.
OPTIMAL A PRIORI ERROR ESTIMATES FOR THE
hp-VERSION OF THE LOCAL DISCONTINUOUS GALERKIN
METHOD FOR CONVECTION{DIFFUSION PROBLEMS

PAUL CASTILLO, BERNARDO COCKBURN, DOMINIK SCHOTZAU,
AND CHRISTOPH
SCHWAB
Abstrat. We study the onvergene properties of the hp-version of the loal
disontinuous Galerkin nite element method for onvetion-diusion problems; we onsider a model problem in a one-dimensional spae domain. We
allow arbitrary meshes and polynomial degree distributions and obtain upper
bounds for the energy norm of the error whih are expliit in the mesh-width
h, in the polynomial degree p, and in the regularity of the exat solution. We
identify a speial numerial ux for whih the estimates are optimal in both h
and p. The theoretial results are onrmed in a series of numerial examples.
Math. Comp., Vol. 71, pp. 455{478, 2002
1. Introdution
This paper ontains the rst a priori error estimate of the hp-version of the so-alled
loal disontinuous Galerkin (LDG) nite element method for onvetion-diusion
problems. Suh an error analysis, whih takes into aount 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 sine, being a loally onservative method that does
not require any inter-element ontinuity, it is ideally suited for hp-adaptivity. In this
paper, we onsider a model onvetion-diusion equation in one spae dimension
with Dirihlet boundary onditions and obtain, for a speial hoie of the numerial
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 onvergene for p = 0.
The LDG method was introdued by Cokburn and Shu in [11℄ as an extension to
general onvetion-diusion problems of the numerial sheme for the ompressible
Navier-Stokes proposed by Bassi and Rebay in [1℄. This sheme was in turn an
extension of the Runge-Kutta disontinuous Galerkin (RKDG) method developed
by Cokburn and Shu [10, 9, 8, 6, 12℄ for non-linear hyperboli systems. For a fairly
omplete set of referenes on RKDG and LDG methods see the short monograph
by Cokburn [4℄; see also the review of the development of disontinuous Galerkin
methods by Cokburn, Karniadakis, and Shu [7℄.
To put our result under proper perspetive, let us briey desribe the relevant
results available in the urrent literature. There are only a few a priori error
1991 Mathematis Subjet Classiation. Primary 65N30; Seondary 65M60.
Key words and phrases. Disontinuous Galerkin Methods, hp-Methods, Convetion-Diusion.
The seond author was partially supported by the National Siene Foundation (Grant DMS9807491) and by the University of Minnesota Superomputer Institute. The third author was
supported by the Swiss National Siene Foundation (Shweizerisher Nationalfonds).
1
2

P. CASTILLO, B. COCKBURN, D. SCHOTZAU,
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
Cokburn 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 hp . This rate of
onvergene was obtained for the general form of the so-alled numerial uxes that
appear in the denition of the LDG method and is sharp sine for the numerial ux
proposed by Bassi and Rebay [1℄ this rate is atually ahieved. Later, this analysis
was extended by Cokburn and Dawson [5℄ to the ase in whih the onvetive
veloity and the diusion tensor depend on x and the domain is bounded; the rate
of onvergene of order hp was one again obtained.
Although the rate of onvergene of order hp is sharp for general uxes, Cokburn
and Shu [11℄ reported numerial experiments in the one-dimensional ase indiating
that, for a speial numerial ux, a rate of onvergene of order hp+1 is ahieved
for very smooth solutions. This indiation was later put on rm mathematial
grounds by Castillo [3℄ who showed, for the model problem of onstant-oeÆient,
linear onvetion-diusion in one spae dimension, that the LDG method with a
partiular numerial ux onverges with the optimal rate of onvergene of order
hp+1 . Castillo's result an be viewed as an extension to the onvetion-diusion
setting of the a priori error estimate for the disontinuous Galerkin (DG) method
for the purely onvetive ase obtained in 1974 by LeSaint and Raviart [15℄ who
prove that the rate of onvergene is of order hp+1 .
In this paper, we obtain an a priori error estimate for the hp-version of the LDG
method for general numerial uxes whih is expliit in the mesh-width h and the
polynomial degree p. Assuming that the (s + 1)-th derivative of the exat solution
in the energy norm plus the L1(0; T ; L2)-norm of the time derivative is nite, we
show that, for general numerial uxes, the energy norm of the error has a rate of
onvergene of order hp+1=2 =ps+1=2 in the purely onvetive ase and of hp =ps 1=2 in
the onvetion-diusion ase. Moreover, by using the speial numerial ux studied
by Castillo [3℄, we obtain the optimal rate of onvergene of order hp+1 =ps+1 for
totally arbitrary meshes and polynomials of degree p in all elements. This result
holds in the purely onvetive ase as well as in the purely paraboli ase.
Let us give an idea of how the error estimate is obtained. First, using the tehnique
employed by Cokburn and Shu [11℄, we nd an upper bound for the energy norm of
a projetion into the nite element spae 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 numerial ux of the LDG method and by suitably hoosing suh a
projetion. Indeed, instead of using the L2 -projetion operator used by Cokburn
and Shu [11℄, the projetions used by Houston, Shwab and Suli [14, 30, 29℄, or the
Lagrange interpolation of Gauss-Radau points used by Castillo [3℄, we pik the more
advantageous projetion used in 1985 by Thomee [31℄ in his study of disontinuous
Galerkin time-disretizations for paraboli problems and reently by Shotzau and
Shwab [23, 22℄ in their study of the hp-version of this method. Indeed, with this
projetion, many terms in the upper bound of the error beome identially zero
whih allows us to obtain an optimal rate of onvergene after a simple appliation
of the sharp hp-approximation results for this projetion.
Reent work on other disontinuous nite element methods for onvetion-diusion
(and for pure diusion) problems has been reviewed by Cokburn, Karniadakis
and Shu [7℄. See, in partiular, the numerial 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 onvetiondiusion 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 Shwab [14, 30, 29℄
of disontinuous Galerkin methods for seond-order problems with non-negative
harateristi form. We also mention the reent work of Wihler and Shwab [32℄
in whih robust exponential rates of onvergene of DG methods for (stationary)
onvetion-diusion problems in one spae dimension are proven.
The organization of the paper is as follows. In setion 2, we desribe the LDG
method. In setion 3, we state and prove the a priori error estimate for the onstantoeÆient onvetion-diusion problem and the speial numerial ux for whih the
estimates are optimal in both h and p. In setion 4, we disuss several extensions
and, in setion 5, we perform numerial experiments that verify the theoretial
results. We end our presentation with some onluding remarks in setion 6.
2. The LDG method
In this setion, we introdue and briey disuss 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 onvetion-diusion equation in one spae dimension:
(2.1)
ut + ( u d ux )x = f
in QT = (a; b) (0; T );
with the initial ondition
(2.2)
ujt=0 = u0
on = (a; b);
and the Dirihlet boundary onditions
(2.3)
u(a) = uD (a);
u(b) = uD (b)
on J = (0; T ):
The unknown funtion u is a salar, and we assume the veloity > 0 to be a
positive and the diusion oeÆient d 0 to be a non-negative number; we hoose
to work with a positive veloity simply to x the loation of the possible boundary
layer at x = b. Note that in the purely onvetive ase (d = 0), only the Dirihlet
boundary ondition at x = a is taken into aount.
