Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids.
In this paper, we present a superconvergence result for the local discontinuous Galerkin (LDG) method for a model elliptic problem on Cartesian grids. We identify a special numerical flux for which the L2-norm of the gradient and the L2-norm of the potential are of orders k+1/2 and k+1, respectively, when tensor product polynomials of degree at most k are used; for arbitrary meshes, this special LDG method gives only the orders of convergence of k and k+1/2, respectively. We present a series of numerical examples which establish the sharpness of our theoretical results.
Introduction
. In this paper, we derive a priori error estimates of the Local
Discontinuous Galerkin (LDG) method on Cartesian grids for the following classical
model elliptic problem:
@n
where\Omega is a bounded domain of R d and n is the outward unit normal to its boundary
we assume that the (d \Gamma 1)-measure of \Gamma D is non-zero.
Recently, Castillo, Cockburn, Perugia and Sch-otzau [3] obtained the first a priori
error analysis of the LDG method for purely elliptic problems. Meshes consisting
of elements of various shapes and with hanging nodes were considered and general
School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, MN 55455
(cockburn@math.umn.edu). Supported in part by the National Science Foundation (Grant DMS-
9807491) and by the University of Minnesota Supercomputing Institute.
y Institut f?r Angewandte Mathematik, Universit?t Heidelberg, INF 293/294, 69120 Heidelberg,
Germany (guido.kanschat@na-net.ornl.gov). This work was supported in part by the ARO
DAAG55-98-1-0335 and by the University of Minnesota Supercomputing Institute. It was carried
out when the author was a Visiting Professor at the School of Mathematics, University of Minnesota.
z Dipartimento di Matematica, Universit'a di Pavia, Via Ferrata 1, 27100 Pavia, Italy
(perugia@dimat.unipv.it). Supported in part by the Consiglio Nazionale delle Ricerche. This work
was carried out when the author was a Visiting Professor at the School of Mathematics, University
of Minnesota.
x School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, MN 55455
(schoetza@math.umn.edu). Supported by the Swiss National Science Foundation (Schweizerischer
Nationalfonds).
2 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
numerical fluxes were studied. It was shown that, for very smooth solutions, the
orders of convergence of the L 2 -norms of the errors in ru and in u are k and k
respectively when polynomials of degree at most k are used. On the other hand,
Castillo [2] and Castillo, Cockburn, Sch-otzau and Schwab [4] proved that, for one-
space dimension transient convection-diffusion problems, the order of convergence of
the error in the energy norm is optimal, that is, k that the so-called
numerical fluxes are suitably chosen. In this paper, we extend these results to the
LDG method on Cartesian grids for the multi-dimensional elliptic model problem
(1.1); we show that the orders of convergence in the L 2 -norm of the error in ru and
respectively, when tensor product polynomials of degree at
least k are used. Our proof of this super-convergence result is a modification of the
analysis carried out in [3]; it takes advantage of the Cartesian structure of the grid
and makes use of a key idea introduced by LeSaint and Raviart [10] in their study of
the original DG method for steady-state linear transport.
Since our analysis is a special modification of that of [3], in order to avoid unnecessary
repetitions, we refer the reader to [3] for a more detailed description of the framework
of our error analysis. The organization of this paper is as follows. In Section 2, we
briefly display the LDG method in compact form, introduce the special numerical flux
on Cartesian grids and present and discuss our main result. In Section 3, the detailed
proofs are given and in Section 4, we present several numerical experiments showing
the optimality of our theoretical results. We end in Section 5 with some concluding
remarks.
2. The main results. In this section we recall the formulation of the LDG
method and identify the special numerical flux we are going to investigate on Cartesian
grids. Then we state and discuss our main results. As pointed out in the introduction,
