapter 6; see Sec. 3.4.2 for a brief outline of the approach.

This chapter considers the shape-linearization approach to derive the lin-earized adjoint and obtain an appropriate dual problem for fluid–structure in-teraction. We illustrate the approach on the fluid–structure interaction modelof the previous chapter. To perform the shape linearization, we need to remove

1Fourth Estate, HarperCollins, 1997.

148 8. Shape-Linearization Approach to Fluid–Structure Interaction

the essential boundary conditions in the fluid velocity test and trial spaces. Thisis achieved by introducing a very weak form of the fluid subproblem. A lin-earization of this very weak form then allows the extraction of a suitable dualproblem.

We show that the dual problem corresponds to a (linearized) dual fluid sub-problem coupled with a dual structure-displacement subproblem. These sub-problems are coupled in a kinematically straightforward manner. The othercoupling condition is nonstandard, but local. This condition is similar to thecurvature-dependent boundary condition encountered in the shape-linearizeddual problem of the Bernoulli free-boundary problem.

We note that the presented shape-linearization approach bears similaritiesto the so-called shape derivative used in shape optimization of fluid problems;see [23, 28, 170] and also Sec. A.2 for a brief introduction to the shape derivative.In dynamic settings, such derivatives are investigated in [77, 158, 170]. Fur-thermore, the shape-linearization approach in this chapter is related to formalmodel linearizations of fluid–structure interaction giving rise to the so-calledtranspiration boundary condition; see [87, 96, 97].

The contents of this chapter are arranged as follows. Sec. 8.2 recalls thefluid–structure interaction model. In Sec. 8.3, we apply the shape linearizationapproach. We first consider linearization at smooth interfaces and then brieflydiscuss the extension to nonsmooth interfaces. Numerical experiments are pre-sented in Sec. 8.4. Finally, Sec. 8.5 contains concluding remarks.


We briefly recall the fluid–structure interaction problem considered in the pre-vious chapter and introduced in Sec. 2.2. For the sake of clarity, we considera Stokes fluid without forcing, and with only Dirichlet boundary conditionson the fixed boundaries. These simplifications allow us to focus on the essen-tials of the shape-linearization procedure. To perform such a linearization, weintroduce a suitable very weak formulation.

8.2.1 Fluid–structure interaction model

We consider a fluid–structure system such as graphically depicted in Fig. 2.2.Consider a (vectorial) structure displacement θ in W1,∞(Γ0) ∩ H1

0(Γ0). This canbe extended to a Lipschitz field, also denoted θ, on the sufficiently-large hold-all domain D, which vanishes on the fixed boundary ΓD := ∂Ω0 \ Γ0. We denotethe space of such Lipschitz fields by

Θ(Ω0) :=

θ = ExtΓ0 η ∈ C0,1(D; R2) , η ∈ W1,∞(Γ0) : θ = 0 on ΓD

,


with ExtΓ0 a (sufficiently smooth) extension operator. For each θ, consider theperturbation of the identity map (or ALE map) Tθ := Id + θ which associatesto θ the fluid domain and current fluid–structure interface via

Ωθ := Tθ(Ω0) and Γθ := Tθ(Γ0) .

On Ωθ , we consider the following Stokes problem for the velocity u : Ωθ →R

2 and pressure p : Ωθ → R:

−ν ∆u + ∇p = 0

−div u = 0

in Ωθ , (8.1a)

u = 0 on Γθ ∪ ΓD , (8.1b)

where ν is the fluid kinematic viscosity. Note that we assumed homogeneousDirichlet boundary conditions for the sake of simplicity.

The structure displacement θ : Γ0 → R2 satisfies the following (vector)

string equation:

−E ∂2s θ = g +

((−ν ∂nu + p n) Tθ

)ωθ in Γ0 , (8.1c)

where the constant E is the string’s Young modulus, ∂s(·) := ∂(·)/∂s is thederivative along Γ0, and g ∈ L2(Γ0)

2. The tangential Jacobian is given by ωθ =Jθ |DT -T

θ n|, with Jθ = det DTθ the Jacobian determinant.

8.2.2 Weak formulation

A weak formulation of the coupled fluid–structure problem is given by:


0(Ωθ) × L2(Ωθ) :

S(θ, η) + N(Ωθ , (u, p); (v + ExtΓθ

η, q))

= G(η)

∀(η, v, q) ∈ H10(Γ0) × H1

0(Ωθ) × L2(Ωθ) ,

(8.2)

where

N(Ω, (u, p); (v, q)

):=

∫

Ω

(ν ∇u · ∇v − p div v − q div u

),

S(θ, η

):=

∫

Γ0

E ∂sθ · ∂sη ,

G(η) :=∫

Γ0

g · η ,

and the spaces incorporating essential boundary conditions are defined as

H10(Γ0) :=

η ∈ H1(Γ0) : η = 0 on ∂Γ0

.

H10(Ωθ) :=

v ∈ H1(Ωθ) : v = 0 on Γθ ∪ ΓD

.


We note that the string equation (8.1c) is enforced in a weak manner by means ofthe extension map ExtΓθ

defined for nonzero θ as the composition of extensionand transformation:

η 7→ ExtΓθη := (ExtΓ0 η) T -1

θ : H10(Γ0)

ExtΓ0−−−→ H10,ΓD

(Ω0)T -1

θ−−→ H10,ΓD

(Ωθ) ,(8.3)

where H10,ΓD

denotes the space of H1-functions with zero trace on ΓD .We shall call a displacement θ admissible if it is a member of W1,∞(Γ0) ∩

H10(Γ0) and if it leads to a Lipschitz fluid domain Ωθ . As usual, we assume

the existence of a (possibly non-unique) admissible displacement θ and a cor-responding velocity and pressure (u, p) ∈ H1

0(Ωθ) × L2(Ωθ) that solve (8.2).

8.2.3 Goal functionals

Our interest will be specific (bounded and continuously differentiable) goalfunctionals Q : (θ, u, p) ∈ H1

0(Γ0) × H10(Ωθ) × L2(Ωθ) → R of the solution.

For simplicity, we will restrict the ensuing analysis to functionals

Q(θ, u) =∫

Γ0

qdisp · θ +∫

Ωθ

qvel · u . (8.4)

where qdisp ∈ L2(Γ0) and qvel ∈ H1(R2). Our objective is to derive a dual-based

estimate of the goal-error Q(θ, u) −Q(θh, uh).

8.2.4 Very-weak fluid formulation

For a linearization of (a weak form of) the fluid–structure-interaction problem,it is important that the involved spaces do not depend on θ. Hence, we needto remove the essential boundary conditions at Γθ from the fluid velocity testand trial spaces in (8.2). The constraint on test functions v can be removedby absorbing ExtΓθ

η, i.e., the structure test function η is eliminated in favor of anonzero v. The constraint on trial velocity functions can be removed by going toa very weak formulation. We remark that it also possible to introduce Lagrangemultipliers to remove the constraints. This is, however, more involved.

Let us define the very weak fluid form Nv as

Nv(Ω, (u, p); (v, q)

):=

∫

Ω

(u ·

(− ν ∆v + ∇q

)− p div v

), (8.5)

We summarize the consistent reformulation in the following proposition.


Proposition 8.1 Let (θ, u, p), viewed as a member of H10(Γ0) × H1

0,ΓD(D) ×

L2(D), denote the solution of (8.2). It holds that

M(θ, (u, p); (v, q)

):=

S(θ, (v Tθ)|Γ0

)+ Nv

(Ωθ , (u, p); (v, q)

)− G

((v Tθ)|Γ0

)= 0 ,

(8.6)

for all v ∈ H2(D) ∩ H10,ΓD

(D) and q ∈ H1(D).

Proof Any v ∈ H2(D) ∩ H10,ΓD

(D) can be split as v = v0 + v1, with v0 = 0 onΓθ and v1 is an extension of v|Γ0 . First consider v0. Then (8.6) implies that

Nv(Ωθ , (u, p); (v0, q)

)= 0 .

Since this is a very weak form of N, a simple integration by parts of (8.5) andusing u = 0 on Γθ yields

N(Ωθ , (u, p); (v0, q)

)= 0 .

which is known to hold: the fluid subproblem in (8.2). Next consider v1. Notethat there exists an η ∈ H1

0(Γ0) such that v1 = η T -1θ on Γθ . Hence,

S(θ, η

)+ Nv

(Ωθ , (u, p); (v1, q)

)− G

(η)

= 0 .

Again, apply an integration by parts on Nv to obtain the structure subproblemin (8.2). ¤

In view of the reformulation, we can linearize M in (8.6) with respect to (θ, u, p).In particular, note that Ω 7→ Nv(Ω, (u, p); (v, q)) is a shape functional for fixedu, p, v and q. Hence, this linearization follows directly from results in Chapter 4on shape differential calculus.

8.3 Goal-oriented error estimation by shape lin-

earization

In this section, we take up the linearization of M (defined in (8.6)) and Q. Let(θh, uh, ph) ∈ H1

0(Γ0)× H10(Ωθh)× L2(Ωθh) denote the approximation at which

the linearization takes place. For convenience, we introduce the notation


We shall extract the dual problem for (z, s) from the linearized-adjoint equation:

M′(θh, (uh, ph); (z, s))(δθ, δu, δp) = Q′(θh, uh)(δθ, δu)

∀(δθ, δu, δp) ∈ H10(Γ0) × H1

0(Ωθh) × L2(Ωθh) .(8.7)

We first consider the case of smooth interfaces. The extension to the more gen-eral case of nonsmooth interfaces is discussed in Sec. 8.3.4.


8.3.1 Linearization with respect to fluid variables

Let us consider the linearization of (u, p) 7→ M(θh, (u, p); (z, s)). This is

straightforward as only Nv depends on it and the dependence is linear. De-noting the derivative by ∂(u,p)(·)(δu, δp), we obtain

∂(u,p)M(θh, (uh, ph); (z, s)

)(δu, δp) =

∫

Ω

(δu ·

(− ν ∆z + ∇s

)+ δp div z

)

Similarly, for the goal functional in (8.4), we obtain

∂uQ(θh, uh)(δu) =∫

Ωqvel · δu

These results already show that the fluid part of the dual problem correspondssimply to a Stokes problem:

−ν ∆z + ∇s = qvel

div z = 0

in Ω . (8.8)

More important are the boundary conditions at the interface, which follow fromthe linearization with respect to θ.