The weak formulation p
we are going to use is obtained as follows. First, we introdue
the new variable q := d ux and the \ux" funtion
p >
p
d q;
d u) ;
h = (hu ; hq )> := ( u
and rewrite (2.1) - (2.3) in the form
ut + (hu )x = f
in QT ;
q + (hq )x = 0
in QT ;
ujt=0 = u0
on ;
u(a) = uD (a); u(b) = uD (b)
on (0; T ):
Next, given the nodes a = x0 < x1 < ::: < xM 1 < xM = b, we dene the mesh
T = fIj = (xj 1 ; xj ); j = 1; :::; M g and set hj := jIj j = xj xj 1 ; furthermore, we
dene h := maxM
j =1 hj . To the mesh T , we assoiate the so-alled broken Sobolev
spae
o
n
H 1 (
; T ) := v : ! lRj v jIj 2 H 1 (Ij ); j = 1; :::; M :
4

P. CASTILLO, B. COCKBURN, D. SCHOTZAU,
AND C. SCHWAB
For a funtion u 2 H 1 (
; T ) the one-sided limits at the nodes fxj g are denoted as
follows:
(2.4)
u = u(x
j ) := lim u(x):
x!xj
Throughout, we assume the exat solution w = (u; q) of (2.1){(2.3) belongs to
H 1 (0; T ; H 1(
; T )) L2 (0; T ; H 1 (
; T )). Then, it satises the following equations
(ut; v)Ij
(q; r)Ij
x
(hu ; vx )Ij + hu vjxj+
j 1
xj
(hq ; rx )Ij + hq r x+
j 1
(u( ; 0); v)Ij
j
= (f; v)Ij ;
= 0;
= (u0 ; v)Ij ;
for all test funtions v; r 2 H 1 (
; T ) and for j = 1; :::;R M . Here, the time derivative
is to be understood in the weak sense and (u; v)I = I u(x) v(x) dx.
2.2. The method. A disrete version of the above mixed formulation is obtained
by restriting the trial and test funtions to nite dimensional subspaes VN H 1 (
; T ) and by replaing the ux funtion h at the nodes by a numerial ux
h^ = (h^ u ; h^ q )> : Find wN = (uN ; qN ) 2 H 1 (0; T ; VN ) L2 (0; T ; VN ) suh that for
all v; r 2 VN and for j = 1; :::; M the following equations hold:
x
j
(hu ; vx )Ij + h^ u v + = (f; v)Ij ;
xj 1
x
j
(2.5)
(qN ; r)Ij (hq ; rx )Ij + h^ q r + = 0;
xj 1
(uN (; 0); v)Ij = (u0 ; v)Ij :
Upon a hoie of basis for the subspaes VN , and, more importantly, of the numerial
uxes, the semi-disrete problem (2.5) beomes an ODE system of dimension 2 N
on J = (0; T ), where N = dim(VN ). In what follows we do not onsider the impat
of the time disretization 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 hoie of the spae VN is the spae of disontinuous, pieewise polynomial
funtions
o
n
u : ! lRj ujIj 2 P pj (Ij ); j = 1; :::; M ;
where P pj (Ij ) denotes the set of all polynomials of degree less or equal than pj on
Ij . Notie that the polynomial degrees an vary from element to element.
To omplete the denition of the LDG method, it remains to dene the numerial
ux h^ .
^ Cruial for the stability as well as for the auray
2.3. The numerial ux h.
of the LDG method is the hoie of the numerial ux h^ . To dene it, we introdue
with the notation in (2.4) the following quantities
[u℄ = u+ u ; u = (u+ + u )=2:
The numerial ux h^ has the following general form:
p
11
12
>
^h(w+ ; w ) = ( u; 0)>
d (q; u)
12
0 [w ℄ ;
((uN )t ; v)Ij
OPTIMAL A PRIORI ERROR ESTIMATES FOR THE
hp
LDG METHOD
5
whih also holds at the boundary if we dene
(u; q)(a ) = (uD (a); q(a+ ));
(u; q)(b+ ) = (uD (b); q(b )):
Let us stress several important points onerning this numerial 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 beause we want to be able to solve for qN
in terms of uN element by element. This loal solvability, whih gives the
name to the LDG method, is not shared by most mixed methods and allows
us to easily eliminate the unknown qN from the equations.
The main purpose of the oeÆient 11 is to enhane the stability of the
method; that is why it must be a positive number. This results in an
improvement of the auray of the method too.
The hoie 21 = 12 ensures the stability of the LDG method.
The main purpose of the oeÆients 12 is to enhane the auray of the
method. Thus, if we take, following Bassi and Rebay [1℄, 12 = 0 the rate
of onvergene of the energy norm is of orderphp for smooth funtions. If
instead, following Castillo [3℄, we take 12 = d=2, we obtain the optimal
rate of order hp+1 .
Let us point out that, if we onsider the general form of the numerial uxes, it is
possible to obtain exponential onvergene for pieewise analyti exat 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℄ hoie of
the numerial ux h^ , namely,
8
p + p
> for j = 0;
>
<( uD (a) pd q (a ); pd uD (a))
(2.6) h^ (xj ) = ( u(xj ) pd q(x+j ); pd u(xj ))> for j = 1; : : : ; M 1;
>
:
( u^(b)
d q (b );
d uD (b))>
for j = M;
where u^(b) = u(b ) maxf=2; maxfp1; pM gd=hM g (uD (b)
This ux is obtained by setting 12 d=2 and
(
11 (xj )
=
=2
maxf=2; maxf1; pM g d=hM g
u(b
)).
for j = 0; : : : ; M
for j = M:
1;
Note again that in the purely onvetive ase, d = 0, this numerial ux is nothing
but the standard upwinding ux used by the original DG method. Note also that
the fat that the oeÆient 11 (b) has a speial form is a reetion of the fat that
at x = b there might be a boundary layer whih requires speial treatment. The
fator maxf1; pM g=hM 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 setion is devoted to our main a priori error estimates. First, we state and
briey disuss the results; the remainder of the setion is devoted to their proof.

P. CASTILLO, B. COCKBURN, D. SCHOTZAU,
AND C. SCHWAB
6
3.1. A priori estimates. Our main result follows naturally from an estimate
of a suitably dened projetion of the error e = w wN and from the hpapproximation properties of the projetion . Eah of these results are ontained
in several lemmas that we state next. To do that, we need to introdue the projetion and the norm jj jjT in whih we measure e.
The projetion is the operator from H 1 (
; T )2 to VN2 that assoiates (u; q) to
( u; + q) where, for eah interval Ij = (xj 1 ; xj ); j = 1; : : : ; M , is dened by
the following pj + 1 onditions:
(3.1)
8v 2 P pj 1 (Ij ); if pj > 0;
( w w; v)I = 0
(3.2)
j
w(xj )
= w(xj );
+ w(xj
1
) = w(x+j 1 ):
Let us now introdue a norm that appears naturally in the analysis of the LDG
method. In what follows, we denote by k kD the L2 -norm in the subdomain D
and omit D when D = . For v = (v; r) the norm jj jjT is dened as follows:
jj v jjT = k v kE;T + T;T (v);
where the energy norm k v kE;T is given by
k v kE;T = k v(T ) k + k r kQT ;
(3.3)
2
2
2
2
2
and
T;T (v) =
Z T
0
11 (a) v 2 (a+ ; t) +
M
X1
j =1
11 (xj ) [v ℄
2
(xj ; t) + 11 (b) v2 (b
; t) dt:
Note that information about the numerial ux is ontained in the norm
only through T;T (). We are now ready to state our results.
