An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems.
In this paper, we present the first a priori error analysis for the local discontinuous Galerkin (LDG)  method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L2-norm of the gradient and the L2-norm of the potential are of order k and k+1/2, respectively, when polynomials of total degree at least k are used; if stabilization parameters of order h-1 are taken, the order of convergence of the potential increases to k+1. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains.
Introduction
. In this paper, we present the first a priori error analysis of the
Local Discontinuous Galerkin (LDG) method 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
for the sake of simplicity, we assume that the (d \Gamma 1)-dimensional measure
of \Gamma D is non-zero.
The LDG method was introduced by Cockburn and Shu in [25] as an extension to
general convection-diffusion problems of the numerical scheme for the compressible
Navier-Stokes equations proposed by Bassi and Rebay in [6]. This scheme was in turn
an extension of the Runge-Kutta Discontinuous Galerkin (RKDG) method developed
Scientific Computing Program, School of Mathematics, University of Minnesota, Vincent Hall,
Minneapolis, MN 55455, email: castillo@math.umn.edu.
y School of Mathematics, University of Minnesota, Vincent Hall, Minneapolis, MN 55455, email:
cockburn@math.umn.edu. Supported in part by the National Science Foundation (Grant DMS-
9806956) and by the University of Minnesota Supercomputing Institute.
z Dipartimento di Matematica, Universit'a di Pavia, Via Ferrata 1, 27100 Pavia, Italy, email:
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, email:
schoetza@math.umn.edu. Supported by the Swiss National Science Foundation (Schweizerischer Na-
tionalfonds).
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
by Cockburn and Shu [19, 21, 23, 24, 26] for nonlinear hyperbolic systems. The LDG
method is one of several discontinuous Galerkin methods which are being vigorously
studied, especially as applied to convection-diffusion problems, because of their applicability
to a wide range of problems, their properties of local conservativity and high
degree of locality, and their flexibility in handling adaptive hp-refinement. The state
of the art of the development of discontinuous Galerkin methods can be found in the
volume [20] edited by Cockburn, Karniadakis and Shu.
Let us give the reader familiar with (classical and stabilized) mixed and mortar finite
element methods for elliptic problems an idea of what kind of method is the LDG
method.
ffl The LDG is obtained by using a space discretization that was originally applied to
first-order hyperbolic systems. Hence, to define the method, we rewrite our elliptic
model problem as a system of first-order equations and then we discretize it. Thus,
we introduce the auxiliary variable obtain the equations
\Gammar
Since these are nothing but the equations used to define classical mixed finite element
methods, the LDG method can be considered as a mixed finite element method.
However, the auxiliary variable q can be eliminated from the equations which is usually
not the case for classical mixed methods.
ffl In the LDG method, local conservativity holds because the conservation laws (1.2)
and (1.3) are weakly enforced element by element. In order to do that, suitable
discrete approximations of the traces of the fluxes on the boundary of the elements
are needed which are provided by the so-called numerical fluxes; these are widely
used in the numerical approximation of non-linear hyperbolic conservation laws and
are nothing but the so-called approximate Riemann solvers; see Cockburn [17]. As in
the case of non-linear hyperbolic conservation laws, these numerical fluxes enhance
the stability of the method and hence the quality of its approximation. This is why the
LDG method is strongly related to stabilized mixed finite element methods; indeed,
the stabilization is associated with the jumps of the approximate solution across the
element boundaries.
ffl The LDG method allows general meshes with hanging nodes and elements of several
shapes since no inter-element continuity is required. This is also a key property
of the mortar finite element method. However, in the LDG method there are no
Lagrange multipliers associated to the continuity constraint; instead, the Lagrange
multiplier is replaced by fixed functions of the unknowns which are nothing but the
above mentioned numerical fluxes.
ffl In the LDG method, on each element, both the approximation to u as well as
the approximations to each of the components of q belong to the same space, which
is very convenient from an implementational point of view. Moreover, the lack of
continuity constraints across element boundaries in the finite element spaces renders
the coding of the hp-version of the LDG method much simpler than that of standard
mixed methods.
us briefly describe the recent work on error analysis of DG methods in order
to put our results into perspective. Analyses of the LDG method in the context of
An analysis of the LDG method for elliptic problems 3
transient convection-diffusion problems have been carried out by Cockburn and Shu
[25], by Cockburn and Dawson [18], by Castillo [14] and more recently by Castillo,
Cockburn, Sch-otzau and Schwab [15].