8.3.2 Shape linearization at smooth interfaces

Next we consider the linearization with respect to θ in the direction δθ. Letus assume that the approximate displacement θh is smooth, in particular, θh ∈W2,∞(Γ0) ∩ H1

0(Γ0). The approximate interface Γ is then a C1,1-boundary.The linearization of the very weak fluid form, i.e., of the map

θ 7→ Nv(Ωθ , (uh, ph); (z, s)

)=

∫

Ωθ

(uh ·

(ν ∆z + ∇s

)+ ph div z

),

corresponds to the shape derivative at Ω in the direction δθ T -1θh . Note that uh

is zero at Γ. Furthermore, we already know from (8.8) that div z = 0. Hence, inview of Prop. 6.1, the shape derivative vanishes:

∂ΩNv(Ω, (uh, ph); (z, s)

)(δθ T -1

θh ) = 0 .

Similarly, one contribution to the derivative of the goal functional vanishes:

∂θQ(θh, uh)(δθ) =∫

Γ0

qdisp · δθ .


Next, consider the structure related maps:

θ 7→ S(θ, (z Tθ)|Γ0) =∫

Γ0

E ∂sθ · ∂s(z Tθ)

θ 7→ G((z Tθ)|Γ0

)=

∫

Γ0

g · (z Tθ)

We collect their derivatives in the following lemma.

Lemma 8.2 Denoting the total derivative by dθ(·)|θh(δθ), it holds that2

dθS(θ, (z Tθ)|Γ0)∣∣θh(δθ)

=∫

Γ0

(E ∂sδθ · ∂s(z Tθh) − E ∂2

s θh · (∇zT Tθh) · δθ)

,

dθG((z Tθ)|Γ0

)∣∣θh(δθ) =

∫

Γ0

g · (∇zT Tθh) · δθ .

Proof To differentiate the (z Tθ)-terms at θh, we first introduce the map

Tθ := Tθ T -1θh : Ω → Ωθ ;

see also Fig. 5.2. We then have from Lemma 5.6:⟨∂θ(zi Tθ), δθ

⟩=

⟨∂θ(zi Tθ Tθh), δθ

⟩= (∇zi Tθh) · δθ ,

for i = 1, 2. Hence,⟨∂θ(z Tθ), δθ

⟩= (∇zT Tθh) · δθ

Note that to arrive at the second term in the derivative of S, we integrated byparts and used δθ = 0 at ∂Γ0. ¤

Collecting the θ-derivatives, we arrive at the following δθ-equation of thelinearized-adjoint equation (8.7):

∫

Γ0

(E ∂sδθ · ∂s(z Tθh) −

(g + E ∂2

s θh)· (∇zT Tθh) · δθ

)=

∫

Γ0

qdisp · δθ .

(8.9)

2The innerproduct of a matrix with a vector is defined as A · v = Ai,j vj and v · A = vi Ai,j. Hence∇zT · v = ∂xj

zi vj.



We are now ready to define the dual problem. Note that the δθ-equation (8.9)can be interpreted as a dual structure problem. This motivates the introduc-tion of the dual structure displacement ζ = (z Tθh)|Γ0 . The aggregated dualproblem is then defined as

− ν ∆z + ∇s

div z

= qvel

= 0

in Ω ,

z = 0 on ΓD ,

z − ζ T -1θh = 0 on Γ ,

−E ∂2s ζ − (∇z Tθh) ·

(g + E ∂2

s θh)

= qdisp in Γ0 ,

ζ = 0 on ∂Γ0 .

(8.10a)

(8.10b)

(8.10c)

(8.10d)

(8.10e)

The dual fluid and dual structure subproblems can be identified as (8.10a–8.10c)and (8.10d–8.10e), respectively. These subproblems are coupled kinematicallyby (8.10c). This coupling condition is the same as obtained by the domain-maplinearization in the previous chapter.

The complementary coupling mechanism, see (8.10d), although nonstan-dard, is local. Recall that in the previous chapter, this complementary cou-pling condition was nonlocal. The condition in (8.10d) appears to be similar tothe curvature-dependent boundary condition obtained in the shape-linearizeddual problem of the Bernoulli free-boundary problem; see Chapter 6. Indeed,the second derivative ∂2

s θh is a measure of the curvature of Γ.A weak formulation of the dual problem (8.10) can be derived by employing

standard formulations. Let us denote the space for the dual velocity z as

H1ζ,Γ(Ω) :=

v ∈ H1(Ω) : v|Γ = ζ T -1

θh

.

A weak formulation of (8.10) then reads

Find ζ ∈ H10(Γ0) and (z, s) ∈ H1

ζ (Ω) × L2(Ω) :∫

Ω

(ν ∇δu · ∇z − s div δu

)=

∫

Ωqvel · δu ,

∫

Ω−δp div z = 0 ,

∫

Γ0

(E ∂sδθ · δζ − δθ · (∇z Tθh) ·

(g + E ∂2

s θh))

=∫

Γ0

qdisp · δθ ,

∀(δθ, δu, δp) ∈ H10(Γ0) × H1

0(Ω) × L2(Ω) .

(8.11)

Unfortunately, this weak formulation, as it stands, contains unbounded func-tionals due to the ∇z-term at Γ0. Furthermore, it is not clear under which con-


ditions (8.11) admits a solution. We note that boundary conditions involvingthe gradient are so-called oblique boundary conditions. Problems with suchboundary conditions are studied in [120, p. 167] and [54, p. 398], for instance.

Since we have derived the dual problem from the linearization of M and Q,any sufficiently smooth solution (z, s) of (8.11) satisfies the linearized adjointequation (8.7). Accordingly, we can present the following error representationformula.

Theorem 8.A (Shape-Linearized Dual-Based Error Representation) Givenany admissible structure approximation θh ∈ W2,∞ ∩ H1

0(Γ0) and correspond-ing fluid domain Ω = Ωθh and fluid approximation (uh, ph) ∈ H1

0(Ω) × L2(Ω)of the fluid–structure solution (θ, u, p) of (8.2), assume that the dual prob-lem (8.11) has a sufficiently smooth solution (ζ, z, s). Then it holds that

Q(θ, u) −Q(θh, uh) = R(θh, uh, ph; ζ, z, s

)+ r ,

where the coupled fluid–structure residual is given by

R(θh, uh, ph; ζ, z, s

)= G(ζ) − S(θh, ζ) −N

(Ω, (uh, ph); (z, s)

).

The remainder r = o(‖eθ‖H1

0 (Γ0), ‖eu‖H1

0 (D), ‖ep‖L2(D)

)and the errors are de-

fined as

eθ := θ − θh , eu := u − uh , ep := p − ph .

Proof The proof follows closely the proof of Theorem 3.H in Sec. 3.3. Considerthe goal error EQ = Q(θ, u) −Q(θh, uh). Using a Taylor-series formula for Q,see Theorem 3.E, and invoking (8.7), we obtain

EQ = Q′(θh, uh)(eθ , eu) + rQ = M′(θh, (uh, ph); (z, s))(eθ , eu) + rQ ,

with rQ = o(‖eθ‖H1

0 (Γ0), ‖eu‖H1

0 (D)

). Applying a Taylor-series formula for M,

we obtain

EQ = M(θ, (u, p); (z, s)

)−M

(θh, (uh, ph); (z, s)

)+ rQ − rM .

with rM = o(‖eθ‖H1

0 (Γ0), ‖eu‖H1

0 (D), ‖ep‖L2(D)

). The first term on the right-hand

side vanishes on account of consistency of the solution with M; see Prop. 8.1.Furthermore, expanding the second term in accordance with (8.6), it followsthat

EQ = G((z Tθh)|Γ0

)− S

(θh, (z Tθh)|Γ0

)−Nv

(Ω, (uh, ph); (z, s)

)+ r .

Applying an integration by parts on Nv thereby using uh = 0 on ∂Ω, we canreplace Nv by N. Finally, note that (z Tθh)|Γ0 = ζ. ¤


8.3.4 Extension to nonsmooth interfaces

In finite-element discretizations, the structure approximation θh is generallyonly continuous and not continuously differentiable. The interface Γ is thenLipschitz continuous. In this case, the dual problem in (8.10) and its weak for-mulation (8.11) are not valid, since the term ∂2

s θh is singular at the singularpoints (kinks) of the interface. This complication can, however, be resolved bytaking the singular points into account during the derivation of the dual prob-lem.

Let us consider, for the sake of simplicity, a piecewise-linear structure ap-proximation, i.e.,

θh ∈ S1(Γ0) :=

η ∈ H10(Γ0) : η|K ∈ P

1(K) , ∀K ∈ τ(Γ0)

,

where τ is a partition of Γ0 into finite elements K and P1(K) is the space of

polynomials up to first order on K. Let us denote the set of inter-element edges(points) x by χ.

The complication can be traced back to the linearization of the map

θ 7→ S(θh, (z Tθ)|Γ0) =∫

Γ0

E ∂sθh · ∂s(z Tθ) ,

in Sec. 8.3.2. Prior to the linearization, we invoked a global integration by partson this integral and obtained ∂2

s θh. For a nonsmooth θh, we have to integrate byparts element-wise prior to linearization, i.e.,

∫

Γ0

E ∂sθh · ∂s(z Tθ) = ∑K∈τ

[E ∂sθh · (z Tθ)

]

∂K.

If we now linearize with respect to θ, we obtain

∑K∈τ

[E ∂sθh · (∇zT Tθh) · δθ

]

∂K= ∑

x∈χ

[[E ∂sθh · (∇zT Tθh)

]]

x· δθ ,

where [[ · ]]x denotes the jump of (·) at x. Hence, the weak form of the dualproblem which is suitable at nonsmooth interfaces is essentially given by (8.11)with the replacement:

∫

Γ0

−δθ · (∇z Tθh) · E ∂2s θh ← ∑

x∈χ

δθ ·[[(∇z Tθh) · E ∂sθh

]]

x.


We consider numerical experiments for Stokes flow coupled with the vectorstring model. Similar to the numerical experiments in Chapter 7, we discretize


0.0

-0.1

-0.2

-0.3

-0.4

Figure 8.1: Parabolic interface testcase. Dual solution on a fine mesh [left] and on thecoarsest 8-element mesh [right]. The illustrations show dual velocity vectors over thedual pressure.

the fluid subproblem using (p2-p1) Taylor-Hood finite elements and linear fi-nite elements for the string problems. The string elements exactly correspondto adjacent fluid-element edges. Dual problems are based on (8.11) with thereplacement as discussed in Sec. 8.3.4 to handle the piecewise linear interface.The dual problems are discretized on the primal-problem mesh with shape-functions that are one order higher. Adaptive mesh-refinement is performed asdescribed in Sec. 7.4.

8.4.1 Parabolic interface testcase

First, we consider the manufactured fluid–structure interaction problem intro-duced in Sec. 7.4.1. We refer to Fig. 7.2 [top] for an illustration of the solutionon a fine mesh and on the coarsest mesh.