jj jjT
Lemma 3.1 (The basi estimate). The error e between the exat solution and
the approximation given by the LDG method with numerial ux (2.6) satises the
inequality
jj e jjT A = (T ) +
1 2
Z T
0
B (t) dt;
where
A(T )
= k
u0
u0
and
k
2
+ k + q
B (t)
k
q 2QT
= k ( (ut )
+
d
11 (b)
k (
+
q
k
ut )( ; t) :
q )(b ;
) k
2
(0;T );
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 introdue on the referene interval I = ( 1; 1) and for
s 2 lN0 the following weighted semi-norm
jujV s I
2
( )
:=
Z
ju s (x)j
( )
I
2
(1 + x)s (1
We an now state our approximation result.
x)s ds:
OPTIMAL A PRIORI ERROR ESTIMATES FOR THE
hp
LDG METHOD
7
Lemma 3.2 (The p-approximation estimates for xed s). Let I = ( 1; 1) be the
referene interval and p a generi polynomial degree on I . Assume w0 2 V s (I ) for
s 2 lN0 and w 2 P p (I ). Then we have the following estimates:
k w wk I (s) maxf1; pg (s+1)jw0 jV s (I ) ;
j ( w w)(1) j (s) maxf1; pg (s+1=2) jw0 jV s (I ) ;
where (s) depends on
s
but is independent of
p
and w.
For standard nite element methods, the introdution of the weighted norms jjV 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 pj inreases unboundedly, yields twie the onvergene
rate than the h-version provided that the singularity lies at a mesh point xj ; see,
e.g., Shwab [24℄. The results in Lemma 3.2 are slightly suboptimal with regard
to these aspets as will be shown in the numerial experiments in setion 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 saling
and interpolation arguments.
Lemma 3.3 (The hp-approximation estimates for xed s). Let w 2 H s+1 (Ij ) for
j = 1; : : : ; M and s 0. Then we have the following estimates:
k w
w Ij
j (
w
w)(xj )
j (s)
w
w)(xj
1 ) j (s)
j (
+
(s)
k
where (s) depends on
s
min(s;pj )+1
hj
maxf1; pj gs+1
min(s;pj )+1=2
hj
maxf1; pj gs+1=2
min(s;pj )+1=2
hj
maxf1; pj gs+1=2
k wkH s+1 Ij ;
(
)
k wkH s+1 Ij ;
(
)
k wkH s+1 Ij ;
(
)
but is independent of pj , Ij and w.
Now, assume that k u(s+1) kE ;T
<
1, where
p
Z T
sup k w(t) k +
k wt (; t) k dt + 3 d k wx kQT :
0tT
0
Our main result is a simple onsequene of the above lemmas. Indeed, sine
k e kE;T jj ejjT + k w w kE;T
jj ejjT + sup k ( u u)(; t) k + k + q q kQT ;
0tT
from Lemmas 3.1 and 3.3, we obtain the following result.
k w kE ;T = 2
Theorem 3.4 (The estimate of the energy norm). Let e be the error between the
exat solution and the approximation given by the LDG method with numerial ux
(2.6) and polynomial degree p on eah interval. Then, for totally arbitrary meshes,
the energy norm of the error satises the inequality
fs;pg+1
min
h
s
k e kE;T (s) max
f1; pgs k u kE ;T :
+1
( +1)
8

P. CASTILLO, B. COCKBURN, D. SCHOTZAU,
AND C. SCHWAB
Remark 3.5. The error estimates in Theorem 3.4 are optimal in h and p for smooth
solutions, even for the ase in whih pieewise onstant approximations (p = 0) are
used. Note also that in the purely onvetive ase, d = 0, the above error estimate
is nothing but the extension of the super-onvergene error estimate of LeSaint and
Raviart [15℄ for the h-version of the DG method for purely onvetive problems.
Remark 3.6. The proof of Lemma 3.3 atually gives us estimates whih are ompletely expliit in the mesh-width h, in the polynomial degree p, and in the regularity of the exat solution (see Proposition 3.12 below). Hene, ompletely expliit
error estimates in the energy norm an be obtained. In onjuntion with geometri
meshes and linearly inreasing polynomial degrees, suh estimates an be used in
the hp-version to prove exponential rates of onvergene in the presene of solution
singularities; see, e.g., the reent monograph by Shwab [24℄ and the referenes
therein. However, sine the orresponding analyti regularity in spae-time still
remains to be found, we do not further pursue these issues here.
Remark 3.7. From Theorem 3.4, we onlude that in the p-version of the LDG
method, where the mesh is kept xed and the polynomial degree p is inreased, we
have k e kE;T (s)p (s+1) k u(s+1) kE ;T . Hene, for smooth solutions onvergene
rates of arbitrarily high algebrai order in p are possible. This is sometimes referred
to as spetral onvergene. Furthermore, for solutions whih are analyti in QT ,
even exponential rates of onvergene are obtained in the p-version, i.e.,
(3.4)
k e kE;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 funtions. 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 typially
p exhibit visous boundary layers (or shok proles) of length sale O(d) or
O( d). In priniple, layer omponents in the solutions are still analyti (see the
work of Melenk [16℄ and Melenk and Shwab [19, 18℄ for a omplete haraterization of boundary layers in stationary problems with analyti input data) and an
thus be approximated at exponential rates of onvergene, in agreement with (3.4).
However, the estimate (3.4) is not robust with respet to the diusivity d and deteriorates 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 reently by
Melenk [16℄, Melenk and Shwab [17, 19℄, Shwab and Suri [25℄ and Wihler and
Shwab [32℄ that the use of these mesh-design priniples yields exponential rates
of onvergene that are robust with respet to the diusivity parameter d. We
demonstrate this robustness in our numerial examples in setion 5.5 below.
3.2. Proof of the basi estimate. This setion is devoted to the proof of Lemma
3.1. To do so, we follow the tehnique used by Cokburn and Shu [11℄; see also
Cokburn [4℄, Castillo [3℄ and Cokburn and Dawson [5℄.
We start by rewriting the denition of the LDG method in ompat form for general
numerial uxes. Integrating (2.5) with respet to t from 0 to T and summing over
all elements, it turns out that the LDG solution is dened as follows:
Find wN = (uN ; qN ) 2 H 1 (0; T ; VN ) L2(0; T ; VN ) suh that
(3.5)
BN (wN ; v) = L(v) 8 v = (v; r) 2 H 1 (0; T ; VN ) L2 (0; T ; VN );
OPTIMAL A PRIORI ERROR ESTIMATES FOR THE
hp
LDG METHOD
9
where the disrete bilinear form BN (; ) is given by
(3.6)
Z
BN (wN ; v)
:= (u(0); v(0)) +
+
Z T
0
Z T j =1
0
+
+
+
Z0
0
(
p
(
T
Z0 T
((uN )t (; t); v(; t)) dt
(h(wN (x; t)); vx (x; t))Ij dt
j =1
Z TM
X1
0
Z0 T
0
(qN (; t); r(; t)) dt
Z TX
M
+
T
(
h^ (wN )(xj ; t)> [v℄ (xj ; t)dt
=2 + 11 (a)) uN (a+ ; t) +
d=2
d qN (a+ ; t)
v (a+ ; t) dt
12 (a)) uN (a+ ; t) r(a+ ; t) dt
(=2 + 11 (b)) uN (b
p
p
d=2
; t)
p
d qN (b ; t)
v (b ; t) dt
12 (b)) uN (b ; t) r(b ; t) dt;
and the disrete linear form L(; ) is given by
LN (v) := (Zu0 ; v(0)) + (f; v)
+
+
(3.7)
+
+
T
Z0
T
Z0
T
Z0 T
0
(=2 + 11 (a)) uD (a; t) v(a+ ; t) dt
p
(
d=2
12 (a)) uD (a; t) r(a+ ; t) dt
(
=2 + 11 (b)) uD (b; t) v (b ; t) dt
(
d=2
p
12 (b)) uD (b; t) r(b ; t) dt:
The basi error estimate now follows by standard manipulations. Indeed, sine we
have that
BN (w; v) = L(v) 8 v 2 H 1 (0; T ; VN ) L2 (0; T ; VN );
we obtain that
BN (e; v) = 0 8 v 2 H 1 (0; T ; VN ) L2 (0; T ; VN );
where e := w wN , and this implies that
(3.8)
BN (N e; N e) = BN (N e e; N e) = BN (N w w; N e):
It only remains to obtain a suitable expression for BN (N e; N e) and an upper
bound for BN (N w w; N e).
