OPTIMALschoetzau/reportsDS/Castillo...]; [4 see also the review of t elopmen dev uous tin discon...

.

OPTIMAL A PRIORI ERROR ESTIMATES FOR THE

hp-VERSION OF THE LOCAL DISCONTINUOUS GALERKIN

METHOD FOR CONVECTIONDIFFUSION PROBLEMS

PAUL CASTILLO, BERNARDO COCKBURN, DOMINIK SCH

OTZAU, AND CHRISTOPH

SCHWAB

Abstra t. We study the onvergen e properties of the hp-version of the lo al

dis ontinuous Galerkin nite element method for onve tion-diusion prob-

lems; we onsider a model problem in a one-dimensional spa e domain. We

allow arbitrary meshes and polynomial degree distributions and obtain upper

bounds for the energy norm of the error whi h are expli it in the mesh-width

h, in the polynomial degree p, and in the regularity of the exa t solution. We

identify a spe ial numeri al ux for whi h the estimates are optimal in both h

and p. The theoreti al results are onrmed in a series of numeri al examples.

Math. Comp., Vol. 71, pp. 455478, 2002

1. Introdu tion

This paper ontains the rst a priori error estimate of the hp-version of the so- alled

lo al dis ontinuous Galerkin (LDG) nite element method for onve tion-diusion

problems. Su h an error analysis, whi h takes into a ount both the mesh-size of

the element, h, and the degree of the approximating polynomial in it, p, is quite

relevant for the LDG method sin e, being a lo ally onservative method that does

not require any inter-element ontinuity, it is ideally suited for hp-adaptivity. In this

paper, we onsider a model onve tion-diusion equation in one spa e dimension

with Diri hlet boundary onditions and obtain, for a spe ial hoi e of the numeri al

uxes dening the LDG method, a priori error estimates that are optimal both in

h and p, even for p = 0; all other error estimates available in the urrent literature

are suboptimal in both h and p and do not give a rate of onvergen e for p = 0.

The LDG method was introdu ed by Co kburn and Shu in [11 as an extension to

general onve tion-diusion problems of the numeri al s heme for the ompressible

Navier-Stokes proposed by Bassi and Rebay in [1. This s heme was in turn an

extension of the Runge-Kutta dis ontinuous Galerkin (RKDG) method developed

by Co kburn and Shu [10, 9, 8, 6, 12 for non-linear hyperboli systems. For a fairly

omplete set of referen es on RKDG and LDG methods see the short monograph

by Co kburn [4; see also the review of the development of dis ontinuous Galerkin

methods by Co kburn, Karniadakis, and Shu [7.

To put our result under proper perspe tive, let us brie y des ribe the relevant

results available in the urrent literature. There are only a few a priori error

1991 Mathemati s Subje t Classi ation. Primary 65N30; Se ondary 65M60.

Key words and phrases. Dis ontinuous Galerkin Methods, hp-Methods, Conve tion-Diusion.

The se ond author was partially supported by the National S ien e Foundation (Grant DMS-

9807491) and by the University of Minnesota Super omputer Institute. The third author was

supported by the Swiss National S ien e Foundation (S hweizeris her Nationalfonds).

1

2 P. CASTILLO, B. COCKBURN, D. SCH

OTZAU, AND C. SCHWAB

estimates for the LDG method and they are all for the h-version of the method.

The rst a priori error estimate for the LDG method was obtained in 1998 by

Co kburn and Shu [11 who proved that, when polynomials of degree p are used,

the LDG method onverges in the energy norm at a rate of order h

p

. This rate of

onvergen e was obtained for the general form of the so- alled numeri al uxes that

appear in the denition of the LDG method and is sharp sin e for the numeri al ux

proposed by Bassi and Rebay [1 this rate is a tually a hieved. Later, this analysis

was extended by Co kburn and Dawson [5 to the ase in whi h the onve tive

velo ity and the diusion tensor depend on x and the domain is bounded; the rate

of onvergen e of order h

p

was on e again obtained.

Although the rate of onvergen e of order h

p

is sharp for general uxes, Co kburn

and Shu [11 reported numeri al experiments in the one-dimensional ase indi ating

that, for a spe ial numeri al ux, a rate of onvergen e of order h

p+1

is a hieved

for very smooth solutions. This indi ation was later put on rm mathemati al

grounds by Castillo [3 who showed, for the model problem of onstant- oeÆ ient,

linear onve tion-diusion in one spa e dimension, that the LDG method with a

parti ular numeri al ux onverges with the optimal rate of onvergen e of order

h

p+1

. Castillo's result an be viewed as an extension to the onve tion-diusion

setting of the a priori error estimate for the dis ontinuous Galerkin (DG) method

for the purely onve tive ase obtained in 1974 by LeSaint and Raviart [15 who

prove that the rate of onvergen e is of order h

p+1

.

In this paper, we obtain an a priori error estimate for the hp-version of the LDG

method for general numeri al uxes whi h is expli it in the mesh-width h and the

polynomial degree p. Assuming that the (s+ 1)-th derivative of the exa t solution

in the energy norm plus the L

1

(0; T ;L

2

)-norm of the time derivative is nite, we

show that, for general numeri al uxes, the energy norm of the error has a rate of

onvergen e of order h

p+1=2

=p

s+1=2

in the purely onve tive ase and of h

p

=p

s1=2

in

the onve tion-diusion ase. Moreover, by using the spe ial numeri al ux studied

by Castillo [3, we obtain the optimal rate of onvergen e of order h

p+1

=p

s+1

for

totally arbitrary meshes and polynomials of degree p in all elements. This result

holds in the purely onve tive ase as well as in the purely paraboli ase.

Let us give an idea of how the error estimate is obtained. First, using the te hnique

employed by Co kburn and Shu [11, we nd an upper bound for the energy norm of

a proje tion into the nite element spa e of the error. Then, following Castillo [3,

we eliminate as many as possible terms in the upper bound of the error by arefully

dening the numeri al ux of the LDG method and by suitably hoosing su h a

proje tion. Indeed, instead of using the L

2

-proje tion operator used by Co kburn

and Shu [11, the proje tions used by Houston, S hwab and Suli [14, 30, 29, or the

Lagrange interpolation of Gauss-Radau points used by Castillo [3, we pi k the more

advantageous proje tion used in 1985 by Thomee [31 in his study of dis ontinuous

Galerkin time-dis retizations for paraboli problems and re ently by S hotzau and

S hwab [23, 22 in their study of the hp-version of this method. Indeed, with this

proje tion, many terms in the upper bound of the error be ome identi ally zero

whi h allows us to obtain an optimal rate of onvergen e after a simple appli ation

of the sharp hp-approximation results for this proje tion.

Re ent work on other dis ontinuous nite element methods for onve tion-diusion

(and for pure diusion) problems has been reviewed by Co kburn, Karniadakis

and Shu [7. See, in parti ular, the numeri al method of Baumann and Oden [2,

OPTIMAL A PRIORI ERROR ESTIMATES FOR THE hp LDG METHOD 3

the optimal error estimates for the method as applied to nonlinear onve tion-

diusion equations by Riviere and Wheeler [20, the analysis of several versions

of the Baumann and Oden method for ellipti problems by Riviere, Wheeler and

Girault [21, and the hp-version analyses by Houston, Suli and S hwab [14, 30, 29

of dis ontinuous Galerkin methods for se ond-order problems with non-negative

hara teristi form. We also mention the re ent work of Wihler and S hwab [32

in whi h robust exponential rates of onvergen e of DG methods for (stationary)

onve tion-diusion problems in one spa e dimension are proven.