Our goal quantity of interest is again the vertical displacement of thestring at a point

√2/4 ≈ 0.3536 from the left-side. Fig. 8.1 displays the

shape-linearized dual solutions on a fine mesh and the coarsest mesh. Theconvergence of the corresponding estimate EstQdisp and exact error EQdisp onuniformly-refined meshes is reported in Table 8.1. Note the consistency of theestimates in view of the effectivity being close to 1. To enable a comparison,Table 8.1 also presents the results obtained in Sec. 7.4.1 for the domain-maplinearization approach.

In Fig. 8.2, we plot the convergence of the error and the dual-based estimateon uniform and adaptive meshes. Note the suboptimal and optimal conver-gence on uniform and adaptive meshes, respectively. Fig. 8.3 displays severaladaptively-refined meshes.


10-0

10-2

10-4

10-6101 102 103

n

slope -1

slope -2


Figure 8.2: Parabolic interface testcase. Convergence of the error and error estimateunder uniform and adaptive mesh refinement versus the total number of degrees offreedom n.

8.4.2 Driven cavity with flexible bottom

Next, we consider the lid-driven cavity problem with a flexible bottom intro-duced in Sec. 7.4.2; see Fig. 7.5. We refer to Figure 7.6 for the solution on a finemesh.

The goal quantity of interest is again the vertical displacement of the stringat the point

√2/4 ≈ 0.3536 from the left-side. Fig. 8.4 displays the shape-

Table 8.1: Convergence of the goal-oriented error estimate EstQdisp under uniform meshrefinement for the shape-linearization approach and the domain-map linearization ap-proach of Chapter 7.

Shape Domain-mapElements Dofs Qdisp EQdisp EstQdisp Effect. EstQdisp Effect.

8 60 0.1843 0.10674 0.10116 0.948 0.10750 1.00716 115 0.2683 0.02280 0.02319 1.017 0.02272 0.99632 187 0.2677 0.02331 0.02432 1.044 0.02396 1.02864 369 0.2842 0.00683 0.00701 1.026 0.00682 0.999

128 657 0.2842 0.00689 0.00702 1.020 0.00708 1.027256 1,309 0.2901 0.00092 0.00063 0.683 0.00093 1.005512 2,461 0.2901 0.00093 0.00072 0.772 0.00098 1.053

1,024 4,917 0.2907 0.00039 0.00041 1.042 0.00040 1.0062,048 9,525 0.2907 0.00039 0.00037 0.953 0.00041 1.0324,096 19,045 0.2910 0.00009 0.00009 0.963 0.00009 1.008

∞ ∞ 0.2911 0


Figure 8.3: Parabolic interface testcase. The initial mesh and adaptively refined meshesafter 8, 16 and 24 iterations, controlling the error in the vertical structure displacementat a point.

linearized dual solution corresponding to this goal functional. Note the dualpressure singularity at the connection point of the Neumann and Dirichletboundary and the weak singularity in the vertical dual velocity at the pointof interest. In Fig. 8.5, we plot the convergence of the dual-based error estimateon uniform and adaptive meshes. We also plot the error with respect to thereference value of Sec. 7.4.2. Unfortunately, the reference value is not accurateenough to compare it with the adaptive results. Nevertheless, we observe anoptimal convergence rate in the estimate on adaptive meshes. We show severaladaptively-refined meshes in Fig. 8.6.

8.4.3 Backward step with flexible bottom

The last application we consider is the test problem discussed in Sec. 7.4.3 in-volving a backward step and a flexible bottom; see Fig. 7.10. We refer to Fig. 7.11for the solution on a fine mesh. The goal quantity of interest is again the integralof the vorticity in the circular domain right after the backward step:

Qω(u) =∫

Ωω∇× u .

In Fig. 8.7, we depict the approximate dual solution for the coarsest mesh. InFig. 8.8, we plot the convergence of the dual-based error estimate on uniformand adaptive meshes. For uniformly refined meshes, we also plot the error withrespect to the reference value used in Sec. 7.4.3. The reference value is veryinaccurate to be compared with the adaptive results. However, we observean optimal convergence rate of O(n−2) for the dual-based error estimates onadaptive meshes. Fig. 8.9 shows several adaptively-refined meshes.


We obtained the dual problem (linearized adjoint) for fluid–structure interac-tion by means of the shape–linearization approach. We introduced a very weak


0.0

1.0

2.0

3.0

4.0

5.0

0.2

0.1

0.0

-0.1

-0.2

0.08

0.04

-0.04

-0.08

0.00

Figure 8.4: Driven cavity with flexible bottom testcase. Dual solution on the coarsestmesh for a vertical point-displacement goal functional: dual velocity vectors [top-left],dual pressure [top-right], horizontal dual velocity [bottom-left] and vertical dual veloc-ity [bottom-right].

10-1

10-2

10-3

10-4

10-5

102 103 104n

slope -1

slope -2



form of the fluid subproblem which allowed the linearization using techniquesfrom shape calculus. We showed that the dual problem corresponds to the same


Figure 8.6: Adaptively refined meshes after 8, 16 and 24 iterations, controlling the errorin the vertical structure displacement at a point.

linear fluid and structure subproblems with the same kinematic coupling asobtained by the domain-map linearization approach in Chapter 7. The comple-mentary coupling is a nonstandard but local condition, contrary to what wasobtain before. This coupling is similar to the curvature-dependent boundarycondition encountered in the shape-linearized dual problem of the Bernoullifree-boundary problem; see Chapter 6. We demonstrated the effectivity of thedual-based error estimate and its usefulness in goal-oriented adaptivity by sev-eral numerical experiments.

We would like to comment on some of the simplifications made for the sakeof clarity. Firstly, we considered the Stokes equations instead of the Navier-Stokes equations, but there is nothing essential about this. Of course such afluid model changes the dual fluid subproblem, but it does not change the dualboundary conditions: Similar to the other fluid terms, the shape derivative ofthe nonlinear convective term vanishes. Another assumption pertained to theabsence of an external external forcing in the momentum equation. Such a forc-ing can however be straightforwardly introduced, resulting in an additionalterm in the dual coupling condition (8.10d), similar to the term due to f in theshape-linearized dual problem in Chapter 6. Indeed, we employed this addi-tional term in the numerical experiment of Sec. 8.4.1.


0.0

0.5

1.0

-1.0

-0.5

0.06

0.03

0.00

-0.03

-0.06 0.06

0.03

0.00

-0.03

-0.06

Figure 8.7: Backward step with a flexible bottom. Dual solution on the coarsest meshfor a local vorticity goal functional. From top to bottom: dual velocity vectors, dualpressure, horizontal dual velocity and vertical dual velocity.


10-1

10-2

10-3

10-4

10-5102 103 104

n

slope -1/2

slope -2



Figure 8.9: Adaptively refined meshes after 10, 20 and 30 iterations, controlling the errorin the local vorticity goal functional.

Chapter 9

Conclusions and Future Prospects

Far better an approximate answer to the right question, which is oftenvague, than an exact answer to the wrong question, which can always bemade precise.1

John Wilder Tukey (1915–2000), 1962

Conclusions In this thesis, we developed techniques for the adaptive dis-cretization of fluid–structure interaction controlling the error in goal function-als of the solution. The foundation for such discretizations is provided bythe theory of goal-oriented error estimation and goal-oriented adaptive meth-ods. We explained that the major difficulty in applying the canonical theory tofluid–structure interaction, or to free-boundary problems in general, pertainsto the determination of a suitable dual problem. That is, the linearization of thedomain-dependent nonlinearity in free-boundary problems is highly nontriv-ial.

We introduced two approaches to derive the linearized operators and obtainthe dual problem, both of which rely heavily on concepts from shape calcu-lus. In the domain-map linearization approach, the free-boundary problem is firsttransformed to a fixed reference domain. Such a reformulation allows the di-rect application of the canonical theory. Essentially, a linearization is performedwith respect to the domain map. In the shape-linearization approach, we write thedomain dependence as shape functionals. Such a dependence is then linearizedusing shape-derivative techniques from shape calculus.

To focus on the free-boundary character, we first applied the linearizationapproaches to a model free-boundary problem. We showed that the domain-map linearized dual problem corresponds to a boundary value problem witha nonlocal boundary condition. This nonlocality is impractical from an imple-mentation point of view. The shape-linearized dual problem is similar, but hasa local boundary condition which depends on the curvature of the boundary.For both approaches, we presented numerical experiments that demonstrate

1The Annals of Mathematical Statistics 33(1) (1962), pp. 1–67.

166 9. Conclusions and Future Prospects

the effectivity of the implied goal-oriented error estimates and their usefulnessin goal-oriented adaptive mesh-refinement.

We then applied the linearization approaches to a steady fluid–structure in-teraction model problem. We considered an incompressible flow coupled witha low-order structural model, viz., a string (membrane) model. The domain-map linearization approach corresponds to a linearization with respect to theALE (arbitrary Lagrangian–Eulerian) map. We showed that the dual problemobtained by this approach corresponds to a linear coupled fluid–structure prob-lem with a kinematically straightforward coupling. The complementary cou-pling condition, however, depends in a nonstandard and nonlocal manner onthe dual fluid solution. This is similar to what was obtained for the modelfree-boundary problem. The shape-linearized dual problem corresponds to thesame linear fluid and structure subproblems but has local coupling conditions.We presented numerical experiments that showed the suitability of the goal-oriented error estimates for the goal-adaptive discretization of fluid-structureinteraction.

Future prospects The presented work can be extended in various directions.Most obvious is the goal-oriented error estimation and adaptive discretizationof other more complex free-boundary problems. For example, realistic exten-sions of the investigated model free-boundary problem are free-surface flowand two-fluid flow. A particular difficulty in these problems is the linearizationof the surface terms. The investigated fluid–structure-interaction model can beextended to three dimensions and/or more realistic structural models, such asdeformable solids. In such extensions, the main challenge is the derivation ofdual coupling conditions.

We remarkt that the analysis simplifies considerably if the approximationshave smooth interfaces. This motivates the use of smooth discretizations meth-ods such as the recently developed isogeometric analysis; see [18, 135, 233].Other problems involving interfaces or discontinuities present, in principle,similar difficulties in deriving suitable dual problems. For example, there hasbeen some research on the linearization of shape functionals involving crackeddomains; see [93, 102]. The extension to unsteady problems is also very impor-tant. We believe that the derivation of unsteady dual problems proceeds simi-larly as presented and can be based on a space-time or ALE point of view [19].

A different extension of our work is goal-oriented adaptive discretizationof fluid–structure interaction using more sophisticated adaptive discretizationstrategies. Indeed, the application of hp-adaptive strategies [61, 181, 203, 209]and anisotropic refinement strategies [107, 154, 227] based on the derived dualproblems is immediate.