we refer to [3] for more details concerning the formulation of the LDG method.
2.1. The LDG method. We assume that the problem
domain\Omega can be covered
by a Cartesian grid. To define the LDG method, we rewrite our elliptic model problem
(1.1) as the following system of first-order equations:
\Gammar
Next, we discretize the above problem on a Cartesian grid T . To obtain the weak
formulation with which the LDG is defined, we multiply equations (2.1) and (2.2)
by arbitrary, smooth test functions r and v, respectively, and integrate by parts over
the d-dimensional rectangle K 2 T . Then we replace the exact solution (q; u) by its
approximation (q N ; uN ) in the finite element space MN \Theta VN , where
and
of degree at most k in each variable on Kg:
The LDG method on Cartesian grids for elliptic problems 3
The method consists in finding (q VN such that
Z
Z
Z
Z
Z
Z
for all test functions (r; v) 2 S(K) d \Theta S(K), for all elements . The functions
uN and b q N in (2.5) and (2.6) are the so-called numerical fluxes. These are nothing
but discrete approximations to the traces of u and q on the boundary of the elements
K and are defined as follows. Consider a face e of the d-dimensional rectangle K. If
e lies inside the
domain\Omega\Gamma we define
uN
- ffq N gg
and, if e lies on the boundary of \Omega\Gamma
g N on \Gamma N ;
and b u :=
g D on \Gamma D ;
Moreover, the stabilization parameter C 11 and the auxiliary parameter C 12 are defined
as follows:
where i is a positive real number and v is an arbitrary but fixed vector v with non
zero components; see Fig. 3.1.
2.2. Error analysis on Cartesian grids. To state our main result, we need
to recall some notation and to introduce new hypotheses. We restrict our analysis
to
domains\Omega such that, for smooth data, the solution u of problem (1.1) belongs to
(\Omega\Gamma5 and such that when f is in L
2(\Omega\Gamma and the boundary data are zero, we have the
elliptic regularity result k u Grisvard [8] or [9]. Since the
domain\Omega
will be triangulated by means of a Cartesian grid, the above requirements hold only
if\Omega is a d-dimensional rectangle.
We denote by hK the diameter of an element K, and set, as usual h := maxK2T hK .
We denote by E I the set of all interior faces of the triangulation T , by ED the set of
faces on \Gamma D , and by EN the set of faces on we assume that \Gamma
e. The Cartesian triangulations we consider are regular, that is, if ae K
denotes the radius of the biggest ball included in K,
ae K
Finally, we denote by EN
ae\Omega a closed set containing the intersection between the
and the set fx 0g. Moreover, we assume
that the triangulation T is such that
where K e denotes, from now on, an element containing the face e.
4 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
We are now ready to state our main result.
Theorem 2.1. Assume that the solution (q; u) of (2.1)-(2.4) belongs to H k+1
H k+2 for k - 0; assume also that if the intersection between \Gamma N and fx
non-empty, u belongs to W k+1;1 (EN ). Assume that the Cartesian
grid T is shape-regular, (2.12), and that it satisfies the condition
VN be the approximation of (q; u) given
by the LDG method with k - 0 and numerical fluxes defined by (2.9), (2.11) and by
(2.10).
Then we have
and
where the constant C solely depends on i, k, d, oe and on the norms k u k k+2 and
Several important remarks are in order before we prove this result in the next Section.
Remark 2.2. This theorem is an extension to the bounded domain case of the
corresponding result by Cockburn and Shu [7] for the LDG method for transient
convection-diffusion problems. It is also an extension to the multi-dimensional case of
the results obtained by Castillo, Cockburn, Sch-otzau and Schwab [4] in the one-space
dimension case. The key ingredient of its proof is a super-convergence result of LeSaint
and Raviart [10] used in their study of the original DG method for steady-state linear
transport in Cartesian grids.
Remark 2.3. Note that Theorem 2.1 holds true in the case
approximate solution is piecewise constant. In [3], all the error estimates obtained for
the corresponding LDG method on general grids are valid only for k - 1; moreover,
no order of convergence is numerically observed for
Remark 2.4. From an approximation point of view, the order of convergence in q,