The organization of the paper is as follows. In se tion 2, we des ribe the LDG

method. In se tion 3, we state and prove the a priori error estimate for the onstant-

oeÆ ient onve tion-diusion problem and the spe ial numeri al ux for whi h the

estimates are optimal in both h and p. In se tion 4, we dis uss several extensions

and, in se tion 5, we perform numeri al experiments that verify the theoreti al

results. We end our presentation with some on luding remarks in se tion 6.

2. The LDG method

In this se tion, we introdu e and brie y dis uss the various key elements of the

LDG method for a simple model problem.

2.1. The model problem and its weak formulation. In this paper we onsider

the following model onve tion-diusion equation in one spa e dimension:

(2.1) u

t

+ ( u d u

x

)

x

= f in Q

T

= (a; b) (0; T );

with the initial ondition

(2.2) uj

t=0

= u

0

on = (a; b);

and the Diri hlet boundary onditions

(2.3) u(a) = u

D

(a); u(b) = u

D

(b) on J = (0; T ):

The unknown fun tion u is a s alar, and we assume the velo ity > 0 to be a

positive and the diusion oeÆ ient d 0 to be a non-negative number; we hoose

to work with a positive velo ity simply to x the lo ation of the possible boundary

layer at x = b. Note that in the purely onve tive ase (d = 0), only the Diri hlet

boundary ondition at x = a is taken into a ount.

The weak formulation we are going to use is obtained as follows. First, we introdu e

the new variable q :=

p

d u

x

and the \ ux" fun tion

h = (h

u

; h

q

)

>

:= ( u

p

d q;

p

d u)

>

;

and rewrite (2.1) - (2.3) in the form

u

t

+ (h

u

)

x

= f in Q

T

;

q + (h

q

)

x

= 0 in Q

T

;

uj

t=0

= u

0

on ;

u(a) = u

D

(a); u(b) = u

D

(b) on (0; T ):

Next, given the nodes a = x

0

< x

1

< ::: < x

M1

< x

M

= b, we dene the mesh

T = fI

j

= (x

j1

; x

j

); j = 1; :::;Mg and set h

j

:= jI

j

j = x

j

x

j1

; furthermore, we

dene h := max

M

j=1

h

j

. To the mesh T , we asso iate the so- alled broken Sobolev

spa e

H

1

(; T ) :=

n

v : ! lRj vj

I

j

2 H

1

(I

j

); j = 1; :::;M

o

:



For a fun tion u 2 H

1

(; T ) the one-sided limits at the nodes fx

j

g are denoted as

follows:

(2.4) u

= u(x

j

) := lim

x!x

j

u(x):

Throughout, we assume the exa t solution w = (u; q) of (2.1)(2.3) belongs to

H

1

(0; T ;H

1

(; T ))L

2

(0; T ;H

1

(; T )). Then, it satises the following equations

(u

t

; v)

I

j

(h

u

; v

x

)

I

j

+ h

u

vj

x

j

x

+

j1

= (f; v)

I

j

;

(q; r)

I

j

(h

q

; r

x

)

I

j

+ h

q

rj

x

j

x

+

j1

= 0;

(u(; 0); v)

I

j

= (u

0

; v)

I

j

;

for all test fun tions v; r 2 H

1

(; T ) and for j = 1; :::;M . Here, the time derivative

is to be understood in the weak sense and (u; v)

I

=

R

I

u(x) v(x) dx.

2.2. The method. A dis rete version of the above mixed formulation is obtained

by restri ting the trial and test fun tions to nite dimensional subspa es V

N

H

1

(; T ) and by repla ing the ux fun tion h at the nodes by a numeri al ux

^

h = (

^

h

u

;

^

h

q

)

>

: Find w

N

= (u

N

; q

N

) 2 H

1

(0; T ;V

N

) L

2

(0; T ;V

N

) su h that for

all v; r 2 V

N

and for j = 1; :::;M the following equations hold:

(2.5)

((u

N

)

t

; v)

I

j

(h

u

; v

x

)

I

j

+

^

h

u

v

x

j

x

+

j1

= (f; v)

I

j

;

(q

N

; r)

I

j

(h

q

; r

x

)

I

j

+

^

h

q

r

x

j

x

+

j1

= 0;

(u

N

(; 0); v)

I

j

= (u

0

; v)

I

j

:

Upon a hoi e of basis for the subspa es V

N

, and, more importantly, of the numeri al

uxes, the semi-dis rete problem (2.5) be omes an ODE system of dimension 2N

on J = (0; T ), where N = dim(V

N

). In what follows we do not onsider the impa t

of the time dis retization and refer to Shu [28, Shu and Osher [26, 27 and Gottlieb

and Shu [13 for the analysis of ertain TVD Runge-Kutta methods for the solution

of the ODE systems.

Our hoi e of the spa e V

N

is the spa e of dis ontinuous, pie ewise polynomial

fun tions

n

u : ! lRj uj

I

j

2 P

p

j

(I

j

); j = 1; :::;M

o

;

where P

p

j

(I

j

) denotes the set of all polynomials of degree less or equal than p

j

on

I

j

. Noti e that the polynomial degrees an vary from element to element.

To omplete the denition of the LDG method, it remains to dene the numeri al

ux

^

h.

2.3. The numeri al ux

^

h. Cru ial for the stability as well as for the a ura y

of the LDG method is the hoi e of the numeri al ux

^

h. To dene it, we introdu e

with the notation in (2.4) the following quantities

[u = u

+

u

; u = (u

+

+ u

)=2:

The numeri al ux

^

h has the following general form:

^

h(w

+

;w

) = ( u; 0)

>

p

d (q; u)

>

11

12

12

0

[w ;


whi h also holds at the boundary if we dene

(u; q)(a

) = (u

D

(a); q(a

+

)); (u; q)(b

+

) = (u

D

(b); q(b

)):

Let us stress several important points on erning this numeri al ux:

In the purely hyperboli ase, i.e., in the ase d = 0, if we take

11

= =2,

we obtain the well known \upwinding" ux of the original DG method; see,

e.g., [4.

Note that

22

= 0. This is so be ause we want to be able to solve for q

N

in terms of u

N

element by element. This lo al solvability, whi h gives the

name to the LDG method, is not shared by most mixed methods and allows

us to easily eliminate the unknown q

N

from the equations.

The main purpose of the oeÆ ient

11

is to enhan e the stability of the

method; that is why it must be a positive number. This results in an

improvement of the a ura y of the method too.

The hoi e

21

=

12

ensures the stability of the LDG method.

The main purpose of the oeÆ ients

12

is to enhan e the a ura y of the

method. Thus, if we take, following Bassi and Rebay [1,

12

= 0 the rate

of onvergen e of the energy norm is of order h

p

for smooth fun tions. If

instead, following Castillo [3, we take

12

=

p

d=2, we obtain the optimal

rate of order h

p+1

.