Finally, we note that the presented linearization approaches are useful for

167

many other applications involving free-boundary problems. We mention thederivations of Newton-type algorithms such as in [98, 100, 141], and the ap-plication of linearized adjoints in modeling error estimation [179] and optimalcontrol [170].

Appendix

A.1 Elements of tangential calculus

The derivation of the shape derivative of a boundary integral in Sec. 4.3.4 re-quires some concepts of tangential calculus. These concepts allow the manipu-lation of the boundary terms that appear during the derivation.

Basically, tangential calculus is concerned with functions and their deriva-tives on manifolds. Although giving an equivalent description, the differentialgeometry approach is avoided here, and intrinsic definitions are introducedthrough smooth extensions on R

N. Most of the results discussed here can befound in [58, 60, 63] and [59, Ch. 8].

A.1.1 Tangential derivatives

Let Γ be a compact manifold of codimension one representing a sufficientlysmooth (piece of) boundary of Ω, e.g., of class C2. Using smooth extensions(·) of functions defined on Γ, we introduce the following (classical) intrinsicdefinition:

Definition A.1 (Tangential Gradient) The tangential gradient of ϕ ∈ C1(Γ; R) isdefined as

∇Γ ϕ := ∇ϕ∣∣Γ− ∂ϕ

∂nn ,

in C0(Γ; R), where ϕ ∈ C1(RN; R) is an extension of ϕ. ¤

Note that the tangential gradient is the component of the gradient in the localtangent space. It can be verified that the tangential gradient is independent ofthe selected extension used. Furthermore, this definition can be extended toSobolev spaces, for example ∇Γ : Ws,p(Γ) → Ws−1,p(Γ), s ≥ 1; see [202, p. 84]or [63]. Intrinsic derivatives for vector functions follow straightforwardly.

170 Appendix

Definition A.2 (Tangential Derivatives for Vectors) The tangential Jacobianmatrix and tangential divergence of w ∈ C1(Γ; R

N), and the Laplace-Beltrami(tangential Laplace) of ϕ ∈ C2(Γ; R) are defined as

[DΓw

]ij

:=[∇Γwi

]j

,

divΓ w := tr DΓw ,

∆Γ ϕ := divΓ ∇Γ ϕ .¤

These expressions can be written in terms of their extensions as

DΓw = Dw − ∂nw ⊗ n ,

divΓ w = div w − ∂nw · n ,

∆Γ ϕ = ∆ϕ − κ ∂n ϕ − ∂2n ϕ ,

where the normal derivative ∂n(·) for vector functions is defined as ∂(·)/∂n :=D(·) n and for scalar functions as ∇(·) · n. Further, κ := divΓ n ∈ C0(Γ; R)coincides with the additive curvature (sum of N − 1 curvatures) of Γ.

A.1.2 Tangential integration by parts

The following counterpart of Stokes’ theorem on manifolds holds; see [202,p. 86] and [63]. We first consider the case where the manifold has a boundary.

Theorem A.A (Tangential Stokes’ Theorem) Let Γ be a C2-manifold withboundary ∂Γ. For w ∈ C1(Γ; R

N), the following identity holds∫

ΓdivΓ w =

∫

Γκ w · n +

∫

∂Γw · τ .

where τ(x) is the unit vector normal to the boundary ∂Γ, and contained in thetangent space to Γ at x, outward pointing with respect to Γ.

An integration by parts identity is obtained by applying this theorem to the vec-tor function ϕ w with ϕ ∈ C1(Γ; R) and w ∈ C1(Γ; R

N). This yields a tangentialGreen’s identity:

∫

Γ

(ϕ divΓ w + ∇Γ ϕ · w

)=

∫

Γκ ϕ w · n +

∫

∂Γϕ w · τ .

If Γ is the boundary of a smooth domain of class C2, then Γ is a closed manifold,i.e., ∂Γ = ∅. The Green’s identity then reduces to

∫

Γ


)=

∫

Γκ ϕ w · n . (A.1)

A.1 Elements of tangential calculus 171

Ω

Γ1x1x2. . .

τ1

Figure A.1: Illustration of the singular points xi, a boundary segment Γi and a vectorτi (composed of the two unit tangent vectors) for a domain Ω with a piecewise smoothfree boundary Γ.

This specific identity can be proven using standard shape derivative concepts.Since such a proof appears to be unfamiliar in the literature, we give this proofat the end of this section.

In case of a piecewise C2 boundary, (A.1) does not hold because the cur-vature κ is singular at singular points of the boundary; see [57, Sec. 5]. Nev-ertheless, an identity involving additional terms at the singular points can beobtained by applying Th. A.A to the smooth parts; see [202, p. 150] and [63].

Let us consider this in R2 for a piecewise C2 boundary Γ of Ω ⊂ R

2, seeFig. A.1. Denote the m singular points in succession by xi ∈ Γ, for i = 1, . . . , m,and the corresponding boundary segments between xi and xi+1 by Γi+1 (be-tween xm and x1 for Γ1). Let τi denote the sum of the two unit tangent vectorsτ|Γi

and τ|Γi+1 at xi, outward with respect to their segment, i.e.,

τi := τ∣∣Γi

(xi) + τ∣∣Γi+1

(xi) .

Then the following tangential Green’s identity holds:

∫

Γ


)=

m

∑i=1

( ∫

Γi

(κ ϕ w · n) + ϕ(xi) w(xi) · τi

), (A.2)

for all ϕ ∈ C1(Γ; R) and w ∈ C1(Γ; R2).

Proof (of identity (A.1)) Using standard shape derivative concepts, we provethe identity

∫

ΓdivΓ w =

∫

Γκ w · n ,

for w ∈ C1(Γ; RN), where Γ is a smooth boundary of a domain Ω ⊂ R

N ofclass C2. We start with the divergence theorem for a domain Ωt, i.e.,

∫

Ωt

div w =∫

Γt

w · nt . (A.3)

172 Appendix

The proof follows by differentiating this identity with respect to t. Using Theo-rem 4.E, we can differentiate the left side to obtain:

∂

∂t

( ∫

Ωt

div w)∣∣∣

t=0=

∫

Γdiv w v · n .

To differentiate the right side, first rewrite it to a boundary integral on Γ:∫

Γt

w · n =∫

Γ(w Tt) · (nt Tt) ωt ;

see also Sec. 4.3.4. Using the following transformation result for normals,see [202, p. 79]:

nt Tt =DT -T

t n

|DT -Tt n| ,

we can differentiate this relation to obtain

∂

∂t

(nt Tt

)∣∣t=0 = (Dv n · n) n − DvT n = −DΓvT n .

Hence, the derivative of the right member of (A.3) is given by

∂

∂t

( ∫

Γt

w · n)∣∣∣

t=0=

∫

Γ

(DwT v · n − w · DΓvT n + w · n divΓ v

).

Putting both sides together and choosing v = n, we obtain∫

Γdiv w =

∫

Γ

(DwT n · n − w · DΓnT n + w · n divΓ n

)

=∫

Γ

(Dw n · n − w · DΓnT n + w · n κ

).

Now, we use the key identity DΓnT n = 0 which can be shown by taking thetangential gradient on both sides of the identity

1 = n · n , on Γ .

Rearranging finally yields∫

Γ

(div w − Dw n · n

)=

∫

Γκ w · n . ¤

A.2 Shape derivatives of shape functions 173

A.2 Shape derivatives of shape functions

In this section, we give a brief introduction to derivatives of shape functions.There are two main notions of derivatives of shape functions. Unfortunately,they are confusingly referred to as the shape derivative and the material deriva-tive in literature. As in the shape functional case, the derivatives are definedthrough one-parameter families of domain perturbations.

Recall that O denotes a set of open bounded Lipschitz domains in the hold-all D. We consider (scalar) domain shape functions. These are scalar functionswhich, given a domain Ω ∈ O, return a function y(Ω) defined on this domain.2

Typically y(Ω) is an element of a smooth Banach space W(Ω), e.g. Wm,p(Ω)with m ≥ 1 and p ≥ 1. We assume W(Ω) to be such that functions in W(Ω)have a trace on Γ. Note that the shape function is a map that can be written as

y : O → W :=⋃

Ω∈O

W(Ω) .

A.2.1 Extension approach to shape derivative

Consider domain perturbations Ωt = Tt(Ω) along a sufficiently smooth admis-sible velocity field v ∈ C

([0, t ]; Θ(D)

)with Θ(D) ⊂ V(D), and Tt the asso-

ciated smooth transformation; see Sec. 4.2.1. Note that it is not straightawaypossible to form a difference quotient of y at Ω and Ωt since they exist on dif-ferent domains. Therefore, suppose we have a function Y ∈ C0([0, t ], W(D)

)

such that its restriction to Ωt coincides with y(Ωt), i.e.,

Y(t)∣∣Ωt

= y(Ωt) , ∀t ∈ [0, t ] . (A.4)

Basically, since Y acts as a smooth extension, we can define the shape derivativeof y through the derivative of t 7→ Y(t). This is the approach given in [35, 63,240]; cf. [185, p. 83]. In other words, if Y satisfies (A.4) and the semiderivative

∂tY(0) = limtց0

Y(t) − Y(0)

t, (A.5)

exists in L1(D), then the shape derivative of shape function y : O → W at Ω ∈ O

in the direction v is defined as

y′(Ω)(v) := ∂tY(0)∣∣Ω

. (A.6)

Note that this definition only provides for the existence of a shape derivative.The question of uniqueness is dealt with in the following.

2Boundary shape functions, returning functions on the boundary of Ω, will not be consideredhere; see for instance [63, 202].

174 Appendix

Proposition A.3 The shape derivative y′(Ω)(v) is unique and independent ofthe choice of the extension Y.

Proof We follow [63, 240]. Let Y1 and Y2 be two extension functions satisfying(A.4) and ϕ ∈ C∞

0 (D) a test function. We have

0 =∫

Ωt

(Y1(t) − Y2(t)

)ϕ ,

since Y1(t) = Y2(t) = y(Ωt) on Ωt. Taking the derivative with respect to tyields

0 =∫

Ω

(∂tY1(0) − ∂tY2(0)

)ϕ +

∫

Γ

(Y1(0) − Y2(0)

)ϕ v(0) · n , (A.7)

where we used Theorem 4.E. Noting that Y1(0) = Y2(0) = y(Ω) at Γ, thesecond integral vanishes, and since (A.7) holds for all ϕ ∈ C∞

0 (Ω) ⊂ C∞0 (D),

we have ∂tY1(0) = ∂tY2(0) in Ω. ¤

As in the shape functional case, if v 7→ y′(Ω)(v) is a continuous map, then theshape derivative only depends on the initial velocity field, i.e.,

y′(Ω)(v) = y′(Ω)(v(0)) ; (A.8)

see [240] or [202, p. 111]. In conformity with the shape functional case, weintroduce the following definition of shape differentiability.