namely, k+1=2, is suboptimal by one half; however, it is confirmed to be sharp by our
numerical experiments in Section 4. For general numerical fluxes and unstructured
grids, an order of convergence in q of only k is obtained; see [3].
Remark 2.5. If we take the more general case
e
e
are constants, we might conceive the possibility that a suitable
tuning of the value of ff could improve the order of convergence in q. However, this
is not true, as will be made clear in the proof of Theorem 2.1 displayed in the next
section. See also [3] for other results about the influence of the value of ff on the
orders of convergence of the general LDG method.
Remark 2.6. In Theorem 2.1 an extra regularity condition on the exact solution u
on the closed set EN containing part of the Neumann boundary is required. If this
condition is dropped, and if 0g is not empty, only an order
of convergence of k in the error in q can be proved by using our technique which
represents a loss of 1=2. Note that whenever it is possible to choose v in such a way
that regularity assumption on the exact
solution is required.
The LDG method on Cartesian grids for elliptic problems 5
3. Proofs. This section is devoted to the proof of Theorem 2.1. For simplicity,
we consider only the case
and\Omega rectangle; see Fig. 3.1. All the arguments we
present in our analysis rely on tensor product structures and can be easily extended
to the case d ? 2.
\GammaE
+Fig. 3.1. The Cartesian grid T and the auxiliary vector v used to define the numerical fluxes.
To prove Theorem 2.1, we follow the approach used by [3]. Thus, we start, in Section
3.1, by briefly reviewing the setting of our error analysis. We proceed in Section
3.2, by introducing the projections \Pi and \Pi which generalize to several space
dimensions the projections used by Castillo, Cockburn, Sch-otzau and Schwab [4] in
their study of the LDG method for transient convection-diffusion problems in one-
space dimension. Then, in Section 3.3, we derive the expressions of the functionals
KA and KB needed in the setting of [3] to get error estimates. To do so, we make
use of a super-convergence result essentially due to LeSaint and Raviart [10], and
whose proof is presented in Section 3.4. The proof of Theorem 2.1 is completed in
Section 3.5.
3.1. The framework of the error analysis. All the following results are collected
from [3]. First, we start by reviewing that, by summation over all elements,
the LDG method can be written in the compact form: Find (q
such that
for all (r; v) 2 MN \Theta VN , by setting
6 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
with
a(q; r) :=
Z
K2T
Z
ur
Z
e
ds \Gamma
Z
e
c(u; v) :=
Z
e
ds
e2ED
Z
e
C 11 uv ds:
The linear forms F , G are defined by
F (r) :=
e2ED
Z
e
Z
e2ED
Z
e
ds
Z
e
We also introduce the semi-norm j (q; u) j 2
A that appears in a natural way in the
analysis of the LDG method and is defined as
Z
e
ds
e2ED
Z
e
To prove error estimates for the LDG method, we follow [3] and introduce two func-
tionals, KA and KB , which capture the approximation properties of the LDG method;
the functionals are related to two suitably chosen projections \Pi and \Pi onto the FE
spaces MN and VN , respectively. Namely, we require KA and KB to satisfy
for any (q; u); (\Phi;
2(\Omega\Gamma3 and
for any (r; v) 2 MN \Theta VN and (q; u) 2 H
By Galerkin orthogonality, all the error estimates can then be solely expressed in
terms of KA and KB as can be seen in the following result.
Lemma 3.1 ([3]). We have
A (q; u; q; u) +KB (q; u):
Furthermore,
KA (q; u; \Phi; ')
with ' denoting the solution of the adjoint problem
@n
and
The LDG method on Cartesian grids for elliptic problems 7
3.2. Projections. In this section we define the projections \Pi and \Pi we are
going to use to prove Theorem 2.1 and list their properties. To this end, we start by
introducing one-dimensional projections. Let I be an arbitrary interval,
and let P k (I) be the space of the polynomials of degree at most k on I . We denote
by - the L 2 (I)-projection onto P k (I), i.e., for a function w 2 L 2 (I) the projection -w
is the unique polynomial in P k (I) satisfying
Z
I
Furthermore, for w
we define the projections - \Sigma w 2 P k (I) by the following
I
On a rectangle we define the following tensor product operators:
with the subscripts indicating the application of the one-dimensional operators - or
- \Sigma with respect to the corresponding variable.
Finally, we define the projections \Pi and \Pi as
In our error analysis, we use key properties of these projections displayed in the
following result.
Lemma 3.2. With the notation indicated in Figure 3.1, we have
Z
Z
2:
We also need several approximation results which we gather in the lemma below.
Lemma 3.3 (Cf. [5]). Let
Furthermore, for any edge e i parallel to the x i -axis, we have
Finally, if u
8 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
3.3. The functionals KA and KB . In this subsection, we obtain the functionals
KA and KB introduced in Section 3.1.
We consider the stabilization parameter C 11 , defined by (2.14), in order to highlight
the fact that any choice of ff 6= 0 in (2.14) deteriorates the rates of convergence of the
estimates of Theorem 2.1.
In [3, Corollary 3.4], KA has been investigated for general projection operators \Pi and
\Pi satisfying the approximation results in Lemma 3.3 with 1. Thus, we we just
report here the final result.
Lemma 3.4 ([3]). Let u
to be given by (2.14). Then, if we set the approximation
property (3.2) holds true with
KA (q; u; \Phi;
\Theta
Furthermore, in the particular case where (\Phi; there holds
KA (q; u; q;
\Theta
In [3], the functional KB was only studied in the case where \Pi and \Pi are L 2 -
projections. Next, we show that a better result for KB can be obtained on Cartesian
grids for the projections defined by (3.4) and the numerical fluxes defined by (2.11).
To obtain such a result, we use the following standard inverse inequality.
Lemma 3.5 (Cf. [5]). There exists a positive constant C solely depending on k, d
and oe such that for all s 2 MN we have
for all K 2 T , e being any side of K.
We set
the sides of \Gamma. We are now ready to state our main lemma.
Lemma 3.6. Let u
to be given by
and let \Pi and \Pi be the operators defined by (3.4). Then, for any (r; w) 2
MN \Theta VN , the approximation property (3.3) holds true, with KB given by
where the constant C solely depends on k, d and oe.
Proof. In order to be able to distinguish the many parts of \Gamma and facilitate the proof
of the above result, we introduce the following notation:
and define E
these boundaries are indicated in Fig. 3.1.
We set - q :=
The LDG method on Cartesian grids for elliptic problems 9
and estimate each of the forms separately.
a. Estimate of T 1 . We have
K2T
Z
K2T
K2T
K2T
b. Estimate of T 2 . We can write
K2T
Z
Z
e
ds
e2ED
Z
e
ds
Taking into account the definition of the fluxes in (2.11) and the properties of the
projection \Pi in Lemma 3.2, we conclude that
Z
Z
e
Z
e
Consequently,
Z
e
ds
Multiplying and dividing each term of the sum by C2
11 , and using the approximation
properties of \Pi, we have
Z
e
ds
0;e
0;e
s+1;Ke
Note that we have used the shape-regularity assumption (2.12) to bound C \Gamma1
11 by
Ke .
c. The estimate of T 4 . We have
Z
e
ds
e2ED
Z
e
ds
e2ED
0;e
e2ED
0;e
K2T
0;e
K2T
G. Kanschat, I. Perugia and D. Sch-otzau
d. Estimate of T 3 . This estimate cannot be obtained as easily as the previous ones
since it is here that the key idea introduced by LeSaint and Raviart [10] has to be
suitably applied.
We start by writing
K2T
Z
Z
e
(ff- u gg +C 12 \Delta [[- u ds \Gamma
Z
e
ds
K2T
Z
K2T
I
Z
e
(ff- ds
Z
e
ds
Again with (2.11), we see that the contribution of an interior element K to this
expression is
Z
Z
ds \Gamma
Z
where the superscript 'out' denotes the traces taken from outside K. Since u out
and [\Piu] out
for the corresponding one-dimensional projection -
i , this
contribution can be written as
Z
Z
ds \Gamma
Z
ds
Z
ds \Gamma
Z
For boundary elements, we add and subtract corresponding terms to obtain
K2T
Z
e
ds
Z
e
ds
ds
K2T
Z
e
ds
Z
e
ds
Z
e
with ZK (r; u) defined in (3.5).
We start by bounding the contributions to T 3 stemming from a boundary edge e
parallel to the x i -axis, 2. Since u 2 H s+2
implies
see [9], by the property (3.3) and the inverse inequality in Lemma 3.5, we get
Z
ds -
The LDG method on Cartesian grids for elliptic problems 11
Here, K e i denotes again the element containing the edge e i . Consequently, the global
contribution to T 3 of the boundary edges belonging to E i n can be bounded
by
0;Ke