Let us point out that, if we onsider the general form of the numeri al uxes, it is

possible to obtain exponential onvergen e for pie ewise analyti exa t solutions,

but not optimality in both h and p. As we shall see, this optimality is guaranteed

for ompletely arbitrary meshes if we take (an extension of) Castillo's [3 hoi e of

the numeri al ux

^

h, namely,

(2.6)

^

h(x

j

) =

8

>

<

>

:

( u

D

(a)

p

d q(a

+

);

p

d u

D

(a))

>

for j = 0;

( u(x

j

)

p

d q(x

+

j

);

p

d u(x

j

))

>

for j = 1; : : : ;M 1;

( u(b)

p

d q(b

);

p

d u

D

(b))

>

for j =M;

where u(b) = u(b

)maxf =2;maxf1; p

M

gd=h

M

g (u

D

(b) u(b

)).

This ux is obtained by setting

12

p

d=2 and

11

(x

j

) =

(

=2 for j = 0; : : : ;M 1;

maxf =2;maxf1; p

M

g d=h

M

g for j =M:

Note again that in the purely onve tive ase, d = 0, this numeri al ux is nothing

but the standard upwinding ux used by the original DG method. Note also that

the fa t that the oeÆ ient

11

(b) has a spe ial form is a re e tion of the fa t that

at x = b there might be a boundary layer whi h requires spe ial treatment. The

fa tor maxf1; p

M

g=h

M

ensures the optimality in both h and p of the energy norm

of the error.

The denition of the LDG method is now omplete.

3. Error Analysis

This se tion is devoted to our main a priori error estimates. First, we state and

brie y dis uss the results; the remainder of the se tion is devoted to their proof.



3.1. A priori estimates. Our main result follows naturally from an estimate

of a suitably dened proje tion of the error e = w w

N

and from the hp-

approximation properties of the proje tion . Ea h of these results are ontained

in several lemmas that we state next. To do that, we need to introdu e the proje -

tion and the norm jjj jjj

T

in whi h we measure e.

The proje tion is the operator from H

1

(; T )

2

to V

2

N

that asso iates (u; q) to

(

u;

+

q) where, for ea h interval I

j

= (x

j1

; x

j

); j = 1; : : : ;M ,

is dened by

the following p

j

+ 1 onditions:

(

w w; v)

I

j

= 0 8v 2 P

p

j

1

(I

j

); if p

j

> 0;(3.1)

w(x

j

) = w(x

j

);

+

w(x

j1

) = w(x

+

j1

):(3.2)

Let us now introdu e a norm that appears naturally in the analysis of the LDG

method. In what follows, we denote by k k

D

the L

2

-norm in the subdomain D

and omit D when D = . For v = (v; r) the norm jjj jjj

T

is dened as follows:

(3.3) jjjv jjj

2

T

= kv k

2

E;T

+

T;T

(v);

where the energy norm kv k

E;T

is given by

kv k

2

E;T

= k v(T ) k

2

+ k r k

2

Q

T

;

and

T;T

(v) =

Z

T

0

11

(a) v

2

(a

+

; t) +

M1

X

j=1

11

(x

j

) [v

2

(x

j

; t) +

11

(b) v

2

(b

; t)

dt:

Note that information about the numeri al ux is ontained in the norm jjj jjj

T

only through

T;T

(). We are now ready to state our results.

Lemma 3.1 (The basi estimate). The error e between the exa t solution and

the approximation given by the LDG method with numeri al ux (2.6) satises the

inequality

jjje jjj

T

A

1=2

(T ) +

Z

T

0

B(t) dt;

where

A(T ) = k

u

0

u

0

k

2

+ k

+

q q k

2

Q

T

+

d

11

(b)

k (

+

q q)(b

; ) k

2

(0;T );

and

B(t) = k (

(u

t

) u

t

)(; t) k:

If we ombine the above result with the estimates of w

w, for w = u and

w = q, we immediately obtain our desired error bound. To state the orresponding

approximation results, we introdu e on the referen e interval I = (1; 1) and for

s 2 lN

0

the following weighted semi-norm

juj

2

V

s

(I)

:=

Z

I

ju

(s)

(x)j

2

(1 + x)

s

(1 x)

s

ds:

We an now state our approximation result.


Lemma 3.2 (The p-approximation estimates for xed s). Let I = (1; 1) be the

referen e interval and p a generi polynomial degree on I. Assume w

0

2 V

s

(I) for

s 2 lN

0

and

w 2 P

p

(I). Then we have the following estimates:

k

w wk

I

(s)maxf1; pg

(s+1)

jw

0

j

V

s

(I)

;

j (

w w)(1) j (s)maxf1; pg

(s+1=2)

jw

0

j

V

s

(I)

;

where (s) depends on s but is independent of p and w.

For standard nite element methods, the introdu tion of the weighted norms jj

V

s

(I)

enables one to show that for singular solutions the p-version of the method, i.e.,

when the mesh T is xed and p

j

in reases unboundedly, yields twi e the onvergen e

rate than the h-version provided that the singularity lies at a mesh point x

j

; see,

e.g., S hwab [24. The results in Lemma 3.2 are slightly suboptimal with regard

to these aspe ts as will be shown in the numeri al experiments in se tion 5 below.

However, for smooth solutions we obtain optimal approximation properties for

in h and p. This an be inferred immediately from Lemma 3.2 and standard s aling

and interpolation arguments.

Lemma 3.3 (The hp-approximation estimates for xed s). Let w 2 H

s+1

(I

j

) for

j = 1; : : : ;M and s 0. Then we have the following estimates:

k

w wk

I

j

(s)

h

min(s;p

j

)+1

j

maxf1; p

j

g

s+1

kwk

H

s+1

(I

j

)

;

j (

+

w w)(x

j

) j (s)

h

min(s;p

j

)+1=2

j

maxf1; p

j

g

s+1=2

kwk

H

s+1

(I

j

)

;

j (

w w)(x

j1

) j (s)

h

min(s;p

j

)+1=2

j

maxf1; p

j

g

s+1=2

kwk

H

s+1

(I

j

)

;

where (s) depends on s but is independent of p

j

, I

j

and w.

Now, assume that ku

(s+1)

k

E;T

<1, where

kw k

E;T

= 2 sup

0tT

kw(t) k+

Z

T

0

kw

t

(; t) k dt+ 3

p

d kw

x

k

Q

T

:

Our main result is a simple onsequen e of the above lemmas. Indeed, sin e

k e k

E;T

jjjejjj

T

+ kww k

E;T

jjjejjj

T

+ sup

0tT

k (

u u)(; t) k+ k

+

q q k

Q

T

;

from Lemmas 3.1 and 3.3, we obtain the following result.

Theorem 3.4 (The estimate of the energy norm). Let e be the error between the

exa t solution and the approximation given by the LDG method with numeri al ux

(2.6) and polynomial degree p on ea h interval. Then, for totally arbitrary meshes,

the energy norm of the error satises the inequality

k e k

E;T

(s)

h

minfs;pg+1

maxf1; pg

s+1

ku

(s+1)

k

E;T

:



Remark 3.5. The error estimates in Theorem 3.4 are optimal in h and p for smooth

solutions, even for the ase in whi h pie ewise onstant approximations (p = 0) are

used. Note also that in the purely onve tive ase, d = 0, the above error estimate

is nothing but the extension of the super- onvergen e error estimate of LeSaint and

Raviart [15 for the h-version of the DG method for purely onve tive problems.