Definition A.4 (Shape Differentiability of Shape Function) The shape func-tion y : O → W is said to be shape differentiable at Ω with respect to Θ(D) if theshape derivative (A.6) exists in all directions v ∈ C

([0, t ]; Θ(D)

)such that (A.8)

holds and the map v(0) 7→ y(Ω)(v(0)) is linear and continuous on Θ(D). ¤

Because of (A.8), we shall from now on assume v to be autonomous, unlessspecified otherwise.

In view of the arguments in Sec. 4.3.1 concerning a domain-preserving trans-formation, it is also expected that y′(Ω)(v) is nonzero only if v is nonzero atthe boundary Γ. This is confirmed by the following proposition.

Proposition A.5 If the shape function y is shape differentiable at Ω with respectto Θ(D), then the map v 7→ y′(Ω)(v) is supported on Γ, i.e.,

y′(Ω)(v) = 0 , ∀v ∈ Θ(D) with v = 0 on Γ .

Furthermore, if Γ is sufficiently smooth, then there exists a linear and continu-ous map g(Ω) : (Γ → R) → L1(Ω), such that

y′(Ω)(v) = g(Ω)(γ(v) · n) . (A.9)

where γ is the trace operator on Γ.


Proof The proof is inspired by [63, Prop. 2.3]. See also [240, Prop. 3.4] and [202,p. 112]. Let ϕ ∈ C∞

0 (D) be a test function and consider the shape functional

J(Ωt) =∫

Ωt

y(Ωt) ϕ .

Since y is shape differentiable, we have that J is shape differentiable; see [202,p. 113]. The shape derivative of J yields

J′(Ω)(v) =∫

Ωy′(Ω)(v) ϕ +

∫

Γy(Ω) ϕ v · n . (A.10)

If v = 0 at Γ, then Tt is domain-preserving, i.e., Ωt = Ω, hence J′(Ω)(v) = 0and (A.10) reduces to

0 =∫

Ωy′(Ω)(v) ϕ . (A.11)

Since this holds for all ϕ ∈ C∞0 (Ω) ⊂ C∞

0 (D), we have y′(Ω)(v) = 0. To provethe second part, recall from the Hadamard-Zolésio structure theorem, Th. 4.C,that if Γ is sufficiently smooth, then J′(Ω)(v) =

⟨j′(Γ), γ(v) · n

⟩. Hence if

v · n = 0 at Γ, (A.10) reduces again to (A.11), implying y′(Ω)(v) = 0. Then,there exists a map g(Ω) such that the identification (A.9) can be made. See theproof of Prop 2.3 in [63] for the exact details of the identification. ¤

A.2.2 Material derivative approach to shape derivative

A different notion of derivative which does not require an extension function Yis obtained by the material derivative approach; see for example [202, p. 98].Essentially, by transporting the shape function at Ωt back onto Ω, a differencequotient on Ω is possible.

Definition A.6 (Material Derivative of Shape Function) The material deriva-tive of y at Ω ∈ O in the direction v is defined as the limit

y(Ω)(v) = limtց0

y(Ωt) Tt − y(Ω)

t,

if it exists in L1(Ω). ¤

The material derivative is, however, not a satisfactory derivative in the follow-ing sense. If Tt is a domain-preserving transformation, i.e. Ωt = Tt(Ω) = Ω,then applying Lemma 4.7 we have

y(Ω)(v) = limtց0

y(Ω) Tt − y(Ω)

t= ∇y(Ω) · v ;

176 Appendix

cf. [202, p. 101]. Consequently, even though the shape function is not changing,since y(Ωt) = y(Ω), the material derivative is generally nonzero. Moreover, itdepends on the specifics of v in the interior of Ω.

On the other hand, the material derivative can be used in the computationof the shape derivative because they are related through the following simpleidentity; see [63], for example.

Proposition A.7 If y has a material derivative y(Ω)(v) at Ω in the direction vand ∇y(Ω) · v(0) ∈ L1(Ω), then y has a shape derivative given by

y′(Ω)(v) = y(Ω)(v) −∇y(Ω) · v . (A.12)

Proof We follow the proof of Prop. 9 in [35]. Let yt := y(Ωt) Tt denote theshape function at Ωt transformed back onto Ω. Furthermore, let E denote asufficiently smooth extension operator mapping L1(Ω) to L1(D) and W(Ω) toW(D). By adding a partition of zero to the definition of the shape derivative,we obtain

y′(Ω)(v) = limtց0

Y(t) − Y(0)

t

∣∣∣Ω

= limtց0

Y(t) − Eyt

t

∣∣∣Ω

+ limtց0

Eyt − Y(0)

t

∣∣∣Ω

The last limit can be recognized as the material derivative:

limtց0

Eyt − Y(0)

t

∣∣∣Ω

= limtց0

yt − y(Ω)

t= y(Ω)(v) .

Next, we choose the following specific function Y satisfying (A.4):

Y(t) = (Eyt) T -1t .

The other limit can then be evaluated using Lemma 4.7:

limtց0

Y(t) − Eyt

t

∣∣∣Ω

= limtց0

(Eyt) T -1t − Eyt

t

∣∣∣Ω

= limtց0

Y(t) − Y(t) Tt

t

∣∣∣Ω

= −∇y(Ω) · v . ¤

The shape derivative identity of Proposition A.7 is important. In the books [170,202], this identity is taken as the definition for the shape derivative.

A.2.3 Shape function involving restriction

Let us next consider example shape functions. As an almost trivial example,we consider the restriction shape function [202, p. 99] defined as

y(Ω) = ψ∣∣Ω

,


where ψ ∈ W1,1(D). In this case, we have W(Ω) = W1,1(Ω) and we can easilyfind an extension function Y such that (A.4) holds: set Y(t) = ψ on D. Hence,the shape derivative is

y′(Ω)(v) = ∂tY(0)∣∣Ω

= 0 ,

that is, it vanishes. This is expected since y does not vary if the domain isperturbed. On the other hand, the material derivative is given by

y(Ω)(v) = ∇ψ∣∣Ω· v ∈ L1(Ω) .

Note that the identity (A.12) is indeed satisfied.Next, let us consider the same restriction shape function, but scaled by the

volume of its domain, i.e.,

y(Ω) = ψ∣∣Ω

∫

ΩdΩ ,

where ψ ∈ W1,1(D). Using the extension Y(t) = ψ∫

ΩtdΩt on D, the shape

derivative follows easily from the shape derivative of the volume integral:

y′(Ω)(v) = ψ∣∣Ω

∫

Γv · n .

In this case, the shape derivative does not vanish, unless the volume re-mains constant. In view of Proposition A.5 concerning the structure of theshape derivative, it is clear that the map g(Ω) in (A.9) can be identified withψ|Ω 〈1, ·〉Γ.

A.2.4 Shape function involving Dirichlet problem

A more typical example is the shape function involving a Dirichlet problem;see [202, p. 153]. For each domain Ω, let y(Ω) be the solution of the boundaryvalue problem

−∆y(Ω) = f in Ω ,

y(Ω) = 0 on Γ ,(A.13)

where f ∈ L2(D). We assume that Ω is of class C1,1, hence it follows fromelliptic regularity theory that y(Ω) ∈ H2(Ω); see [120], for example. For thisnontrivial example, it follows that the shape derivative is also the solution of aboundary value problem. We summarize this in the following proposition.

178 Appendix

Proposition A.8 The shape function y defined as the solution of (A.13) is shapedifferentiable at Ω. Its shape derivative y′(Ω)(v) ∈ H1(Ω) is the weak solutionof the boundary value problem

−∆y′(Ω)(v) = 0 in Ω ,

y′(Ω)(v) = −∂ny(Ω) v · n on Γ .

Before we provide a proof, note that the shape derivative conforms to the struc-ture as stated in Prop. A.5. That is, it depends only on v · n at Γ.

Proof A weak formulation of the Dirichlet problem at Ωt is given by: Findy(Ωt) ∈ H1

0(Ωt) such that∫

Ωt

∇y(Ωt) · ∇φ =∫

Ωt

f φ ∀φ ∈ H10(Ωt) .

To obtain the equation for the shape derivative, it is common practice to gothrough the material derivative and use identity (A.12), see [170] and [202,p. 155]. Here, we derive the shape derivative using the main definition (A.6).We start by noting that the test function φ ∈ H1

0(Ωt) depends on t through itszero trace at Γt. To bypass this complication, we note that (·) Tt transportsall functions in H1

0(Ωt) onto H10(Ω) and (·) T -1

t vice versa; see Lemma 4.7.Hence,

∫

Ωt

∇y(Ωt) · ∇(φ T -1t ) =

∫

Ωt

f (φ T -1t )

for all φ ∈ H10(Ω). Denoting by Y(t) an extension in H1

0(D) of y(Ωt), we clearlyhave

∫

Ωt

∇Y(t) · ∇(φ T -1t ) =

∫

Ωt

f (φ T -1t ) .

At this point, we cannot differentiate this expression with respect to t usingReynolds’ theorem, see Corollary 4.9, since the integrand of the left integral isnot smooth enough. Therefore, we apply an integration by parts, and considersufficiently smooth test functions φ ∈ H1

0(Ω) ∩ C∞(Ω) and sufficiently smoothtransformations Tt. Furthermore, noting that Y(t) = y(Ωt) = 0 on Γt, it holdsthat ∫

Ωt

−Y(t) ∆(φ T -1t ) =

∫

Ωt

f (φ T -1t ) .

Although the integrand of the right integral is not smooth enough to use Corol-lary 4.9, we can differentiate this integral using Prop. 6.1. Hence, differentiatingthe equation on both sides gives

∫

Ω

(− ∂tY(0) ∆φ + Y(0) ∆(∇φ · v)

)−

∫

ΓY(0) ∆φ v · n =

∫

Ω− f ∇φ · v ,


where we used Lemma 4.7. Noting that Y(0) = y(Ω) = 0 on Γ, we obtain

−∫

Ωy′(Ω)(v) ∆φ =

∫

Ω

(∇y(Ω) · ∇(∇φ · v) − f ∇φ · v

).