For the edges e in EN "E \Gamma , we have to use a different argument. Thus, by Lemma 3.5,
we have
Z
e
ds -k - u k L 1
Hence, by the Cauchy-Schwarz inequality,
sup
and so
Finally, we estimate the contribution ZK (r; u), by using the following super-convergence
result, essentially due to LeSaint and Raviart [10], whose proof is postponed to Section
3.4.
Lemma 3.7. Let ZK (r; u) be defined by (3.5). Then we have for s - 0
By combining the result of Lemma 3.7 with the above estimates of the contribution
of boundary edges, we obtain
Conclusion. The result now follows by simply gathering the estimates for T i ,
obtained above. This completes the proof.
12 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
3.4. Proof of Lemma 3.7. We can write
where
Z
Z
Z
and
Z
Z
Z
The proof of the approximation results for Z K;1 and Z K;2 are analogous; therefore,
we just present the one for Z K;1 , essentially following the same arguments as in [10].
First, we consider Z K;1 on the reference square (\Gamma1; 1) 2 . We claim that
To prove (3.6), fix r are polynomial preserving operators,
holds true for every u 2 Q k (K). Therefore, we just have to consider the cases
.
Let us start with
1 . On
2 we have
1, and on
2 we have
is a polynomial of degree at most k \Gamma 1 in x 1 , we
obtain Z
Z
Thus, Z K;1 (r
1 .
In the case u(x 1
2 , we integrate by parts and obtain
Z
Z
Z
Z
due to the special form of u, we conclude
that Z K;1 (r
2 . This completes the proof of (3.6).
For fixed r 1 2 Q k (K), the linear functional u 7! Z K;1 (r 1 ; u) is continuous on H s+2 (K)
with norm bounded by Ckr 1 k 0;K . Due to (3.6), it vanishes over P s+1 (K) for
k. Hence, by applying Bramble-Hilbert's Lemma (see [6, Lemma 6], for instance), we
obtain for
This proves the assertion on the reference element (\Gamma1; 1) 2 . The general case follows
from a standard scaling argument.
The LDG method on Cartesian grids for elliptic problems 13
3.5. Proof of Theorem 2.1. If the exact solution of our model problem, (q; u),
belongs to H k+1 2 \Theta H k+2
KA (q; u; q; u) - h 2k+1+ff kuk 2
and
with
. The estimate of the error j follows now from
Lemma 3.1. Notice that gives the best order of convergence in h equal to k
.
Our assumptions on the domain imply that the solution ' of the adjoint problem in
Lemma 3.1 belongs to H 2
(\Omega\Gamma and that we have k'k 2 - Ck-k 0 ; see [8, 9]. Hence, we
conclude that
The estimate of ku \Gamma uN k 0 thus follows from Lemma 3.1. Notice that
again the best order of convergence in h which is k + 1.
4. Numerical Experiments. In this section, we display a series of numerical
experiments showing the computed orders of convergence of the LDG method; we
show (i) that the orders given by our theoretical results are sharp, (ii) that they
can deteriorate when the stabilization parameter C 11 is not of order one, (iii) that
the exact capture of the boundary conditions induces an unexpected increase of 1in
the order of convergence of the gradient, and (iv) that the orders of convergence are
independent of the dimension.
In all experiments, we estimate the orders of convergence of the LDG method as
follows. We consider successively refined Cartesian grids T ' , ' - 0, consisting of 2 d '
uniform d-dimensional cubes with corresponding mesh size 2 \Gamma'+1 ; we present results
in two and three space dimensions. If e(T ' ) denotes the error on the '-th mesh, then
the numerical order of convergence is computed as follows:
log
The results have been obtained with the object-oriented C++ library deal.II developed
by Bangerth and Kanschat [1].
4.1. The sharpness of the orders of convergence of Theorem 2.1. We
consider the two-dimensional model problem (1.1) on the
f and boundary conditions chosen in such a way that the exact solution is given by
We consider two cases: In the first, we impose inhomogeneous
Dirichlet boundary conditions on the whole boundary, and in the second, we also impose
inhomogeneous Neumann boundary conditions on the edge f\Gamma1g \Theta (\Gamma1; 1). The
results are contained in Tables 4.1 and 4.2 where the numerical orders of convergence
in the L 2 - and L 1 -norm in u, q 1 and q 2 of the LDG method with Q k elements for
are shown. We take C and the coefficients C 12 as in (2.11) with
14 B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
In