Lemma 3.9. For any v = (v; r) 2 H 1 (0; T ; VN ) L2 (0; T ; VN ), there holds
1
1
BN (v; v) = jj v jj2T + j v j2T ;
2
2
where
j v jT = k v(0) k
2
2
+
Z T
0
k r(; t) k dt + T;T (v)
2

P. CASTILLO, B. COCKBURN, D. SCHOTZAU,
AND C. SCHWAB
10
and
jj jjT
is dened by (3.3).
Proof. This is a diret onsequene of the denition of the form BN .
1
2
Lemma 3.10. For any v 2 H (0; T ; VN ) L (0; T ; VN ) and the numerial ux
(2.6), we have,
Z T
1
BN ( w w; v) k (ut ) ut )(; t) k k v(; t) k dt
k
u0 u0 k2 +
2
0
Z
1 T
+
k (+ q q)(; t) k2 dt
2 0
Z T
d
j (+ q q)(b ; t) j2 dt + 12 j v j2T :
+
2
(
b
)
11
0
Proof. Taking into aount the denition of the form BN , (3.6), and the denition
of the projetion , (3.1) and (3.2), we easily get that
BN ( w
w; v) =
u(0)
+
Z T
u(0); v (0)
0
p
0
Z T
( (ut )
0
(+ q
Z T
+
ut )( ; t); v ( ; t) dt
q )( ; t); r( ; t) dt
d ( + q
q )(b ; t) v (b ; t) dt:
The result follows from simple appliations of Cauhy-Shwarz's and Young's inequalities and from the denition of the funtional j jT dened in Lemma 3.9. Now, inserting the results of Lemmas 3.9 and 3.10 into (3.8), we get the inequality
jj e jj k 2
u0
+2
Z T
0
whih is of the form
(3.9)
k
2
u0
k (
+ k + q
(T ) + R(T )
with
(T )
= k (
R(T )
=
A(T )
= k
Z T
0
d
11 (b)
k (
+
q
q )(b ;
) k
2
(0;T )
ut )( ; t)
A(T ) + 2
Z T
0
B (t) (t) dt;
k
uN )( ; T ) ;
u
k (
u0
+
k k v(; t) k dt;
(ut )
2
k
q 2QT
+
qN )
q
u0
k
2
k
2
dt + T;T ( u
+ k + q
k
q 2QT
+
uN );
d
11 (b)
k (
+
q
q )(b ;
) k
2
(0;T ) ;
= k ( (ut ) ut )(; t) k:
Sine inequality (3.9) holds true for all T > 0, Lemma 3.1 now follows after a simple
appliation of the following result.
Lemma 3.11. Suppose that for all t > 0 we have
B (t)
(t) + R(t)
2
A(t) + 2
Z t
0
B (s) (s) ds;
OPTIMAL A PRIORI ERROR ESTIMATES FOR THE
where
R, A,
and
B
p
hp
LDG METHOD
are nonnegative funtions. Then, for any
2 (T ) + R(T )
0
R
sup A1=2 (t) +
tT
Z T
0
T >
11
0,
B (t) dt:
Proof. Dene (t) = 2 0t B (s)(s)ds and x T > 0. Setting ST = sup0tT A(t),
the hypothesis implies that for 0 t T
p
p
0 (t) = 2B (t)(t) 2B (t) A(t) + (t) 2B (t) ST + (t):
Integrating over (0; T ) yields
Z T
Z T
0 (t)
p
dt 2
B (t) dt:
ST + (t)
0
0
Hene,
Z
p
+ (T ) ST
p
p
p
ST
+
T
0
B (t) dt:
Sine 2 (T ) + R(T ) ST + (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 setion is devoted to the
proof of Lemma 3.2 whih follows from the subsequent ner approximation results
after an appliation of Stirling's formula.
Let I = ( 1; 1) and reall that
jujV s I
2
(
) :=
Z
ju s (x)j (1 + x)s (1
( )
I
2
x)s dx:
We have:
Proposition 3.12. Let w0 2 V s (I ) for s 2 lN0 . Let 2 P p (I ) dened by
(3.10)
( w w; v)I = 0 8v 2 P p 1(I );
w(1) = w(1):
Then we have
k w
and
j ( w
k 2
w I
w)(
1) j 2
for any 0 k min(p; s).
(p k)! 0 2
6
jw j ;
(2p + 1)2 (p + k)! V k (I )
(p k)! 0 2
2
jw j ;
2p + 1 (p + k)! V k (I )
Proof. The rst estimate was obtained by Shotzau [22℄ and Shotzau and Shwab
[23℄. Nevertheless, we present a detailed proof for the sake of ompleteness. We
onsider only sine the proof for + is similar. We proeed in several steps.
Step 1: First, we derive bounds on the dierene w w in terms of the Legendre
oeÆients of w. To do so, denote by Li (x), i 0, the LegendrePpolynomial of
degreeR i on I and expand the funtion w into the series w = 1
i=0 wi Li with
wi = I w(x)Li (x)dx=k Li k2I . Sine Li (+1) = 1, it an be seen from (3.10) that
w is uniquely given by the series
X
pX1
1 wi Lp :
wi Li +
w=
i=0
i=p
12

P. CASTILLO, B. COCKBURN, D. SCHOTZAU,
AND C. SCHWAB
Hene, the dierene w
an be written as
1
1
X
X
wi Li
w=
w
w
i=p+1
i=p+1
Lp :
wi
P
Let Pp be the L2-projetion from L2 (I ) onto P p (I ). Sine w Pp w = 1
i=p+1 wi Li ,
2
i
2
k Lik I = 2i+1 and Li (1) = (1) , we obtain from the denition of the following bounds:
1
2
X
2
2
2
wi k w w k I = k w Pp w k I + (3.11)
;
2
p+1
i=p+1
1
2
X
(w w)( 1) 2 = 4
w
(3.12)
p+1+2i :
i=0
0
Step 2: To estimate
P the sums in the above equalities, we start by expanding w into
the series w0 =
1
i=0 bi Li .