Note that the right member does not generally vanish, because ∇φ · v 6= 0on Γ. In fact, as −∆y(Ω) = f ∈ L2(Ω), an integration by parts reveals that theright side can be identified as a duality product (pairing H−1/2(Γ) and H1/2(Γ))involving the normal derivative of y(Ω). Moreover, for y(Ω) ∈ H2(Ω) we have∂ny(Ω) ∈ H1/2(Γ) and we obtain

−∫

Ωy′(Ω)(v) ∆φ =

∫

Γ∂ny(Ω) ∂nφ v · n .

where we used that on Γ, ∇φ = ∂nφ n on account of φ = 0 on Γ. The proof isobtained by applying integrations by parts. ¤

180 Appendix

Ω

Γ1

x0

x1x2x3

τ1

Figure A.2: Illustration of the singular points xi, a boundary segment Γi and a vectorτi (composed of the two unit tangent vectors) for a domain Ω with a piecewise smoothfree boundary Γ.

A.3 Additional analysis of the dual problem of

Sec. 6.4

In this section we provide auxiliary results concerning the dual problemin (6.20). We consider the two-dimensional case; the extension to three (andhigher) dimensions follows similarly.

A.3.1 Boundedness of the tangential divergence term

Here, we verify the boundedness of the functional v 7→ K(v) :=∫Γ

divΓ(g v δρ m) (see (6.18)) on H10,ΓD

(Ω). Observe that, in two dimensions, any

v ∈ H10,ΓD

(Ω) satisfies the constraint |v(xi)|2 < ∞ for singular points xi ∈ Γ,i = 0, 1, . . . . The integral in K(v) can be integrated by parts (piecewise) re-sulting in contributions at these singular points. This requires the tangentialStokes theorem on piecewise smooth boundaries; see Sec. A.1. Let us denotethe boundary segments between xi and xi+1 by Γi+1; see Fig. A.2. Note thatΓ = int∪iΓi. Furthermore, let τi denote the vector at xi composed of the twounit tangent vectors τ|Γi

and τ|Γi+1at xi outward with respect to their segment,

i.e.,

τi := τ∣∣Γi+1

(xi) + τ∣∣Γi

(xi) .

For points xi at the boundary of Γ, τi is equal to the outward tangent vector.Then, the following tangential identity holds

∫

ΓdivΓ θ = ∑

i

θ(xi) · τi + ∑i

∫

Γi

κ θ · n ; (A.14)

A.3 Additional analysis of the dual problem of Sec. 6.4 181

for suitable θ : Γ → R2; cf. (A.2). Applying this to the integral in K(v), we have

K(v) =∫

ΓdivΓ(g v δρ m) = ∑

i

(g v δρ)(xi) m(xi) · τi + ∑i

∫

Γi

κ g v δρ ,

where we used m · n = 1 on Γi. This leads to the bound

K(v) ≤ ∑i

|g(xi) δρ(xi)| |v(xi)| |m(xi) · τi| + ∑i

∣∣∣∫

Γi

κ g v δρ∣∣∣ ,

It now follows that K is bounded on H10,ΓD

(Ω) because the first sum is bounded

in view of |v(xi)|2 < ∞ and the second sum is bounded for v ∈ H1(Ω).

A.3.2 Wellposedness by coercivity

On account of the Lax-Milgram theorem, wellposedness of the dual prob-lem (6.20) follows from coercivity with respect to H1

0,ΓD(Ω) of the correspond-

ing bilinear form3

B(δu, z) :=∫

Ω∇δu · ∇z +

∫

Γ


).

Using the tangential identity (A.14), we can rewrite the tangential divergenceterm as

∫

ΓdivΓ(z δu m) = ∑

i

(z δu

)(xi) m(xi) · τi + ∑

i

∫

Γi

κ z δu . (A.15)

It follows that

B(z, z) =∫

Ω|∇z|2 + ∑

i

∫

Γi

1g ( f + ∂ng + κ g) z2 + ∑

i

z2(xi) m(xi) · τi .

In view of the results in Sec. A.3.1 a suitable norm on H10,ΓD

(Ω) is given by

‖z‖2H1

0,ΓD(Ω) :=

∫

Ω|∇z|2 + ∑

i

|z(xi)|2 .

It is now clear how to obtain sufficient conditions to ensure that B is coerciveon H1

0,ΓD(Ω). If the domain is convex at the singular points in the sense that

m(xi) · τi ≥ C > 0 for all i, and if furthermore ( f + ∂ng)/g + κ ≥ 0 on Γi forall i, then

B(z, z) ≥ C ‖z‖2H1

0,ΓD(Ω) ∀z ∈ H1

0,ΓD(Ω) .

3The necessary continuity of the linear form is straightforward and continuity of the bilinearform follows using similar arguments as in Sec A.3.1 for the tangential divergence term.

182 Appendix

Hence, under these conditions, the dual problem (6.20) has a unique solutionin H1

0,ΓD(Ω).

We remark that similar conditions on the data are given in [78, 79]. Relatedboundary value problems with so-called oblique boundary conditions involv-ing tangential derivatives are analyzed in [120, p. 167] and [54, p. 398].

A.4 Proofs 183

A.4 Proofs

A.4.1 Equivalent inf-sup conditions

Proposition A.9 Let U and V be reflexive Banach spaces and let B : U ×V → R

denote a continuous bilinear form. Assume that B satisfies the inf-sup condi-tions, i.e., there exists an inf-sup constant cB such that

infu∈U\0

supv∈V\0

B(u, v)

‖u‖U ‖v‖V

≥ cB > 0 , (A.16a)

∀v ∈ V ,(∀u ∈ U , B(u, v) = 0

)⇒

(v = 0

). (A.16b)

Then B satisfies the adjoint inf-sup conditions with the same adjoint inf-supconstant, i.e.,

infv∈V\0

supu∈U\0

B(u, v)

‖u‖U ‖v‖V

≥ cB > 0 , (A.17a)

∀u ∈ U ,(∀v ∈ V , B(u, v) = 0

)⇒

(u = 0

). (A.17b)

Proof We follow the proof given by Melenk and Schwab in an ETH researchreport; see [162, Prop. A.2]. To show that (A.16a) implies (A.17b), we first notethat (A.17b) is equivalent to

supv∈V

B(u, v) > 0 ∀u ∈ U \ 0 . (A.17b*)

Then, (A.17b*) follows directly from (A.16a). To show (A.17a), we shall con-struct a specific u for each v. Consider an arbitrary v ∈ V \ 0. Then by astandard corollary to the Hahn–Banach Theorem, see [4, p. 82] or [193, p. 58],there exists a continuous linear functional Lv ∈ V∗ such that ‖Lv‖V∗ = 1 andLv(v) = ‖v‖V . Now let uv ∈ U be the solution of

B(uv, w) = ‖v‖V Lv(w) ∀w ∈ V .

The solution uv exists and is unique since B satisfies the inf-sup conditionsin (A.16), and uv satisfies the a priori estimate (see Theorem 3.A):

‖uv‖U ≤ 1cB

‖v‖V ‖Lv‖V∗ = 1cB

‖v‖V .

Furthermore, note that

B(uv, v) = ‖v‖V Lv(v) = ‖v‖2V .

Hence, using these results, we obtain (A.17a):

infv∈V\0

supu∈U\0

B(u, v)

‖u‖U ‖v‖V

≥ infv∈V\0

B(uv, v)

‖uv‖U ‖v‖V

≥ cB . ¤

184 Appendix

Γt

Γθh

Γ0

αθh

(x + θh

1 (x))αt

(x + θh

1 (x) + t δθ1(x))

θh(x)t δθ(x)

x

Figure A.3: Perturbation of the free boundary and associated perturbation of the eleva-tion.

A.4.2 Linearization of the elevation goal

The elevation goal functional is defined in Sec. 5.2.4 as

Qelev(θ) =∫

Γ0

qelev αθ ,

where αθ is, for a specific domain Ωθ , the vertical deviation of the free boundaryΓθ from Γ0. In this section, we present the linearization of Qelev at θh ∈ Θ bydifferentiation of αθ .4 For the sake of clarity, we consider the two-dimensionalcase. However, the derivation extends without difficulty to three (and higher)dimensions.

Given Γ = Γθh and its associated elevation function αθh , consider the pertur-bation Γt := Γθh+t δθ and αt := αθh+t δθ ; see Fig. A.3. Without loss of generality,we view θh and δθ as functions of the horizontal coordinate x only. Next, fix-ing a particular x, we observe from Fig. A.3 that the perturbed elevation can begiven at a shifted location:

αt

(x + θh

1(x) + t δθ1(x))

= α0(x + θh

1(x))+ t δθ2(x) .

This is the key identity to obtain the derivative of αθ . By introducing the Γ0transformation Tθh

1 ,t(x) := x + θh1(x) + t δθ1(x) and denoting Tθh

1:= Tθh

1 ,0, itholds that

⟨∂θαθh , δθ

⟩= lim

t→01t

(αt − α0

)= δθ2 T -1

θh1

+ limt→0

1t

(α0 Tθh

1 T -1

θh1 ,t

− α0)

As Tθh1 T -1

θh1 ,t

=(

Id + t (δθ1 Tθh1)) -1, the limit on the right hand side is equal to

−α′θh δθ1 T -1

θh1

(for a.e. x). Hence, we obtain

⟨∂θαθh , δθ

⟩=

(δθ T -1

θh1

)·(−α′

θh , 1)

.

4Alternatively, one could reformulate Qelev as a shape functional involving a domain integraland use a standard shape derivative.

A.4 Proofs 185

The elevation derivative, α′θh , can be written in terms of θh by differentiating

the identity αθh Tθh1

= θh2 , yielding α′

θh = θh2′/(1 + θh

1′) T -1

θh1

. Subsequently,

we have Qelev′(θh)(δθ) =∫

Γ0qelev (

δθ · (−θh2′/(1 + θh

1′), 1)

) T -1

θh1

. Performing

a change of variables through the Γ0-map Tθh1, thereby picking up a Jacobian

(1 + θh1′), we obtain the expression


Γ0

(qelev Tθh

1

)δθ ·

(−θh

2′, 1 + θh

1′) .

This Γ0-supported integral can be transformed to Γ by the transformationTθh . Carrying out this transformation, we pick up the Jacobian

((θh

2′)2 + (1 +

θh1′)2)−1/2 T -1

θh which nicely combines with(−θh

2′, 1 + θh

1′) T -1

θh to form theunit normal vector n. Furthermore, recalling δθ = δθ T -1

θh , we finally obtainthe concise result:


Γqelev(x) δθ · n .