Remark 3.6. The proof of Lemma 3.3 a tually gives us estimates whi h are om-

pletely expli it in the mesh-width h, in the polynomial degree p, and in the regu-

larity of the exa t solution (see Proposition 3.12 below). Hen e, ompletely expli it

error estimates in the energy norm an be obtained. In onjun tion with geometri

meshes and linearly in reasing polynomial degrees, su h estimates an be used in

the hp-version to prove exponential rates of onvergen e in the presen e of solution

singularities; see, e.g., the re ent monograph by S hwab [24 and the referen es

therein. However, sin e the orresponding analyti regularity in spa e-time still

remains to be found, we do not further pursue these issues here.

Remark 3.7. From Theorem 3.4, we on lude that in the p-version of the LDG

method, where the mesh is kept xed and the polynomial degree p is in reased, we

have k e k

E;T

(s)p

(s+1)

ku

(s+1)

k

E;T

. Hen e, for smooth solutions onvergen e

rates of arbitrarily high algebrai order in p are possible. This is sometimes referred

to as spe tral onvergen e. Furthermore, for solutions whi h are analyti in Q

T

,

even exponential rates of onvergen e are obtained in the p-version, i.e.,

(3.4) k e k

E;T

C exp(bp);

with onstants C; b > 0 independent of p. This result an immediately be derived

from Lemma 3.1, properties of

(see, e.g., (3.15) and (3.16) below) and standard

approximation theory for analyti fun tions. We note that (3.4) holds true for

general uxes as well.

Remark 3.8. For small diusivities d, i.e., for d ! 0 in (2.1), the solutions typ-

i ally exhibit vis ous boundary layers (or sho k proles) of length s ale O(d) or

O(

p

d). In prin iple, layer omponents in the solutions are still analyti (see the

work of Melenk [16 and Melenk and S hwab [19, 18 for a omplete hara teriza-

tion of boundary layers in stationary problems with analyti input data) and an

thus be approximated at exponential rates of onvergen e, in agreement with (3.4).

However, the estimate (3.4) is not robust with respe t to the diusivity d and dete-

riorates as d! 0. A remedy is to employ needle-element of the appropriate width

or geometri mesh renement near the boundary. It has been shown re ently by

Melenk [16, Melenk and S hwab [17, 19, S hwab and Suri [25 and Wihler and

S hwab [32 that the use of these mesh-design prin iples yields exponential rates

of onvergen e that are robust with respe t to the diusivity parameter d. We

demonstrate this robustness in our numeri al examples in se tion 5.5 below.

3.2. Proof of the basi estimate. This se tion is devoted to the proof of Lemma

3.1. To do so, we follow the te hnique used by Co kburn and Shu [11; see also

Co kburn [4, Castillo [3 and Co kburn and Dawson [5.

We start by rewriting the denition of the LDG method in ompa t form for general

numeri al uxes. Integrating (2.5) with respe t to t from 0 to T and summing over

all elements, it turns out that the LDG solution is dened as follows:

Find w

N

= (u

N

; q

N

) 2 H

1

(0; T ;V

N

) L

2

(0; T ;V

N

) su h that

(3.5) B

N

(w

N

;v) = L(v) 8v = (v; r) 2 H

1

(0; T ;V

N

) L

2

(0; T ;V

N

);


where the dis rete bilinear form B

N

(; ) is given by

(3.6)

B

N

(w

N

;v) := (u(0); v(0)) +

Z

T

0

((u

N

)

t

(; t); v(; t)) dt

+

Z

T

0

(q

N

(; t); r(; t)) dt

Z

T

0

M

X

j=1

(h(w

N

(x; t));v

x

(x; t))

I

j

dt

Z

T

0

M1

X

j=1

^

h(w

N

)(x

j

; t)

>

[v (x

j

; t)dt

+

Z

T

0

( =2 +

11

(a))u

N

(a

+

; t) +

p

d q

N

(a

+

; t)

v(a

+

; t) dt

+

Z

T

0

(

p

d=2

12

(a))u

N

(a

+

; t) r(a

+

; t) dt

+

Z

T

0

( =2 +

11

(b))u

N

(b

; t)

p

d q

N

(b

; t)

v(b

; t) dt

+

Z

T

0

(

p

d=2

12

(b))u

N

(b

; t) r(b

; t) dt;

and the dis rete linear form L(; ) is given by

(3.7)

L

N

(v) := (u

0

; v(0)) + (f; v)

+

Z

T

0

( =2 +

11

(a))u

D

(a; t) v(a

+

; t) dt

+

Z

T

0

(

p

d=2

12

(a))u

D

(a; t) r(a

+

; t) dt

+

Z

T

0

( =2 +

11

(b))u

D

(b; t) v(b

; t) dt

+

Z

T

0

(

p

d=2

12

(b))u

D

(b; t) r(b

; t) dt:

The basi error estimate now follows by standard manipulations. Indeed, sin e we

have that

B

N

(w;v) = L(v) 8v 2 H

1

(0; T ;V

N

) L

2

(0; T ;V

N

);

we obtain that

B

N

(e;v) = 0 8v 2 H

1

(0; T ;V

N

) L

2

(0; T ;V

N

);

where e := w w

N

, and this implies that

(3.8) B

N

(

N

e;

N

e) = B

N

(

N

e e;

N

e) = B

N

(

N

w w;

N

e):

It only remains to obtain a suitable expression for B

N

(

N

e;

N

e) and an upper

bound for B

N

(

N

w w;

N

e).

Lemma 3.9. For any v = (v; r) 2 H

1

(0; T ;V

N

) L

2

(0; T ;V

N

), there holds

B

N

(v;v) =

1

2

jjjv jjj

2

T

+

1

2

jv j

2

T

;

where

jv j

2

T

= k v(0) k

2

+

Z

T

0

k r(; t) k

2

dt+

T;T

(v)



and jjj jjj

T

is dened by (3.3).

Proof. This is a dire t onsequen e of the denition of the form B

N

.

Lemma 3.10. For any v 2 H

1

(0; T ;V

N

) L

2

(0; T ;V

N

) and the numeri al ux

(2.6), we have,

B

N

(w w;v)

1

2

k

u

0

u

0

k

2

+

Z

T

0

k

(u

t

) u

t

)(; t) k k v(; t) k dt

+

1

2

Z

T

0

k (

+

q q)(; t) k

2

dt

+

Z

T

0

d

2

11

(b)

j (

+

q q)(b

; t) j

2

dt+

1

2

jv j

2

T

:

Proof. Taking into a ount the denition of the form B

N

, (3.6), and the denition

of the proje tion , (3.1) and (3.2), we easily get that

B

N

(w w;v) =

u(0) u(0); v(0)

+

Z

T

0

(

(u

t

) u

t

)(; t); v(; t)

dt

+

Z

T

0

(

+

q q)(; t); r(; t)

dt

Z

T

0

p

d (

+

q q)(b

; t) v(b

; t) dt:

The result follows from simple appli ations of Cau hy-S hwarz's and Young's in-

equalities and from the denition of the fun tional j j

T

dened in Lemma 3.9.