Integrating this expression yields
1 Zx
X
bi
Li (s)ds;
w(x) = w( 1) +
i=0
and employing for i 1 the identity
Z x
1
Li (s)ds
=
1
(L (x)
2i + 1 i+1
and rearranging terms, we obtain
w(x)
=
w(
1) + b0
1
L0 (x) +
1 b
X
i
i=1
1
2i 1
Li
Li (x)
1
(x));
1 b
X
i+1
i=0
2i + 3
Li (x):
Comparing oeÆients in the Legendre expansions, we onlude that
wi
=
bi
bi+1
1
;
i
1:
2i 1 2i + 3
Hene, after some simple algebrai manipulations,
1
X
bp
b
wi =
+ p+1 ;
2
p+1
2
p+3
i=p+1
and
1
X
As a onsequene,
i=0
wp+1+2i
1
X
w
i i=p+1
1
X
wp+1+2i i=0
= ( 1)p+1
p2p1+ 1
=
1
b :
2p + 1 p
2 b2p
2 b2
+ p+1
2p + 1 2p + 3
1
j b j;
2p + 1 p
1=2
;
OPTIMAL A PRIORI ERROR ESTIMATES FOR THE
hp
LDG METHOD
13
P
2 2
and sine k w0 k I2 = 1
i=0 bi 2i+1 , we get
1
X
1
wi p
(3.13)
k w0 k I ;
2
p+1
i=p+1
1
X
1
(3.14)
k w0 k I :
w
p+1+2i = p
2(2
p
+
1)
i=0
Step 3: Now, note that after inserting the estimates (3.13) and (3.14) into (3.11)
and (3.12), respetively, we get
k w wk 2I k w Pp wk 2I + (2p +2 1)2 k w0 k I2 ;
j(w w)( 1)j2 = 2p 2+ 1 k w0 k I2 :
Replaing in these inequalities w by w q, where q is an arbitrary polynomial of
degree p, and taking into aount that Pp (q) = q and (q) = q gives
(3.15)
k w wk 2I k w Pp wk 2I + (2p +2 1)2 k w0 q0 k I2 ;
j(w w)( 1)j2 2p 2+ 1 k w0 q0 k 2I :
(3.16)
Shwab [24℄ proved that
k w Pp wk 2I (p(+p 2 +k)!k)! jw0 j2V k (I ) ;
for any 0 k min(p; s), and the existene of a polynomial q 2 P p (I ) suh that
k w0 q0 k 2I ((pp + kk)!)! jw0 j2V k (I ) ;
for any 0 k min(p; s). We now simply insert these estimates into (3.15) and
(3.16) to onlude that
k w wk 2I (p + 2 + k)(1 p + 1 + k) + (2p +2 1)2 ((pp + kk)!)! jw0 j2V k (I ) ;
j(w w)( 1)j2 2p 2+ 1 ((pp + kk)!)! jw0 j2V k (I ) ;
for any 0 k min(p; s).
The orresponding estimates for + are obtained by symmetry. Sine
(p + 2 + k)(p + 1 + k) (2p + 1)2 =4;
this proves Proposition 3.12.
From Proposition 3.12 we obtain by standard saling and interpolation arguments
the following hp-approximation properties of :
Corollary 3.13. For eah interval Ij = (xj 1 ; xj ), j = 1; : : : ; M , we have for
w 2 H sj +1 (Ij ), sj 0 real, the estimates
k w
k C
w 2Ij
hj
2
2kj +2 1
p2j
(pj kj + 1)
k wk H2 kj +1 (Ij ) ;
(pj + kj + 1)

P. CASTILLO, B. COCKBURN, D. SCHOTZAU,
AND C. SCHWAB
14
and
j (
j (
+
w
w)(xj )
w
w)(xj
for any 0 kj
1
j )j
2
2
min(pj ; sj ).
hj
C
2
hj
C
2kj +1 2kj +1 2
1
pj
(pj kj + 1)
k wk 2H kj +1 (Ij ) ;
(pj + kj + 1)
(pj kj + 1)
k wk 2H kj +1 (Ij ) ;
(pj + kj + 1)
>
0 is independent of hj , pj and kj .
pj
1
The onstant C
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 numerial uxes.
4.1. Other boundary onditions. Theorem 3.4 holds unhanged for Neumann,
Robin or mixed boundary onditions. To see this, let us onsider, for example, the
following Neumann boundary onditions:
p
d ux (a)
p
= qN (a);
d ux (b)
= qN (b):
First, we take
(u; q)(a ) = (u(a+ ); qN (a));
(u; q)(b+ ) = (u(b ); qN (b)):
Then we redene the numerial ux as follows:
8
p + >
p
+
>
for j = 0;
<( u(a ) pd qN (a); pd u(a ))
h^ (xj ) = ( u(xj ) p d q(x+j ); p d u(xj ))> for j = 1; : : : ; M 1;
>
:
( u(b )
d qN (b);
d u(b ))>
for j = M:
Note that this ux is obtained by setting 12 (
(11 ; 12 )(xj ) =
p
(=2p; d=2)
(=2;
d=2)
p
d=2
and
for j = 0;
for j = 1; : : : ; M:
Now, we simply have to go through the proof of Lemma 3.1 in setion 3.2 to verify
that the basi error estimate of Lemma 3.1 still holds with b as before and
A(T ) = k u0 u0 k2 + k + q q k2QT :
This, and the approximation result of Lemma 3.3, imply that the estimate of Theorem 3.4 holds in this ase.
4.2. General numerial uxes. In the ase of general numerial uxes, the optimality of the estimate of Theorem 3.4 is lost in both h and p. Theoretially, the
main reason is that now there are new terms in B (N w w; N e) whih were equal
to zero for the ux (2.6).
Indeed, in the purely onvetive ase, d = 0, the new term is
:=
Z TM
X1
0
+
j =1
Z T
0
=2 + 11 (xj )
=2 + 11 (a)
(
(
u
u
u)(x+
j ; t) [ u)(a; t) [
(uN
(uN
u)℄(x+
j ; t) dt
u)℄(a+ ; t) dt;
hp
OPTIMAL A PRIORI ERROR ESTIMATES FOR THE
LDG METHOD
15
whih is estimated as follows:
jj Z TM
X1
0
+
=2 + 11 (xj ))2
2 11 (xj )
j =1
Z T
( =2 + 11 (a))2
1
2
Z TM
X1
0
Z T
j =1
(
(
2 11(a)
0
+
(
11 (xj )[
u
(uN
u)2 (x+
j ; t) dt
u
u)2 (a+ ; t) dt
u)℄2 (x+
j ; t) dt
2
1
11 (a) (uN
u) (a+ ; t) dt:
2 0
The last two terms are absorbed by the term j N e j2T and the rst two are bounded,
using Lemma 3.3, by
hminfs;pg+1=2
k u(s+1) kQT ;
C11 (s)
maxf1; pgs+1=2
where
( =2 + 11 (xj ))2
:
C11 := max
0j M 1
2 11 (xj )
Hene, the error estimate is
hminfs;pg+1=2
h 1 =2
(s+1)
(s+1)
k e kE;T (s) max
k
u
k
+
C
k
u
k
E ;T
11
QT :
f1; pgs+1=2 maxf1; pg1=2
Note the loss of half a power in both h and p.