A.4.3 Vanishing shape derivative of domain integral

In this section, we provide a proof of Prop. 6.1. The proof follows by showingthat the limit t → 0 of a suitable bound on the difference quotient (J(Ωt) −J(Ω))/t vanishes. First, we note that the difference in J can be written as

J(Ωt) − J(Ω) =∫

Ωt

φ1 φ2 −∫

Ωφ1 φ2=

∫

∆Ωt

β φ1 φ2

where ∆Ωt := (Ωt ∪ Ω) \ (Ωt ∩ Ω) is the t-dependent set consisting of thenonoverlapping parts of Ωt and Ω; see Fig. A.4 for an illustration in two di-mensions. Furthermore, β is a scalar function that is −1 for the Ω part and 1 forthe Ωt part. Applying the Cauchy-Schwartz inequality, we have

∣∣J(Ωt) − J(Ω)∣∣ ≤ ‖φ1‖L2(∆Ωt)

‖φ2‖L2(∆Ωt)≤ ‖φ1‖L2(D) ‖φ2‖L2(∆Ωt)

.

An upper bound to the t-dependence of the second norm follows from the fol-lowing Poincaré inequality (cf. [53, Eq. 2.21]):

Lemma A.10 For all φ2 ∈ H1(D) with φ2 = 0 on Γ, it holds that

‖φ2‖L2(∆Ωt)≤ C t ‖∇φ2‖L2(∆Ωt)

,

for some constant C independent of t.

186 Appendix

Ω

∆Ωt

Γ

Γtt δθ

Figure A.4: The perturbation of Ω by t δθ creates the t-dependent strip set ∆Ωt.

∆Ωt∆Ωt

Γ Γ

S

Figure A.5: The boundary Γ of the strip set ∆Ωt is flattened under the Lipschitz-continuous map S.

The proof now follows straightforwardly:

limt→0

∣∣J(Ωt) − J(Ω)∣∣/t ≤ lim

t→0C ‖φ1‖L2(D) ‖∇φ2‖L2(∆Ωt)

= 0 .

Proof (of Lemma A.10) Let S : D → D denote a bounded Lipschitz-continuoustransformation that maps Γ to the flat surface Γ and ∆Ωt to ∆Ωt; see Fig. A.5.Define φ2 := φ2 S -1. Then φ ∈ H1(D) with φ = 0 on Γ (see [120, p. 21] or [59,p. 406]). Note that the domain ∆Ωt is bounded by a t-dependent cartesian box.The following Poincaré inequality with a t-dependent Poincaré constant holdsfor such slender domains:

‖φ2‖L2(∆Ωt)≤ C t ‖∇φ2‖L2(∆Ωt)

;

see [124], for example. Substituting φ2 := φ2 S -1 and transforming back to∆Ωt yields

( ∫

∆Ωt

φ22 det DS

)1/2

≤ C t

( ∫

∆Ωt

|DS -T∇φ2|2 det DS

)1/2

.

Noting that det DS and the components of DS -T are in L∞(D) we finally obtainthe Poincaré inequality (with a different constant C). ¤

List of Theorems

Theorem 3.A Banach–Necas–Babuška Theorem on Well-Posedness . . . . 25Theorem 3.B Céa’s Lemma on Norm Convergence . . . . . . . . . . . . . 26Theorem 3.C Error–Residual Equivalence . . . . . . . . . . . . . . . . . . 27Theorem 3.D Babuška–Miller Theorem on Quantity Convergence . . . . 29Theorem 3.E Generalized Taylor’s Theorem . . . . . . . . . . . . . . . . . 43Theorem 3.F Error Estimates near Nonsingular Solutions . . . . . . . . . 46Theorem 3.G Quantity Convergence near Nonsingular Solutions . . . . . 48Theorem 3.H Linearized-Adjoint-Based Error Representation . . . . . . . 50Theorem 4.A Transformation by Velocity Field . . . . . . . . . . . . . . . . 59Theorem 4.B Transformation by Perturbation of Identity . . . . . . . . . . 61Theorem 4.C Hadamard–Zolésio Structure Theorem . . . . . . . . . . . . 65Theorem 4.D Equivalent Shape Functional Derivatives . . . . . . . . . . . 67Theorem 4.E Shape Derivative of Domain Integral . . . . . . . . . . . . . 69Theorem 4.F Shape Derivative of Boundary Integral . . . . . . . . . . . . 71Theorem 5.A Error Representation Based on z0 . . . . . . . . . . . . . . . 82Theorem 5.B Error Representation Based on z . . . . . . . . . . . . . . . . 85Theorem 5.C Dual Boundary Condition . . . . . . . . . . . . . . . . . . . 90Theorem 6.A Dual Consistency: Smooth Free Boundaries . . . . . . . . . 110Theorem 6.B Dual Consistency . . . . . . . . . . . . . . . . . . . . . . . . 115Theorem 7.A Domain-Map Linearized Dual-Based Error Representation 135Theorem 8.A Shape-Linearized Dual-Based Error Representation . . . . . 155Theorem A.A Tangential Stokes’ Theorem . . . . . . . . . . . . . . . . . . . 170

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Summary

Goal-Adaptive Discretization of Fluid–Structure Interaction

Kristoffer G. VAN DER ZEE

The simulation of complex physical phenomena, such as fluid–structure in-teraction, appears to be within reach in view of the significant progress incomputing power over the last decades. Yet, we are still far away from whatis desirable in an evermore-demanding, science-and-technology-based so-ciety. If, however, we are modest with what we need of a physical systemin that we ask for specific goal quantities instead of the entire solution, thenwe are able to save tremendous amounts of computing effort.

Fluid–structure interaction is pertinent to many engineering applications. Classically, themain motivation for the numerical simulation of fluid–structure interaction has been theprediction of aeroelastic phenomena in aerospace and civil engineering applications. Animportant recent application concerns hemodynamics (vascular fluid dynamics or bloodflow) in biomechanical applications. To enable the simulation of complex practical ap-plications, many computational techniques have been developed and analyzed since the1990s.

Numerical procedures for fluid–structure interaction problems generally requirevast computational effort. Typically, most of the computational resources are consumed bythe fluid subsystem while practical interest is restricted to a prescribed response quantityof the structure subsystem, rather than full resolution of the complete coupled problem.In fact, one is often not so much interested in the solution itself, but uses the computed solution

to quantify a certain goal. Example goals in aeroelastic computations are global forces onthe structure such as the lift or drag, or the energy that is transferred from the fluid tothe structure, which is the critical factor in aeroelastic stability. Examples in biomechan-ics are the wall shear stress in the vicinity of aneurysms or, the volumetric flow rate atarterial cross sections.

Goal-oriented adaptive discretization strategies can offer a significant efficiency improve-ment in simulations, where interest is restricted to a particular goal-quantity of interest.Developed since the late 1990s, these strategies start with a coarse discretization, af-ter which refinements are made which substantially benefit to the accuracy of the goalquantity. This adaptive-refinement procedure results in an optimal discretization of the

208 Summary

solution variables specifically for the goal under consideration. Goal-oriented adaptivestrategies rely on local refinement indicators to guide the adaptive procedure. Theserefinement indicators are obtained from duality-based a-posteriori error estimates for thegoal functional of interest. To compute such goal-oriented error estimates, one requiresthe solution of a dual problem. For nonlinear problems, this dual problem is based onthe linearized adjoint operator.

A crucial complication in the application of goal-oriented adaptive discretization tofluid–structure-interaction problems is that the derivation of the linearized adjoint andthe corresponding dual problem are highly nontrivial on account of the free-boundary

character of fluid–structure interaction. Indeed, the fluid domain in fluid–structure-interaction problems is unknown a priori and is to be determined as part of the solution.This domain dependence induces a complex nonstandard nonlinearity.

The aim of this thesis is the rigorous derivation of dual problems for the goal-oriented adaptive discretization of fluid–structure interaction and, in general, free-boundary problems. To derive the corresponding linearized-adjoint operator of free-boundary problems, we present two linearization approaches to deal with the domaindependence, both of which rely on concepts from shape differential calculus. In the domain-

map linearization approach, the free-boundary problem is first transformed to a fixed ref-erence domain. Essentially, a linearization is then performed with respect to the domainmap. In the shape-linearization approach, the domain dependence is included by meansof shape functionals. This dependence is then linearized using shape-derivative tech-niques.

After providing reviews of the theory of goal-oriented error analysis and adaptiv-ity, and of the fundamentals of shape differential calculus, we consider applicationsto model problems. To focus on the free-boundary character, we first apply the lin-earization approaches to the Bernoulli free-boundary problem. We show that the domain-map linearized dual problem corresponds to a boundary value problem with a nonlo-

cal boundary condition. The shape-linearized dual problem is similar, but has a local

boundary condition which depends on the curvature of the boundary. We then applythe linearization approaches to a steady fluid–structure interaction problem consistingof an incompressible flow coupled to a string (membrane) model. We show that the dualproblem by domain-map linearization corresponds to a linear coupled fluid–structureproblem with a kinematically straightforward coupling. The complementary couplingcondition, however, is nonstandard and nonlocal, similar to what is obtained for theBernoulli free-boundary problem. On the other hand, the shape-linearized dual prob-lem again has local coupling conditions. We present numerical experiments for bothmodel problems and both linearization approaches, that illustrate the effectivity of theimplied goal-oriented error estimates and prove their usefulness to guide goal-orientedadaptive mesh refinement.

Samenvatting

Doel-Adaptieve Discretisatie van Stroming–Constructie Interactie

(Goal-Adaptive Discretization of Fluid–Structure Interaction)

Kristoffer G. VAN DER ZEE

De simulatie van complexe fysische fenomenen, zoals de interactie tus-sen stromingen en constructies, lijkt binnen handbereik vanwege de enor-me vooruitgang in computerkracht gedurende de laatste decennia. Maarwe zijn nog ver weg van wat gewenst is in de immer-meer-eisendewetenschap-en-technologie-gebaseerde gemeenschap. Echter, als we be-scheiden zijn met wat we nodig hebben van een fysisch systeem, door ge-noegen te nemen met specifieke grootheden van interesse in plaats de volle-dige oplossing, dan kunnen we enorme hoeveelheden rekenkracht en -tijdbesparen.

De interactie tussen gas- of vloeistofstromingen en constructies, ofwel stoming–constructie interactie (fluid–structure interaction), kent verscheidene technologische toe-passingen. De klassieke motivatie voor de numerieke simulatie van stoming–constructieinteractie is de voorspelling van aeroelastische verschijnselen in de luchtvaart-,ruimtevaart- en civiele techniek. Een belangrijke recente toepassing is hemodynami-ca (stroming in bloedvaten) in biomechanische toepassingen. Om dergelijke complexepraktische toepassingen mogelijk te maken, zijn sinds de jaren ’90 veel rekentechniekenontwikkeld en onderzocht.