Now, inserting the results of Lemmas 3.9 and 3.10 into (3.8), we get the inequality

jjje jjj

2

k

u

0

u

0

k

2

+ k

+

q q k

2

Q

T

+

d

11

(b)

k (

+

q q)(b

; ) k

2

(0;T )

+ 2

Z

T

0

k (

(u

t

) u

t

)(; t) k k v(; t) k dt;

whi h is of the form

(3.9)

2

(T ) +R(T ) A(T ) + 2

Z

T

0

B(t)(t) dt;

with

(T ) = k (

u u

N

)(; T ) k;

R(T ) =

Z

T

0

k (

+

q q

N

) k

2

dt+

T;T

(

u u

N

);

A(T ) = k

u

0

u

0

k

2

+ k

+

q q k

2

Q

T

+

d

11

(b)

k (

+

q q)(b

; ) k

2

(0;T )

;

B(t) = k (

(u

t

) u

t

)(; t) k:

Sin e inequality (3.9) holds true for all T > 0, Lemma 3.1 now follows after a simple

appli ation of the following result.

Lemma 3.11. Suppose that for all t > 0 we have

2

(t) +R(t) A(t) + 2

Z

t

0

B(s)(s) ds;


where R, A, and B are nonnegative fun tions. Then, for any T > 0,

p

2

(T ) +R(T ) sup

0tT

A

1=2

(t) +

Z

T

0

B(t) dt:

Proof. Dene (t) = 2

R

t

0

B(s)(s)ds and x T > 0. Setting S

T

= sup

0tT

A(t),

the hypothesis implies that for 0 t T

0

(t) = 2B(t)(t) 2B(t)

p

A(t) + (t) 2B(t)

p

S

T

+ (t):

Integrating over (0; T ) yields

Z

T

0

0

(t)

p

S

T

+ (t)

dt 2

Z

T

0

B(t) dt:

Hen e,

p

S

T

+ (T )

p

S

T

+

Z

T

0

B(t) dt:

Sin e

p

2

(T ) +R(T )

p

S

T

+ (T ), the assertion in Lemma 3.11 follows.

This ompletes the proof of Lemma 3.1.

3.3. Proof of the hp-approximation results. This se tion is devoted to the

proof of Lemma 3.2 whi h follows from the subsequent ner approximation results

after an appli ation of Stirling's formula.

Let I = (1; 1) and re all that

juj

2

V

s

(I)

:=

Z

I

ju

(s)

(x)j

2

(1 + x)

s

(1 x)

s

dx:

We have:

Proposition 3.12. Let w

0

2 V

s

(I) for s 2 lN

0

. Let

2 P

p

(I) dened by

(3.10) (

w w; v)

I

= 0 8v 2 P

p1

(I);

w(1) = w(1):

Then we have

k

w wk

2

I

6

(2p+ 1)

2

(p k)!

(p+ k)!

jw

0

j

2

V

k

(I)

;

and

j (

w w)(1) j

2

2

2p+ 1

(p k)!

(p+ k)!

jw

0

j

2

V

k

(I)

;

for any 0 k min(p; s).

Proof. The rst estimate was obtained by S hotzau [22 and S hotzau and S hwab

[23. Nevertheless, we present a detailed proof for the sake of ompleteness. We

onsider only

sin e the proof for

+

is similar. We pro eed in several steps.

Step 1: First, we derive bounds on the dieren e w

w in terms of the Legendre

oeÆ ients of w. To do so, denote by L

i

(x), i 0, the Legendre polynomial of

degree i on I and expand the fun tion w into the series w =

P

1

i=0

w

i

L

i

with

w

i

=

R

I

w(x)L

i

(x)dx=kL

i

k

2

I

. Sin e L

i

(+1) = 1, it an be seen from (3.10) that

w is uniquely given by the series

w =

p1

X

i=0

w

i

L

i

+

1

X

i=p

w

i

L

p

:



Hen e, the dieren e w

w an be written as

w

w =

1

X

i=p+1

w

i

L

i

1

X

i=p+1

w

i

L

p

:

Let P

p

be the L

2

-proje tion from L

2

(I) onto P

p

(I). Sin e wP

p

w =

P

1

i=p+1

w

i

L

i

,

kL

i

k

2

I

=

2

2i+1

and L

i

(1) = (1)

i

, we obtain from the denition of

the follow-

ing bounds:

kw

wk

2

I

= kw P

p

wk

2

I

+

1

X

i=p+1

w

i

2

2

2p+ 1

;(3.11)

(w

w)(1)

2

= 4

1

X

i=0

w

p+1+2i

2

:(3.12)

Step 2: To estimate the sums in the above equalities, we start by expanding w

0

into

the series w

0

=

P

1

i=0

b

i

L

i

. Integrating this expression yields

w(x) = w(1) +

1

X

i=0

b

i

Z

x

1

L

i

(s)ds;

and employing for i 1 the identity

Z

x

1

L

i

(s)ds =

1

2i+ 1

(L

i+1

(x) L

i1

(x));

and rearranging terms, we obtain

w(x) =

w(1) + b

0

L

0

(x) +

1

X

i=1

b

i1

2i 1

L

i

(x)

1

X

i=0

b

i+1

2i+ 3

L

i

(x):

Comparing oeÆ ients in the Legendre expansions, we on lude that

w

i

=

b

i1

2i 1

b

i+1

2i+ 3

; i 1:

Hen e, after some simple algebrai manipulations,

1

X

i=p+1

w

i

=

b

p

2p+ 1

+

b

p+1

2p+ 3

;

and

1

X

i=0

w

p+1+2i

= (1)

p+1

1

2p+ 1

b

p

:

As a onsequen e,

1

X

i=p+1

w

i

1

p

2p+ 1

2 b

2

p

2p+ 1

+

2 b

2

p+1

2p+ 3

1=2

;

1

X

i=0

w

p+1+2i

=

1

2p+ 1

j b

p

j;


and sin e kw

0

k

2

I

=

P

1

i=0

b

2

i

2

2i+1

, we get

1

X

i=p+1

w

i

1

p

2p+ 1

kw

0

k

I

;(3.13)

1

X

i=0

w

p+1+2i

=

1

p

2(2p+ 1)

kw

0

k

I

:(3.14)

Step 3: Now, note that after inserting the estimates (3.13) and (3.14) into (3.11)

and (3.12), respe tively, we get

kw

wk

2

I

kw P

p

wk

2

I

+

2

(2p+ 1)

2

kw

0

k

2

I

;

j(w

w)(1)j

2

=

2

2p+ 1

kw

0

k

2

I

:

Repla ing in these inequalities w by w q, where q is an arbitrary polynomial of

degree p, and taking into a ount that P

p

(q) = q and

(q) = q gives

kw

wk

2

I

kw P

p

wk

2

I

+

2

(2p+ 1)

2

kw

0

q

0

k

2

I

;(3.15)

j(w

w)(1)j

2

2

2p+ 1

kw

0

q

0

k

2

I

:(3.16)

S hwab [24 proved that

kw P

p

wk

2

I

(p k)!