If the ase in whih d 6= 0, the following additional term appears:
+
:=
p
Z TM
X1
0
+
+
j =1
Z TM
X1
0
Z T
d=2
p
j =1
p
0
d=2
12 (xj )
d=2
12 (xj )
12 (a)
(
(+ q
q )(xj ; t) (
u
u
(uN
u)
+
u)(x+
j ; t) (qN
u)(a+ ; t) + (qN
q)
(xj ; t) dt
q)
(xj ; t) dt
(a+ ; t) dt:
By using the approximation results of Lemma 3.3, we see that we an bound as
follows:
Z T
hminfs;pg+1=2
j j C12 (s) max
f1; pgs+1=2 0 (t) dt;
where
p
C12 := max
j
d=2 12 (xj ) j;
0j M 1
and
(t) :=
p
k
d u(xs+1) (t)
k
+ k u(s+1) (t) k
M 1
X j =1
M
X1
j =0
(uN
+ (qN
2
u)
2
q)
(xj ; t)
(xj ; t)
1=2
1 =2
;

P. CASTILLO, B. COCKBURN, D. SCHOTZAU,
AND C. SCHWAB
16
Next, we use the following inverse inequality:
p
maxfj w(x+j 1 ) j; j w(xj ) jg Ci p
hj
for w 2 VN in order to get
j j Ci C
12
(s)
hminfs;pg
maxf
g
1; p s
Z T
1 =2
p
k
k w kIj ;
(s+1)
(t)
d ux
0
kk (uN
k
u)(t)
+ k u(s+1) (t) kk +(qN
q )(t)
k
dt;
Note that this produes an additional loss of half power in h and a full power in
Thus, after a few simple manipulations, we obtain the following estimate for
general numerial uxes:
minfs;pg
k e kE;T (C11 ; C12 ; s) maxhf1; pgs 1=2 k u(s+1) kE ;T :
5. Numerial results
The purpose of this setion is to numerially validate the a priori error estimates
given in setion 3. In all our experiments, we use a TVD Runge{Kutta time stepping
method, see Shu and Osher [26, 27℄, with suÆiently small time steps, suh that
the overall error is governed by the spatial error.
5.1. Exponential onvergene. Our rst example illustrates the exponential
onvergene in p for analyti solutions. We solve (2.1) on the spae-time domain
QT = J = (0; 1) (0; 1), with exat solution u(x; t) = exp( dt) sin(2 (x t)).
We use a xed grid onsisting of a uniform mesh with 4 elements and inrease the
polynomial degree p. The orresponding errors in the energy norm at time T = 1
are shown in gure 1. The diusion oeÆient is d = 0:1 and the onvetion oeÆient is hosen as = 0:1 (left) and = 1:0 (right). The urves learly show
exponential rates of onvergene as predited in (3.4) of setion 3. Sine the quadrature points and weights used the determine the LDG solution are omputed only
with an auray of 10 12 , the urves bottom out for p 10.
p.
Exponential convergence
Exponential convergence
0
−1
−1
−2
−2
−3
−3
log10(Energy error)
log10(Energy error)
0
−4
−4
−5
−5
−6
−6
−7
−7
−8
−8
−9
−9
−10
−10
−11
−12
−11
0
1
2
3
4
5
6
7
8
Polynomial degree p
9
10
11
12
−12
0
1
2
3
4
5
6
7
8
9
10
Polynomial degree p
Exponential onvergene in p for an analyti exat
solution. In both examples, the diusion oeÆient is d = 0:1.
The onvetion oeÆient is = 0:1 (left) and = 1:0 (right).
Figure 1.
11
12
OPTIMAL A PRIORI ERROR ESTIMATES FOR THE
hp
LDG METHOD
17
5.2. Optimal order of onvergene in h. In these examples, we show that an
optimal order of onvergene of p + 1 is ahieved when using the numerial ux in
(2.6). For this set of tests, we solve (2.1) on QT = J = ( 1; 1) (0; 1), again
with exat solution u(x; t) = exp( dt) sin(2(x t)). To determine numerially
the onvergene order we onsider the two sequenes of suessively rened meshes
fTi g shown in gure 2. Sine our analysis is valid for arbitrary meshes, we hoose
the seond sequene to onsist of non-uniform meshes whereas the rst one ontains
uniform meshes. Note that in both ases the mesh-size parameter of Ti+1 is half of
the one of Ti .
Non−uniform meshes
Uniform meshes
3
3
2.5
2.5
mesh 1
mesh 1
2
2
mesh 2
mesh 2
1.5
1.5
mesh 3
mesh 3
1
1
mesh 4
mesh 4
0.5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Ω = (−1, 1)
Figure 2.
0.4
0.6
0.8
1
0.5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Ω = (−1, 1)
Sequene of uniform and non-uniform meshes in = ( 1; 1).
If e(Ti ) denotes the error on the i-th mesh in the energy norm, then the numerial
rate of onvergene ri is dened as
e(Ti+1 )
= log(0:5):
ri = log
e(Ti )
In tables 1, 2 and 3, we present these numerial orders fri g in the energy norm at
T = 1:0 for polynomials of degree 0 to 6 on the above two mesh-sequenes. In all
the experiments we use the same onvetion oeÆient and inrease the diusion
oeÆient from 0:01 to 1:0. The results show that our estimates are optimal in
h not only for onvetion dominated problems, but also for diusion dominated
problems. In all the ases the numerial orders agree with the theoretial orders of
our error estimates in Theorem 3.4.
5.3. Non-smooth solutions. In this subsetion, we present some numerial results to illustrate the performane of the LDG method for a solution that is nonsmooth in spae.
We onsider rst the h-version and start by solving the purely onvetive (d = 0)
problem (2.1), on QT = J = (0; 1) (0; 1) with = 0:1 and with data hosen
in suh a way that the exat solution is u(x; t) = x t. The orresponding uniform
and non-uniform spatial disretizations are similar to those used in setion 5.2 (f.,
gure 2). In this purely onvetive problem an order of onvergene of minf +
0:5; p +1g is expeted from our error estimate in setion 3. These orders an learly
be seen in the table 4. Again, they are alulated 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 suh a way that the exat solution is u(x; t) = x t.
18

P. CASTILLO, B. COCKBURN, D. SCHOTZAU,
AND C. SCHWAB
p
0
1
2
3
4
5
6
Non uniform grid
r1
r2
r3
r1
Uniform grid
r2
r3
0.6733 0.6999 0.8801
0.4964 0.7817 0.8728
1.5527 2.0295 1.9846
1.8123 1.8739 1.9658
2.6972 2.8663 2.9384
2.5580 2.9190 2.9504
3.6891 4.1849 3.9948
4.0393 3.9472 3.9840
4.8392 4.9388 4.9562
4.7086 4.9489 4.9681
5.8042 6.2034 5.9937
6.1268 5.9660 5.9880
6.8673 6.9757 6.9365
6.7812 6.9566 6.9652
Table 1.
Orders of onvergene for the h-version and a smooth
exat solution with = 0:1; d = 0:01.
p
0
1
2
3
4
5
6
Non uniform grid
r1
r2
r3
r1
Uniform grid
r2
r3
0.6425 1.1482 0.9394
0.7124 0.9293 0.9405
1.6591 1.7262 2.0137
1.4881 1.9553 1.9914
3.0588 3.1997 2.9655
3.1656 2.9458 2.9738
3.8242 3.6791 3.9919
3.5969 3.9727 3.9888
4.9699 5.2152 4.9785
5.1662 4.9545 4.9840
5.8760 5.6978 5.9926
5.6391 5.9782 5.9915
6.9644 7.2217 6.9748
7.1813 6.9659 6.9860
Table 2.