Numerieke methoden voor stroming–constructie interactie vereisen in het algemeenenorme hoeveelheden rekenkracht. Doorgaans wordt een groot deel van deze hoeveel-heid gebruikt door het stroming subsysteem, terwijl de interesse zich vaak beperkt totvoorschreven grootheden van het constructie subsyteem, in plaats van de volledige op-lossing van het gekoppelde probleem. In feite is men vaak niet zo geïnteresseerd inde oplossing zelf, maar gebruikt men de berekende oplossing om een bepaald doel ofgrootheid te quantificeren. Voorbeelden van dergelijke interesse grootheden in de ae-roelasticiteit zijn globale krachten op de constructie zoals de lift of weerstand, of deenergie-overdracht tussen de stroming en de constructie, welke van cruciaal belang isbij aeroelastische stabiliteit. Voorbeelden in de biomechanica zijn de schuifspanningenaan de bloedvatwand in de buurt van aneurismen of de massastroom door bloedvaten.

210 Samenvatting

Doel-geörienteerde adaptieve discretisatie technieken (goal-oriented adaptive discreti-

zation strategies) kunnen een significante verbetering in efficiëntie opleveren van simu-laties waarin de interesse zich beperkt tot een bepaalde grootheid van interesse (goal-

quantity of interest). Doel-geörienteerde adaptieve technieken zijn eind jaren ’90 ontwik-keld. In dergelijke technieken wordt een initieel grove discretisatie alleen daar verfijndwaar het de nauwkeurigheid van de interesse grootheid substantieel doet toenemen.Deze adaptieve verfijningsprocedure leidt tot een optimale discretisatie van de oplos-singsvariabelen specifiek voor de betreffende interesse. Doel-geörienteerde adaptievetechnieken gaan uit van lokale verfijningsindicatoren die de procedure aansturen. Dezeverfijningsindicatoren worden verkregen uit duaal-gebaseerde a posteriori foutschattin-gen (duality-based a-posteriori error estimates). Om dergelijke foutschattingen te berekenenmoet men de oplossing van een duaal probleem oplossen. Voor niet-lineaire problemenis het duale probleem gebaseerd op de gelineariseerde geadjungeerde (linearized adjoint)operator.

Een cruciale complicatie in de toepassing van doel-geörienteerde adaptieve discreti-saties op stroming–constructie interactie is de afleiding van de gelineariseerde geadjun-geerde en het geassocieerde duale probleem. Deze afleiding is zeker niet voor de handliggend vanwege het vrije-rand karakter van stroming–constructie interactie. Het stro-mingsdomein in stroming–constructie interactie is van te voren namelijk onbekend enmaakt deel uit van de oplossing. Deze domeinafhankelijkheid induceert een complexeen ongebruikelijke niet-lineariteit.

Het doel van deze dissertatie is de rigoreuze afleiding van duale problemen ten be-hoeve van doel-geörienteede adaptieve discretisatie van stroming–constructie interactieen vrije-rand problemen in het algemeen. Om de geassocieerde gelineariseerde geadjun-geerde af te leiden, introduceren we twee linearisatie methoden om de domeinafhanke-lijkheid aan te pakken. Beide methoden gaan uit van concepten van de geometrischedifferentiaalrekening. In de domein-afbeelding linearisatie methode (domain-map linea-

rization approach) wordt het vrije-rand probleem eerst getransformeerd naar een vast re-ferentie domein. In wezen vindt de linearisatie dan plaats met betrekking to de domeinafbeelding. In de geometrie-linearisatie methode (shape linearization approach), wordt dedomainafhankelijkheid inbegrepen middels geometrische functionalen. Deze afhanke-lijkheid kan dan worden gelineariseerd met technieken van geometrische afgeleiden.

Na het behandelen van overzichten van de theorie van doel-geörienteerde foutana-lyse en adaptiviteit, en van de grondbeginselen van de geometrische differentiaalre-kening, beschouwen we toepassingen op modelproblemen. Om nadruk te leggen ophet vrije-rand karakter passen we eerst de linearisatie methoden toe op het Bernoullivrije-rand probleem. We laten zien dat het duale probleem verkregen met de domein-afbeelding linearisatie methode overeenkomt met een randwaarde probleem met eenniet-lokale randconditie. Het duale probleem verkregen met de geometrie-linearisatiemethode is vergelijkbaar, maar heeft een lokale randconditie die afhangt van de krom-ming van de domeinrand. We passen dan de linearisatie methoden toe op een stationairstroming–constructie interactie-probleem bestaande uit een incompressibele strominggekoppeld aan een snaar (membraan) model. We laten zien dat het duale probleem,verkregen met domein-afbeelding linearisatie, overeenkomt met een lineair gekoppelde

Samenvatting 211

stroming–constructie probleem met een kinematisch eenvoudige koppeling. De comple-mentaire koppelingsconditie is echter niet-standaard en niet-lokaal, vergelijkbaar methet resultaat bij het Bernoulli vrije-rand probleem. Het duale probleem verkregen metgeometrie linearisatie bevat weer lokale koppelingscondities. We presenteren resultatenvan numerieke experimenten voor beide modelproblemen en beide linearisatie metho-den. Deze resultaten illustreren de effectiviteit van de geïmpliceerde doel-geörienteerdefoutschattingen en demonstreren hun toepasbaarheid in doel-geörienteerde adaptieveroosterverfijning.

Curriculum Vitae

Kristoffer George van der Zeeborn on May 21, 1980 in Lelystad, The Netherlands

1992 – 1998 VWO (cum laude),I.S.G. Arcus, Lelystad, The Netherlands

1998 – 2001 B.Sc. Aerospace Engineering (cum laude),Delft University of Technology, The Netherlands

2001 – 2004 M.Sc. Aerospace Engineering (cum laude),Delft University of Technology, The Netherlands

Aug. 2002– Dec. 2002

Visiting researcher,Center for Aerospace Structures,University of Colorado at Boulder, USA

2004 – 2009 Ph.D.-candidate,Delft University of Technology, The Netherlands

2007 SIAM Student Paper Prize

Febr. 2009– June 2009

Postdoctoral researcher,Department of Mechanical Engineering,Eindhoven University of Technology, The Netherlands

≥ July 2009 Postdoctoral researcher,Institute for Computational Engineering and Sciences,University of Texas at Austin, USA

Stellingenbehorende bij het proefschrift

Goal-Adaptive Discretization of Fluid–Structure InteractionK.G. van der Zee, 5 juni 2009

1. Het is mogelijk om interesse grootheden van vrije-rand problemen optimaal teberekenen met doel-georienteerde adaptieve methoden, mits gebruik wordt gemaaktvan exact-gelineariseerde geadjungeerden.

2. Als een probleem (in zwakke zin) kan worden geformuleerd in termen van geome-trie functionalen, dan bestaat er een elegante linearisatie die kan worden afgeleidmiddels technieken van de geometrische analyserekening.

3. Voor vrije-rand problemen is het niet aan te raden om zeer nauwkeurige dualeoplossingen te berekenen voor het aansturen van doel-georienteerde adaptieve pro-cedures. Omdat linearisatie plaatsvindt rond domeingeometrieën die doorgaandsniet glad zijn, laten de duale oplossingen zich slecht benaderen.

4. Om volledig inzicht in een primaal probleem te krijgen, dient men het duale pro-bleem te bestuderen.

5. Implementaties met optimale rekencomplexiteit zijn nutteloos zonder een onder-liggende optimale discretisatie. Optimale discretisaties zijn nutteloos zonder eenzorgvuldige specificatie van het doel van de berekening.

6. Het doel van numerieke wetenschappers is niet te rekenen.

7. Wetenschappelijk onderzoek is het abstract maken van ideeën. Mocht je dromen vanabstractie, dan zou je wetenschap moeten doen. Doe je niet aan abstractie, dan kunje slechts dromen van wetenschap.

8. Goede wetenschappers komen in twee varianten voor: grondverleggers enopruimers. Grondverleggers lossen de meest ingewikkelde problemen op en pu-bliceren snel, artikel na artikel. Opruimers doorgronden stof volledig, zoeken naarunificatie en publiceren overzichtsartikelen en boeken. Het zou beter zijn als grond-verleggers af en toe hun eigen rommel opruimden.

9. Voor een toegewijde promovendus geldt dat er een grote positieve constante c bestaatzo dat de volgende ongelijkheid geldt: Uren te laat op werk ≤ c × Productiviteit.

10. In de huidige wereld waar kennis wordt geroemd, wordt het toegeven van wat menniet weet ondergewaardeerd.

Deze stellingen worden opponeerbaar en verdedigbaar geacht en zijn als zodanig goedgekeurddoor de promotor, Prof.dr.ir. R. de Borst.

Propositionsappended to the dissertation

Goal-Adaptive Discretization of Fluid–Structure InteractionK.G. van der Zee, June 5, 2009

1. Goal quantities of interest of free-boundary problems can be computed optimally bygoal-oriented adaptive methods, only if one uses exact linearized adjoints.

2. If a problem can be (weakly) formulated using shape functionals, then an elegantlinearization exists which can be derived using shape-calculus techniques.

3. For free-boundary problems, it is not recommended to compute highly accurate dualsolutions to drive goal-oriented adaptive procedures. Since the linearization gen-erally takes place at nonsmooth geometries, the dual solutions are poorly approx-imable.

4. To fully comprehend a primal problem, one should study the dual problem.

5. Implementations with optimal computational complexity are meaningless withoutan underlying optimal discretization. Optimal discretizations are meaningless with-out a precise specification of the goal of the computation.

6. The goal of computational scientists is not to compute.

7. Scientific research is the abstraction of ideas. If you dream of abstraction, you shoulddo science. If you don’t abstract, you can only dream of doing science.

8. Great scientists come in two variants: groundbreakers and cleaners. Groundbreakerssolve the most complicated problems publishing one paper after the other in a rapidfashion. Cleaners delve on subjects, search for unification and publish reviews andbooks. It would be better to have groundbreakers once in a while clean up afterthemselves.

9. For a dedicated PhD-candidate, there is a large positive constant c such that the fol-lowing a priori estimate holds: Hours too late at work ≤ c × Productivity.

10. In our world where knowledge is praised, admitting not to know is undervalued.

These propositions are considered opposable and defendable and as such have been approvedby the supervisor, Prof.dr.ir. R. de Borst.

The simulation of complex physical phenomena, such as fluid–structure interaction, appears to be within reach in view of the significant progress in computing power over the last decades.Yet, we are still far away from what is desirable in an evermore-demanding, science-and-technology-based society. If, however, we are modest with what we need of a physical system in that we ask for specific goal quantities instead of the entire solution, then we are able to save tremendous amounts of computing effort.

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· Dit proefschrift is goedgekeurd door de promotor: Prof.dr.ir. R. de Borst Copromotor: Dr.ir....

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