Table

4.1, we report the results for Dirichlet boundary conditions imposed on the
whole boundary. Note that, because of the symmetry of the problem, the orders of
convergence are exactly the same for q 1 and q 2 . For we see the optimal order
of convergence of 1 in the L 2 -norm of the error of both u and q; note that Theorem
2.1 predicts an order of convergence of 1
2 only for q. However, for k - 1 the L 2 -rates
are of order k +1 in u and k
2 in q, in full agreement with Theorem 2.1. The orders
on convergence in the L 1 -norm of the error in u and q appear to be k
respectively.
The results displayed in Table 4.2 are those for the case of inhomogeneous Neumann
boundary conditions on part of the boundary. We see that the orders of convergence
in this case are the same as the ones in the previous case.
Thus, the above experiments show that the orders of convergence given by Theorem
2.1 are sharp.

Table
Orders of convergence for the LDG method with C 11
element
6 0.9683 0.9624 0.9724 0.3856
6 2.9661 2.8316 2.4678 1.9658
6 3.9661 3.8249 3.4676 2.9724
4.2. The effect of the choice of C 11 . Next, we test the effect of the choice of
the coefficients C 11 on the orders of convergence of the LDG method. We consider
the same problem as in the previous experiments, case use Q 1 and Q 2
elements. We only show the numerical orders of convergence for the finest grids.
The LDG method on Cartesian grids for elliptic problems 15

Table
Orders of convergence of the LDG method with C 11
element
6 0.9795 0.9555 1.0303 0.9283 0.9954 0.6270
5 2.9563 2.9428 2.5042 1.9392 2.4631 1.9392
6 2.9770 2.8316 2.5044 1.9658 2.4806 1.9658
6 3.9805 3.8264 3.5024 2.9722 3.4815 2.9721
The results are displayed in Tables 4.3 and 4.4. We must compare all these results
with those with C obtained in the first set of experiments. We see that when
C 11 is of order h \Gamma1 , the order of convergence in u remains but the order of
convergence in q degrades from k
2 to only k, as predicted by our analysis; see
section 3.5.
We also see that taking C at the outflow boundary and C 11 of order one
elsewhere only results in a slight reduction of the L 1 -orders of convergence.
In the remaining cases, we take C 11 to be of order h in all the domain and then in all
but the outflow boundary where it is taken to be of order h \Gamma1 . We observe a slight
degradation of all the orders of convergence.
These results indicate that the best choice of the stabilization parameter C 11 for the
LDG method is to take it of order one, as predicted by our analysis.
4.3. Piecewise polynomial boundary conditions. The purpose of these numerical
experiments is to show that if the boundary data are piecewise polynomials
of degree k, the order of convergence of the L 2 -norm of the error in q is optimal, that
1, and not only k
2 as predicted by Theorem 2.1 and shown to be sharp in
sub-section 4.1.
B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau

Table
Orders of convergence of the LDG method with Q 1 elements.
1=h 5 1.9607 1.9550 1.1409 0.8816
6 1.9792 1.9057 1.1019 0.9366
1=h on
elsewhere 6 1.9646 1.7914 1.4605 0.9268
6 1.8603 1.7887 1.4564 0.9701
1=h on
elsewhere 6 1.8563 1.7887 1.4556 0.9319

Table
Orders of convergence of the LDG method with Q 2 elements.
1=h 5 2.9555 2.9541 2.2223 1.8475
6 2.9754 2.9584 2.1685 1.9228
1=h on
elsewhere 6 2.9634 2.7424 2.4663 1.9427
6 2.8240 2.5482 2.4656 1.9742
1=h on
elsewhere 6 2.8211 2.4554 2.4643 1.9365
We consider two test problems. In the first, we take homogeneous Dirichlet boundary
conditions and f such that the exact solution is
In the
second, we take piecewise quadratic Dirichlet boundary conditions and f such that
the exact solution is
The results of the first problem are reported in Table 4.5 where we can see that the
optimal order of convergence of k for the L 2 - and L 1 -norms of the errors in both
u and q are obtained; the results for are displayed.
The results of the second problem are reported in Table 4.6, where we see that the
optimal order of convergence of k for the L 2 - and L 1 -norms of the errors in both
u and q are obtained for k - 2, as claimed. For 2, the order of convergence in the
-norm of the error in q is
2 only which nothing but the order of convergence
predicted by Theorem 2.1.
To better understand this phenomenon, we plot the errors in q 1 for Q 1 and Q 2 elements
in Figs. 4.1 and 4.2, respectively; the triangulation has 16 \Theta 16 elements and
corresponds to the index We immediately see the oscillatory behavior of the
error typical of finite element methods. In Fig. 4.1, we see that the error obtained
with elements is bigger at the boundary than at the interior. This, together with
the fact that the order of convergence in L 2 is 3
2 whereas the order of convergence in
L 1 is only 1, suggests that the error at the boundary is a factor of order
bigger
The LDG method on Cartesian grids for elliptic problems 17
than the error at the interior of the domain. On the other hand, the behavior of the
error with Q 2 elements is rather different, as can be seen in Fig. 4.2. Indeed, the
error behaves in the same way at the boundary and at the interior; this is further
confirmed by the fact that both the the order of convergence in L 2 and the one in L 1
are equal to k + 1.
These experiments justify our contention that the optimal order of convergence in q
can be reached if the boundary conditions are piecewise polynomials of degree k. Our
theoretical analysis does not explain this phenomenon.