Orders of onvergene for the h-version and a smooth
exat solution with = 0:1; d = 0:1.
p
0
1
2
3
4
5
6
Non uniform grid
r1
r2
r3
r1
Uniform grid
r2
r3
0.7198 1.2008 0.9773
0.9099 0.9330 0.9768
1.6489 1.6060 1.9993
1.2495 1.9696 1.9920
3.0070 3.2280 2.9858
3.2297 2.9582 2.9875
3.5119 3.5757 3.9939
3.1284 3.9728 3.9926
5.0340 5.2264 4.9905
5.2742 4.9704 4.9917
5.4512 5.5980 5.9949
5.0866 5.9790 5.9944
7.0632 7.2251 6.9689
7.3031 6.9774 6.9817
Table 3.
Orders of onvergene for the h-version and a smooth
exat solution with = 0:1; d = 1:0.
The results at T = 1:0 are shown in the table 5. From our a priori error estimate, an
order of onvergene of minf 0:5; p +1g is expeted. This is what we atually see
for all values of p exept for p = 2. In this ase, we observe an order of onvergene
of 3 instead of the predited 0:5 2:6416. Sine the order of onvergene for
p > 2 is smaller than 3, we believe that an error anellation might be taking plae
whih ours only for p = 2.
Sine the x t solution is singular at the mesh point x = 0, we expet a doubling of
the onvergene rate in the p-version where p is inreased on a xed mesh T ; see,
OPTIMAL A PRIORI ERROR ESTIMATES FOR THE
p
0
1
2
3
4
5
6
Non uniform grid
r1
r2
r3
r1
hp
LDG METHOD
Uniform grid
r2
19
r3
0.0932 1.0417 1.0029
0.9751 0.9926 0.9976
0.2705 1.8839 1.9163
1.8082 1.8966 1.9524
0.4419 2.8961 2.9684
2.8795 2.9708 2.9776
1.3895 3.4832 3.5605
3.5243 3.6076 3.6162
2.8854 3.6059 3.6209
3.6067 3.6224 3.6310
3.5372 3.6250 3.6316
3.6230 3.6252 3.6332
3.6194 3.6288 3.6298
3.6252 3.6292 3.6343
Table 4.
Orders of onvergene for the h-version and the nonsmooth exat solution x t for the purely onvetive ase =
0:1; d = 0.
p
0
1
2
3
4
5
6
Non uniform grid
r1
r2
r3
r1
Uniform grid
r2
r3
0.7546 0.8327 0.9669
0.8405 0.9604 0.9957
1.8273 1.9542 1.9992
1.9383 1.9829 1.9956
2.9891 3.0066 3.0025
2.9853 2.9811 2.9704
3.0040 2.7869 2.6861
2.6750 2.6493 2.6430
2.6493 2.6426 2.6413
2.6424 2.6408 2.6409
2.6399 2.6403 2.6408
2.6397 2.6403 2.6408
2.6393 2.6403 2.6407
2.6393 2.6403 2.6408
Table 5.
Orders of onvergene for the h-version and the nonsmooth exat solution x t for the onvetion-diusion ase =
0:1; d = 0:1.
e.g., Shwab [24℄. This is shown in gure 3 for the same model problems as above.
We an see a onvergene rate of 2 +1 in the purely hyperboli ase (d = 0) and of
2 1 in the onvetion-diusion ase, respetively, whih orresponds to an exat
doubling of the rates. However, in our theoretial 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 onvetion-diusion ase, resulting in a loss of
one power of p and indiating the suboptimality of Lemma 3.2 with respet to the
weighted spaes j jV s (I ) .
5.4. Testing the optimality of the smoothness requirement. To test if the
smoothness on the exat 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 Dirihlet
boundary onditions and initial data u0 (x) = x(1 x). The results at T = 1:0 are
given in the table 6. Theorem 3.4 predits an order of onvergene of minf1:5; p +
1g but we atually see an order of onvergene of minf2:5; p + 1g. This gives a
strong indiation that, to obtain optimal orders of onvergene at least in h, less
smoothness of the exat solution than required Theorem 3.4 for d 6= 0 is suÆient.
However, obtaining optimality in the smoothness of the exat solution seems to ask
for more sophistiated theoretial tehniques than the ones available in the urrent
literature and has to be addressed in future work.
20

P. CASTILLO, B. COCKBURN, D. SCHOTZAU,
AND C. SCHWAB
−2
log10(Energy error)
−3
−4
d = 0.1
−5
−6
−7
d=0
−8
−9
−10
0
0.2
0.4
0.6
0.8
log (Polynomial degree p)
1
1.2
10
The p-version for the non-smooth exat solution x t.
The onvetion oeÆient is = 0:1 and the diusion oeÆient is
d = 0:1 (top urve) and d = 0 (bottom urve).
Figure 3.
p
0
1
2
3
4
5
6
Non uniform grid
r1
r2
r3
r1
Uniform grid
r2
r3
0.8765 0.9020 0.8555
0.9235 0.9546 0.9370
1.7059 1.8495 1.8887
1.8491 1.9158 1.9044
2.4844 2.4854 2.5047
2.4822 2.4672 2.5089
2.4984 2.4953 2.4908
2.4884 2.4979 2.3841
2.5001 2.4961 2.4808
2.4893 2.4716 2.0190
2.5001 2.5034 2.4875
2.4911 2.4666 2.4900
2.5022 2.5022 2.4932
2.4963 2.4949 2.4618
Table 6.
Orders of onvergene for the h-version and a nonsmooth exat solution orresponding to the initial data u0 (x) =
x(1 x) with = 0:1; d = 0:1.
5.5. Robust exponential onvergene. Our last example shows that robust
exponential onvergene an be obtained in the presene 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 suh that the exat solution is u(x; t) =
t 1 e(1 x)=" . For small ", this solution has an exponential boundary layer of
strength O(") at the outow 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 suh 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 twie 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 depited in gure 4.
All the urves show exponential rates of onvergene. However, the robustness of
OPTIMAL A PRIORI ERROR ESTIMATES FOR THE
hp
LDG METHOD
21
the rates for the geometri boundary layer meshes an learly be observed, whereas
the uniform mesh performs orders of magnitude worse for " = 0:01.
Boundary layer approximation
1
−1
Uniform meshes
−2
−3
10
log (Energy error)
0
−4
Geometric meshes
−5
ε = 0.1
−6
0
1
ε = 0.01
2
3
4
5
6
Polynomial degree p
Exponential rates of onvergene in the presene of a
boundary layer on uniform and geometri meshes for " = 0:1 and
" = 0:01.
Figure 4.
6. Conluding 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 onvetion-diusion equation. We have shown that this is possible
by a areful hoie of the numerial uxes and the assoiated projetions + and
; we have also shown how this optimality in h and p is lost, at least theoretially,
when general numerial uxes are used.
Our numerial results onrm the optimality in h of our main result and the exponential onvergene that follows when the solution is analyti. These results also
indiate that the smoothness requirement on the exat solution is too stringent.
The problem of obtaining optimality in the smoothness of the exat solution seems
to ask for more sophistiated theoretial tehniques than the ones available in the
urrent literature and onstitute the subjet of ongoing work. Also, extensions of
our main result to the more hallenging ases of non-onstant oeÆients and d,
and to the multi-dimensional ase will be onsidered elsewhere.
Referenes
[1℄ F. Bassi and S. Rebay, A high-order aurate disontinuous nite element method for the
numerial solution of the ompressible Navier-Stokes equations, J. Comput. Phys. 131 (1997),
267{279.
[2℄ C.E. Baumann and J.T. Oden, A disontinuous hp nite element method for onvetiondiusion problems, Comput. Methods Appl. Meh. Engrg. 175 (1999), 311{341.