(p+ 2 + k)!

jw

0

j

2

V

k

(I)

;

for any 0 k min(p; s), and the existen e of a polynomial q 2 P

p

(I) su h that

kw

0

q

0

k

2

I

(p k)!

(p+ k)!

jw

0

j

2

V

k

(I)

;

for any 0 k min(p; s). We now simply insert these estimates into (3.15) and

(3.16) to on lude that

kw

wk

2

I

1

(p+ 2 + k)(p+ 1 + k)

+

2

(2p+ 1)

2

(p k)!

(p+ k)!

jw

0

j

2

V

k

(I)

;

j(w

w)(1)j

2

2

2p+ 1

(p k)!

(p+ k)!

jw

0

j

2

V

k

(I)

;

for any 0 k min(p; s).

The orresponding estimates for

+

are obtained by symmetry. Sin e

(p+ 2 + k)(p+ 1 + k) (2p+ 1)

2

=4;

this proves Proposition 3.12.

From Proposition 3.12 we obtain by standard s aling and interpolation arguments

the following hp-approximation properties of

:

Corollary 3.13. For ea h interval I

j

= (x

j1

; x

j

), j = 1; : : : ;M , we have for

w 2 H

s

j

+1

(I

j

), s

j

0 real, the estimates

k

w wk

2

I

j

C

h

j

2

2k

j

+2

1

p

2

j

(p

j

k

j

+ 1)

(p

j

+ k

j

+ 1)

kwk

2

H

k

j

+1

(I

j

)

;



and

j (

+

w w)(x

j

) j

2

C

h

j

2

2k

j

+1

1

p

j

(p

j

k

j

+ 1)

(p

j

+ k

j

+ 1)

kwk

2

H

k

j

+1

(I

j

)

;

j (

w w)(x

j1

) j

2

C

h

j

2

2k

j

+1

1

p

j

(p

j

k

j

+ 1)

(p

j

+ k

j

+ 1)

kwk

2

H

k

j

+1

(I

j

)

;

for any 0 k

j

min(p

j

; s

j

). The onstant C > 0 is independent of h

j

, p

j

and k

j

.

4. Extensions

The a priori error estimate of Theorem 3.4 an be easily extended to the ase of

general boundary onditions and to general numeri al uxes.

4.1. Other boundary onditions. Theorem 3.4 holds un hanged for Neumann,

Robin or mixed boundary onditions. To see this, let us onsider, for example, the

following Neumann boundary onditions:

p

d u

x

(a) = q

N

(a);

p

d u

x

(b) = q

N

(b):

First, we take

(u; q)(a

) = (u(a

+

); q

N

(a)); (u; q)(b

+

) = (u(b

); q

N

(b)):

Then we redene the numeri al ux as follows:

^

h(x

j

) =

8

>

<

>

:

( u(a

+

)

p

d q

N

(a);

p

d u(a

+

))

>

for j = 0;

( u(x

j

)

p

d q(x

+

j

);

p

d u(x

j

))

>

for j = 1; : : : ;M 1;

( u(b

)

p

d q

N

(b);

p

d u(b

))

>

for j =M:

Note that this ux is obtained by setting

12

p

d=2 and

(

11

;

12

)(x

j

) =

(

( =2;

p

d=2) for j = 0;

( =2;

p

d=2) for j = 1; : : : ;M:

Now, we simply have to go through the proof of Lemma 3.1 in se tion 3.2 to verify

that the basi error estimate of Lemma 3.1 still holds with b as before and

A(T ) = k

u

0

u

0

k

2

+ k

+

q q k

2

Q

T

:

This, and the approximation result of Lemma 3.3, imply that the estimate of The-

orem 3.4 holds in this ase.

4.2. General numeri al uxes. In the ase of general numeri al uxes, the op-

timality of the estimate of Theorem 3.4 is lost in both h and p. Theoreti ally, the

main reason is that now there are new terms in B(

N

ww;

N

e) whi h were equal

to zero for the ux (2.6).

Indeed, in the purely onve tive ase, d = 0, the new term is

:=

Z

T

0

M1

X

j=1

=2 +

11

(x

j

)

(

u u)(x

+

j

; t) [

(u

N

u)(x

+

j

; t) dt

+

Z

T

0

=2 +

11

(a)

(

u u)(a; t) [

(u

N

u)(a

+

; t) dt;


whi h is estimated as follows:

j j

Z

T

0

M1

X

j=1

( =2 +

11

(x

j

))

2

2

11

(x

j

)

(

u u)

2

(x

+

j

; t) dt

+

Z

T

0

( =2 +

11

(a))

2

2

11

(a)

(

u u)

2

(a

+

; t) dt

+

1

2

Z

T

0

M1

X

j=1

11

(x

j

)[

(u

N

u)

2

(x

+

j

; t) dt

+

1

2

Z

T

0

11

(a)

(u

N

u)

2

(a

+

; t) dt:

The last two terms are absorbed by the term j

N

e j

2

T

and the rst two are bounded,

using Lemma 3.3, by

C

11

(s)

h

minfs;pg+1=2

maxf1; pg

s+1=2

ku

(s+1)

k

Q

T

;

where

C

11

:= max

0jM1

( =2 +

11

(x

j

))

2

2

11

(x

j

)

:

Hen e, the error estimate is

k e k

E;T

(s)

h

minfs;pg+1=2

maxf1; pg

s+1=2

h

1=2

maxf1; pg

1=2

ku

(s+1)

k

E;T

+ C

11

ku

(s+1)

k

Q

T

:

Note the loss of half a power in both h and p.

If the ase in whi h d 6= 0, the following additional term appears:

:=

Z

T

0

M1

X

j=1

p

d=2

12

(x

j

)

(

+

q q)(x

j

; t)

(u

N

u)

(x

j

; t) dt

+

Z

T

0

M1

X

j=1

p

d=2

12

(x

j

)

(

u u)(x

+

j

; t)

+

(q

N

q)

(x

j

; t) dt

+

Z

T

0

p

d=2

12

(a)

(

u u)(a

+

; t)

+

(q

N

q)

(a

+

; t) dt:

By using the approximation results of Lemma 3.3, we see that we an bound as

follows:

j j C

12

(s)

h

minfs;pg+1=2

maxf1; pg

s+1=2

Z

T

0

(t) dt;

where

C

12

:= max

0jM1

j

p

d=2

12

(x

j

) j;

and

(t) :=

p

d ku

(s+1)

x

(t) k

M1

X

j=1

(u

N

u)

2

(x

j

; t)

1=2

+ ku

(s+1)

(t) k

M1

X

j=0

+

(q

N

q)

2

(x

j

; t)

1=2

;



Next, we use the following inverse inequality:

maxfjw(x

+

j1

) j; jw(x

j

) jg C

i

p

p

h

j

kw k

I

j

;

for w 2 V

N

in order to get

j j C

i

C

12

(s)

h

minfs;pg

maxf1; pg

s1=2

Z

T

0

p

d ku

(s+1)

x

(t) kk

(u

N

u)(t) k

+ ku

(s+1)

(t) kk

+

(q

N

q)(t) k

dt;

Note that this produ es an additional loss of half power in h and a full power in

p. Thus, after a few simple manipulations, we obtain the following estimate for

general numeri al uxes:

k e k

E;T

(C

11

; C

12

; s)

h

minfs;pg

maxf1; pg

s1=2

ku

(s+1)

k

E;T

:

5. Numeri al results

The purpose of this se tion is to numeri ally validate the a priori error estimates

given in se tion 3. In all our experiments, we use a TVD RungeKutta time stepping

method, see Shu and Osher [26, 27, with suÆ iently small time steps, su h that

the overall error is governed by the spatial error.