Table
Orders of convergence for the LDG method with C
element
6 0.9456 0.9658 0.9662 0.9483
6 2.0213 1.9878 2.0003 1.9858
6 2.9815 3.0150 2.9855 3.0161
6 4.0247 3.9918 4.0041 3.9748

Table
Orders of convergence for the LDG method with C 11
element
6 2.0015 1.9775 1.4976 1.0091
6 2.9815 3.0150 2.9855 3.0162
6
B. Cockburn, G. Kanschat, I. Perugia and D. Sch-otzau
-2e-022e-026e-02Fig. 4.1. The error in the first component of the gradient for
Fig. 4.2. The error in the first component of the gradient for
The LDG method on Cartesian grids for elliptic problems 19
4.4. A three-dimensional example. In this experiment, we consider the model
problem (1.1) on the three-dimensional
We take Dirichlet
boundary conditions and f such that the exact solution is
The results are displayed in Table 4.7; the computation on level 5 with Q 2 did not
fit into the computers available to us. We can see that the orders of convergence
are similar to those obtained in the corresponding two-dimensional test problem in
the previous sub-section, cf. Table 4.6. This gives an indication that the orders of
convergence of the LDG method in three space dimension behave in the same way
they do in the two-dimensional case.

Table
Orders of convergence for the LDG method with C
element
3 2.9204 2.8642
5. Concluding remarks. In this paper we have shown that the LDG method
on Cartesian grids and with a special numerical flux super-converges; the proof of
this result is based on suitable defined projections \Pi and \Pi exhibiting a tensor product
structure. This work extends the corresponding result by LeSaint and Raviart
[10] for the DG method for linear hyperbolic problems and that by Castillo [2] and
Castillo, Cockburn, Sch-otzau and Schwab [4] for the LDG method applied to the
one-dimensional transient convection-diffusion. Extensions of this work to more general
elliptic and both steady and transient convection-diffusion problems can easily
be made.



--R

Concepts for object-oriented finite element software - the deal
An optimal error estimate for the local discontinuous Galerkin method
An a priori error analysis of the local discontinuous Galerkin method for elliptic problems
An optimal a priori error estimate for the hp-version of the local discontinuous Galerkin method for convection-diffusion prob- lems
The finite element method for elliptic problems
General Lagrange and Hermite interpolation in R n with applications to finite element methods
The local discontinuous Galerkin finite element method for convection-diffusion systems
Elliptic problems in nonsmooth domains

On a finite element method for solving the neutron transport equa- tion
--TR

--CTR
Paul Castillo, A review of the local discontinuous Galerkin (LDG) method applied to elliptic problems, Applied Numerical Mathematics, v.56 n.10, p.1307-1313, October 2006
Ilaria Perugia , Dominik Schtzau, On the Coupling of Local Discontinuous Galerkin and Conforming Finite Element Methods, Journal of Scientific Computing, v.16 n.4, p.411-433, December 2001
Ilaria Perugia , Dominik Schtzau, Thehp-local discontinuous Galerkin method for low-frequency time-harmonic Maxwell equations, Mathematics of Computation, v.72 n.243, p.1179-1214, July
Bernardo Cockburn , Chi-Wang Shu, RungeKutta Discontinuous Galerkin Methods for Convection-Dominated Problems, Journal of Scientific Computing, v.16 n.3, p.173-261, September 2001
finite elements;elliptic problems;discontinuous Galerkin methods;cartesian grids;superconvergence