[3℄ P. Castillo, An optimal error estimate for the loal disontinuous Galerkin method, Disontinuous Galerkin Methods: Theory, Computation and Appliations (B. Cokburn, G.E. Karniadakis, and C.-W. Shu, eds.), Leture Notes in Computational Siene and Engineering,
vol. 11, Springer Verlag, 2000, pp. 285{290.
22

P. CASTILLO, B. COCKBURN, D. SCHOTZAU,
AND C. SCHWAB
[4℄ B. Cokburn, Disontinuous Galerkin methods for onvetion-dominated problems, HighOrder Methods for Computational Physis (T. Barth and H. Deonink, eds.), Leture Notes
in Computational Siene and Engineering, vol. 9, Springer Verlag, 1999, pp. 69{224.
[5℄ B. Cokburn and C. Dawson, Some extensions of the loal disontinuous Galerkin method
for onvetion-diusion equations in multidimensions, The Proeedings of the Conferene on
the Mathematis of Finite Elements and Appliations: MAFELAP X (J.R. Whiteman, ed.),
Elsevier, 2000, pp. 225{238.
[6℄ B. Cokburn, S. Hou, and C.-W. Shu, TVB Runge-Kutta loal projetion disontinuous
Galerkin nite element method for onservation laws IV: The multidimensional ase, Math.
Comp. 54 (1990), 545{581.
[7℄ B. Cokburn, G.E. Karniadakis, and C.-W. Shu, The development of disontinuous Galerkin
methods, Disontinuous Galerkin Methods: Theory, Computation and Appliations (B. Cokburn, G.E. Karniadakis, and C.-W. Shu, eds.), Leture Notes in Computational Siene and
Engineering, vol. 11, Springer Verlag, 2000, pp. 3{50.
[8℄ B. Cokburn, S.Y. Lin, and C.-W. Shu, TVB Runge-Kutta loal projetion disontinuous
Galerkin nite element method for onservation laws III: One dimensional systems, J. Comput. Phys. 84 (1989), 90{113.
[9℄ B. Cokburn and C.-W. Shu, TVB Runge-Kutta loal projetion disontinuous Galerkin nite
element method for salar onservation laws II: General framework, Math. Comp. 52 (1989),
411{435.
, The Runge-Kutta loal projetion P 1 -disontinuous Galerkin method for salar on[10℄
servation laws, RAIRO Model. Math. Anal.Numer. 25 (1991), 337{361.
[11℄
, The loal disontinuous Galerkin nite element method for onvetion-diusion systems, SIAM J. Numer. Anal. 35 (1998), 2440{2463.
[12℄
, The Runge-Kutta disontinuous Galerkin nite element method for onservation
laws V: Multidimensional systems, J. Comput. Phys. 141 (1998), 199{224.
[13℄ S. Gottlieb and C.-W. Shu, Total variation diminishing Runge-Kutta shemes, Math. Comp.
67 (1998), 73{85.
[14℄ P. Houston, C. Shwab, and E. Suli, Disontinuous hp nite element methods for advetion{
diusion problems, in preparation.
[15℄ P. LeSaint and P.A. Raviart, On a nite element method for solving the neutron transport
equation, Mathematial aspets of nite elements in partial dierential equations (C. de Boor,
ed.), Aademi Press, 1974, pp. 89{145.
[16℄ J.M. Melenk, On the robust exponential onvergene of hp-nite element methods for problems with boundary layers, IMA J. Num. Anal. 17 (1997), 577{601.
[17℄ J.M. Melenk and C. Shwab, hp-FEM for reation-diusion equations I: Robust exponential
onvergene, SIAM J. Numer. Anal. 35 (1998), 1520{1557.
, Analyti regularity for a singularly perturbed problem, SIAM J. Math. Anal. 30
[18℄
(1999), 379{400.
[19℄
, An hp nite element method for onvetion-diusion problems in one spae dimension, IMA J. Num. Anal. 19 (1999), 425{453.
[20℄ B. Riviere and M.F. Wheeler, A disontinuous Galerkin method applied to nonlinear paraboli equations, Disontinuous Galerkin Methods: Theory, Computation and Appliations
(B. Cokburn, G.E. Karniadakis, and C.-W. Shu, eds.), Leture Notes in Computational
Siene and Engineering, vol. 11, Springer Verlag, 2000, pp. 231{244.
[21℄ B. Riviere, M.F. Wheeler, and V. Girault, Improved energy estimates for interior penalty,
onstrained and disontinuous Galerkin methods for ellipti problems. Part I, Comput.
Geosi. 3 (1999), 337{360.
[22℄ D. Shotzau, hp-DGFEM for paraboli evolution problems - appliations to diusion and visous inompressible uid ow, Ph.D. thesis No. 13041, Swiss Federal Institute of Tehnology,
Zurih, 1999.
[23℄ D. Shotzau and C. Shwab, Time disretization of paraboli problems by the hp-version of
the disontinuous Galerkin nite element method, SIAM J. Numer. Anal., to appear.
[24℄ C. Shwab, p- and hp-FEM. Theory and Appliation to Solid and Fluid Mehanis, Oxford
University Press, New York, 1998.
[25℄ C. Shwab and M. Suri, The p- and hp-version of the nite element method for problems
with boundary layers, Math. Comp. 65 (1996), 1403{1429.
OPTIMAL A PRIORI ERROR ESTIMATES FOR THE
hp
LDG METHOD
23
[26℄ C.-W. Shu and S. Osher, EÆient implementation of essentially non-osillatory shokapturing shemes, J. Comput. Phys. 77 (1988), 439{471.
, EÆient implementation of essentially non-osillatory shok apturing shemes, II,
[27℄
J. Comput. Phys. 83 (1989), 32{78.
[28℄ C.-W. Shu, TVD time disretizations, SIAM J. Si. Stat. Comput. 9 (1988), 1073{1084.
[29℄ E. Suli, P. Houston, and C. Shwab, hp-nite element methods for hyperboli problems, The
Proeedings of the Conferene on the Mathematis of Finite Elements and Appliations:
MAFELAP X (J.R. Whiteman, ed.), Elsevier, 2000, pp. 143{162.
[30℄ E. Suli, C. Shwab, and P. Houston, hp-DGFEM for partial dierential equations with nonnegative harateristi form, Disontinuous Galerkin Methods: Theory, Computation and
Appliations (B. Cokburn, G.E. Karniadakis, and C.-W. Shu, eds.), Leture Notes in Computational Siene and Engineering, vol. 11, Springer Verlag, 2000, pp. 221{230.
[31℄ V. Thomee, Galerkin Finite Element Methods for Paraboli Equations, Springer Verlag, 1997.
[32℄ T. Wihler and C. Shwab, Robust exponential onvergene of the hp disontinuous Galerkin
FEM for onvetion-diusion problems in one spae dimension, East West J. Num. Math. 8
(2000), 57{70.
Shool of Mathematis, University of Minnesota, 206 Churh Street S.E., Minneapolis,
MN 55455, USA.
E-mail address :
astillomath.umn.edu
Shool of Mathematis, University of Minnesota, 206 Churh Street S.E., Minneapolis,
MN 55455, USA.
E-mail address :
okburnmath.umn.edu
Shool of Mathematis, University of Minnesota, 206 Churh Street S.E., Minneapolis,
MN 55455, USA.
E-mail address :
shoetzamath.umn.edu
 rih, Switzerland
Seminar of Applied Mathematis, ETHZ, 8092 Zu
E-mail address :
shwabsam.math.ethz.h