5.1. Exponential onvergen e. Our rst example illustrates the exponential

onvergen e in p for analyti solutions. We solve (2.1) on the spa e-time domain

Q

T

= J = (0; 1) (0; 1), with exa t solution u(x; t) = exp(dt) sin(2(x t)).

We use a xed grid onsisting of a uniform mesh with 4 elements and in rease the

polynomial degree p. The orresponding errors in the energy norm at time T = 1

are shown in gure 1. The diusion oeÆ ient is d = 0:1 and the onve tion o-

eÆ ient is hosen as = 0:1 (left) and = 1:0 (right). The urves learly show

exponential rates of onvergen e as predi ted in (3.4) of se tion 3. Sin e the quad-

rature points and weights used the determine the LDG solution are omputed only

with an a ura y of 10

12

, the urves bottom out for p 10.

0 1 2 3 4 5 6 7 8 9 10 11 12−12

−11

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

Exponential convergence

Polynomial degree p

log

10(E

ne

rgy

err

or)

0 1 2 3 4 5 6 7 8 9 10 11 12−12

−11

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

Exponential convergence

Polynomial degree p

log

10(E

ne

rgy

err

or)

Figure 1. Exponential onvergen e in p for an analyti exa t

solution. In both examples, the diusion oeÆ ient is d = 0:1.

The onve tion oeÆ ient is = 0:1 (left) and = 1:0 (right).


5.2. Optimal order of onvergen e in h. In these examples, we show that an

optimal order of onvergen e of p+ 1 is a hieved when using the numeri al ux in

(2.6). For this set of tests, we solve (2.1) on Q

T

= J = (1; 1) (0; 1), again

with exa t solution u(x; t) = exp(dt) sin(2(x t)). To determine numeri ally

the onvergen e order we onsider the two sequen es of su essively rened meshes

fT

i

g shown in gure 2. Sin e our analysis is valid for arbitrary meshes, we hoose

the se ond sequen e to onsist of non-uniform meshes whereas the rst one ontains

uniform meshes. Note that in both ases the mesh-size parameter of T

i+1

is half of

the one of T

i

.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

Uniform meshes

Ω = (−1, 1)

mesh 1

mesh 2

mesh 3

mesh 4

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

Ω = (−1, 1)

Non−uniform meshes

mesh 1

mesh 2

mesh 3

mesh 4

Figure 2. Sequen e of uniform and non-uniform meshes in = (1; 1).

If e(T

i

) denotes the error on the i-th mesh in the energy norm, then the numeri al

rate of onvergen e r

i

is dened as

r

i

= log

e(T

i+1

)

e(T

i

)

= log(0:5):

In tables 1, 2 and 3, we present these numeri al orders fr

i

g in the energy norm at

T = 1:0 for polynomials of degree 0 to 6 on the above two mesh-sequen es. In all

the experiments we use the same onve tion oeÆ ient and in rease the diusion

oeÆ ient from 0:01 to 1:0. The results show that our estimates are optimal in

h not only for onve tion dominated problems, but also for diusion dominated

problems. In all the ases the numeri al orders agree with the theoreti al orders of

our error estimates in Theorem 3.4.

5.3. Non-smooth solutions. In this subse tion, we present some numeri al re-

sults to illustrate the performan e of the LDG method for a solution that is non-

smooth in spa e.

We onsider rst the h-version and start by solving the purely onve tive (d = 0)

problem (2.1), on Q

T

= J = (0; 1) (0; 1) with = 0:1 and with data hosen

in su h a way that the exa t solution is u(x; t) = x

t. The orresponding uniform

and non-uniform spatial dis retizations are similar to those used in se tion 5.2 ( f.,

gure 2). In this purely onve tive problem an order of onvergen e of minf +

0:5; p+1g is expe ted from our error estimate in se tion 3. These orders an learly

be seen in the table 4. Again, they are al ulated for the energy norm at T = 1:0.

Now, we onsider the linear problem with diusion, i.e., with = 0:1 and d = 0:1.

Again, we hoose the data in su h a way that the exa t solution is u(x; t) = x

t.



p Non uniform grid Uniform grid

r

1

r

2

r

3

r

1

r

2

r

3

0 0.6733 0.6999 0.8801 0.4964 0.7817 0.8728

1 1.5527 2.0295 1.9846 1.8123 1.8739 1.9658

2 2.6972 2.8663 2.9384 2.5580 2.9190 2.9504

3 3.6891 4.1849 3.9948 4.0393 3.9472 3.9840

4 4.8392 4.9388 4.9562 4.7086 4.9489 4.9681

5 5.8042 6.2034 5.9937 6.1268 5.9660 5.9880

6 6.8673 6.9757 6.9365 6.7812 6.9566 6.9652

Table 1. Orders of onvergen e for the h-version and a smooth

exa t solution with = 0:1; d = 0:01.


r

1

r

2

r

3

r

1

r

2

r

3

0 0.6425 1.1482 0.9394 0.7124 0.9293 0.9405

1 1.6591 1.7262 2.0137 1.4881 1.9553 1.9914

2 3.0588 3.1997 2.9655 3.1656 2.9458 2.9738

3 3.8242 3.6791 3.9919 3.5969 3.9727 3.9888

4 4.9699 5.2152 4.9785 5.1662 4.9545 4.9840

5 5.8760 5.6978 5.9926 5.6391 5.9782 5.9915

6 6.9644 7.2217 6.9748 7.1813 6.9659 6.9860




r

1

r

2

r

3

r

1

r

2

r

3

0 0.7198 1.2008 0.9773 0.9099 0.9330 0.9768

1 1.6489 1.6060 1.9993 1.2495 1.9696 1.9920

2 3.0070 3.2280 2.9858 3.2297 2.9582 2.9875

3 3.5119 3.5757 3.9939 3.1284 3.9728 3.9926

4 5.0340 5.2264 4.9905 5.2742 4.9704 4.9917

5 5.4512 5.5980 5.9949 5.0866 5.9790 5.9944

6 7.0632 7.2251 6.9689 7.3031 6.9774 6.9817



The results at T = 1:0 are shown in the table 5. From our a priori error estimate, an

order of onvergen e of minf0:5; p+1g is expe ted. This is what we a tually see

for all values of p ex ept for p = 2. In this ase, we observe an order of onvergen e

of 3 instead of the predi ted 0:5 2:6416. Sin e the order of onvergen e for

p > 2 is smaller than 3, we believe that an error an ellation might be taking pla e

whi h o urs only for p = 2.

Sin e the x

t solution is singular at the mesh point x = 0, we expe t a doubling of

the onvergen e rate in the p-version where p is in reased on a xed mesh T ; see,



r

1

r

2

r

3

r

1

r

2

r

3

0 0.0932 1.0417 1.0029 0.9751 0.9926 0.9976

1 0.2705 1.8839 1.9163 1.8082 1.8966 1.9524

2 0.4419 2.8961 2.9684 2.8795 2.9708 2.9776

3 1.3895 3.4832 3.5605 3.5243 3.6076 3.6162

4 2.8854 3.6059 3.6209 3.6067 3.6224 3.6310

5 3.5372 3.6250 3.6316 3.6230 3.6252 3.6332

6 3.6194 3.6288 3.6298 3.6252 3.6292 3.6343

Table 4. Orders of onvergen e for the h-version and the non-

smooth exa t solution x

t for the purely onve tive ase =

0:1; d = 0.


r

1

r

2

r

3

r

1

r

2

r

3

0 0.7546 0.8327 0.9669 0.8405 0.9604 0.9957

1 1.8273 1.9542 1.9992 1.9383 1.9829 1.9956

2 2.9891 3.0066 3.0025 2.9853 2.9811 2.9704

3 3.0040 2.7869 2.6861 2.6750 2.6493 2.6430

4 2.6493 2.6426 2.6413 2.6424 2.6408 2.6409

5 2.6399 2.6403 2.6408 2.6397 2.6403 2.6408

6 2.6393 2.6403 2.6407 2.6393 2.6403 2.6408

Table 5. Orders of onvergen e for the h-version and the non-

smooth exa t solution x

t for the onve tion-diusion ase =

0:1; d = 0:1.

e.g., S hwab [24. This is shown in gure 3 for the same model problems as above.

We an see a onvergen e rate of 2+1 in the purely hyperboli ase (d = 0) and of

2 1 in the onve tion-diusion ase, respe tively, whi h orresponds to an exa t

doubling of the rates. However, in our theoreti al results, if we insert the weighted

bounds from Lemma 3.2 in the proof of Theorem 3.4, we obtain rates of 2 in the

hyperboli ase and of 2 2 in the onve tion-diusion ase, resulting in a loss of

one power of p and indi ating the suboptimality of Lemma 3.2 with respe t to the

weighted spa es j j

V

s

(I)

.

5.4. Testing the optimality of the smoothness requirement. To test if the

smoothness on the exa t solution required by our Theorem 3.4 when d 6= 0 is

optimal, we onsider problem (2.1) with = 0:1, d = 0:1, homogeneous Diri hlet

boundary onditions and initial data u

0

(x) = x(1 x). The results at T = 1:0 are

given in the table 6. Theorem 3.4 predi ts an order of onvergen e of minf1:5; p+

1g but we a tually see an order of onvergen e of minf2:5; p + 1g. This gives a

strong indi ation that, to obtain optimal orders of onvergen e at least in h, less

smoothness of the exa t solution than required Theorem 3.4 for d 6= 0 is suÆ ient.

However, obtaining optimality in the smoothness of the exa t solution seems to ask

for more sophisti ated theoreti al te hniques than the ones available in the urrent

literature and has to be addressed in future work.



0 0.2 0.4 0.6 0.8 1 1.2−10

−9

−8

−7

−6

−5

−4

−3

−2

log10

(Polynomial degree p)

log

10(E

ne

rgy

err

or)

d = 0.1

d = 0

Figure 3. The p-version for the non-smooth exa t solution x

t.

The onve tion oeÆ ient is = 0:1 and the diusion oeÆ ient is

d = 0:1 (top urve) and d = 0 (bottom urve).


r

1

r

2

r

3

r

1

r

2

r

3

0 0.8765 0.9020 0.8555 0.9235 0.9546 0.9370

1 1.7059 1.8495 1.8887 1.8491 1.9158 1.9044

2 2.4844 2.4854 2.5047 2.4822 2.4672 2.5089

3 2.4984 2.4953 2.4908 2.4884 2.4979 2.3841

4 2.5001 2.4961 2.4808 2.4893 2.4716 2.0190

5 2.5001 2.5034 2.4875 2.4911 2.4666 2.4900

6 2.5022 2.5022 2.4932 2.4963 2.4949 2.4618

Table 6. Orders of onvergen e for the h-version and a non-

smooth exa t solution orresponding to the initial data u

0

(x) =

x(1 x) with = 0:1; d = 0:1.

5.5. Robust exponential onvergen e. Our last example shows that robust

exponential onvergen e an be obtained in the presen e of a boundary layer, when

suitable meshes are used; see Remark 3.8. We solve (2.1) on (0; 1) (0; 1) with

= 0:1, d = 0:1 and right-hand side su h that the exa t solution is u(x; t) =

t

1 e

(1x)="

. For small ", this solution has an exponential boundary layer of

strength O(") at the out ow boundary x = 1. In gure 4, we ompare the p-version

of the LDG method when using uniform and geometri meshes. Both meshes are

hosen in su h a way that they have the same number of elements, however, the

distribution of the grid points is dierent: In the geometri mesh the size of the

rst element near x = 1 is in the order of the length of the boundary layer, O("),

the size of the next element is twi e the size of the previous and so forth. The errors

in the energy norm at T = 1:0 for " = 0:1 and " = 0:01 are depi ted in gure 4.

All the urves show exponential rates of onvergen e. However, the robustness of


the rates for the geometri boundary layer meshes an learly be observed, whereas

the uniform mesh performs orders of magnitude worse for " = 0:01.

0 1 2 3 4 5 6−6

−5

−4

−3

−2

−1

0

1

Boundary layer approximation

Polynomial degree p

log

10(E

ne

rgy

err

or)

Geometric meshes

Uniform meshes

ε = 0.1 ε = 0.01

Figure 4. Exponential rates of onvergen e in the presen e of a

boundary layer on uniform and geometri meshes for " = 0:1 and

" = 0:01.

6. Con luding remarks

In this paper, we have obtained optimal error estimates for the hp-version of the

LDG method for the model problem of the initial boundary value problem for a

one-dimensional onve tion-diusion equation. We have shown that this is possible

by a areful hoi e of the numeri al uxes and the asso iated proje tions

+

and

; we have also shown how this optimality in h and p is lost, at least theoreti ally,

when general numeri al uxes are used.

Our numeri al results onrm the optimality in h of our main result and the expo-

nential onvergen e that follows when the solution is analyti . These results also

indi ate that the smoothness requirement on the exa t solution is too stringent.

The problem of obtaining optimality in the smoothness of the exa t solution seems

to ask for more sophisti ated theoreti al te hniques than the ones available in the

urrent literature and onstitute the subje t of ongoing work. Also, extensions of

our main result to the more hallenging ases of non- onstant oeÆ ients and d,

and to the multi-dimensional ase will be onsidered elsewhere.

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S hool of Mathemati s, University of Minnesota, 206 Chur h Street S.E., Minneapolis,

MN 55455, USA.

E-mail address: astillomath.umn.edu


MN 55455, USA.

E-mail address: o kburnmath.umn.edu


MN 55455, USA.

E-mail address: s hoetzamath.umn.edu

Seminar of Applied Mathemati s, ETHZ, 8092 Z

uri h, Switzerland

E-mail address: s hwabsam.math.ethz. h

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