The DG method of Baumann and Oden [7, 8, 9, 32] has also been analyzed by several
authors. Oden, Babu-ska and Baumann [31] studied this method for one dimensional
elliptic problems and later Wihler and Schwab [41] proved robust exponential rates
of convergence of the Oden and Baumann DG method for stationary convection-diffusion
problems also in one space dimension. Rivi'ere, Wheeler and Girault [35] and
Rivi'ere and Wheeler [34] analyzed several variations of the DG method of Baumann
and Oden (involving interior penalty techniques) as applied to non-linear convection-diffusion
problems and, finally, S-uli, Schwab and Houston [38] synthesized the self-adjoint
elliptic, parabolic, and hyperbolic theory by extending the analysis of these
DG methods to general second-order linear partial differential equations with non-negative
characteristic form.
As applied to purely elliptic problems, the LDG method and the method of Baumann
and Oden are strongly related to the so-called interior interior penalty (IP) methods
explored mainly by Babu-ska and Zl'amal [3], Douglas and Dupont [27], Baker [4],
Wheeler [39], Arnold [2] and later by Baker, Jureidini and Karakashian [5], by Rusten,
Vassilevski, and Winther [36] and by Becker and Hansbo [10]. All of these DG methods
for elliptic problems can be recast within a single framework as shown by Arnold,
Brezzi, Cockburn and Marini [1]; this framework should provide a basis for a better
understanding of the connections among them and lead to a unified error analysis of
these methods. As a contribution to this effort, we present in this paper an a priori
error analysis of the LDG method for purely elliptic problems.
We show that if polynomials of degree at least k are used in all the elements, the rate
of convergence of the LDG method in the L 2 -norm of u and q are of order k
and k, respectively, when the stabilization or penalization parameter C 11 is taken to
be of order one. When the stabilization parameter C 11 is taken to be of order h \Gamma1 ,
the order of convergence of u is proven to be k + 1, as expected. Indeed, this is what
happens for the interior penalty methods and for the modifications of the method of
Bassi and Rebay [6] studied by Brezzi, Manzini, Marini, Pietra and Russo [13]; the
penalization parameters of these methods are also of order h \Gamma1 . These results are
summarized in the Table 1.1.

Table
Orders of convergence for k - 1.
method penalization
interior penalty O(h
Brezzi et al. O(h
Finally, let us point out that the order of convergence of u for the DG method for
purely convective problems is k +1=2. This order of convergence was proven by Johnson
and Pitk-aranta [30] and later confirmed by Peterson [33] to be sharp. Whether
4 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
or not a similar phenomenon is actually taking place for the LDG method, with the
stabilization parameter C 11 of order one, as applied to elliptic problems remains to
be investigated. Our numerical experiments for the LDG method have all been performed
on structured and unstructured triangulations without hanging nodes and give
the optimal orders of convergence of k for u and q, respectively, with C 11
of order h \Gamma1 and, remarkably, with C 11 of order one.
The organization of the paper is as follows. In section 2, we present the LDG methods
and state and discuss our main a priori error estimates. We also give a brief sketch
of the proofs in order to display the ideas of our analysis. The analysis is carried out
in full detail in section 3 and several possible extensions are indicated in section 4.
In section 5, we present several numerical experiments testing the sharpness of our
theoretical results. We end in section 6 with some concluding remarks.
2. The main results. In this section, we formulate the LDG method and show
that it possesses a well-defined solution. We then state and discuss our main result
and, finally, we display the main ideas of our error analysis.
We assume, to avoid unnecessary technicalities, that the exact solution u of our model
problem (1.1) belongs to H 2
(\Omega\Gamma and that the solution of the so-called adjoint problem
satisfies the standard ellipticity regularity property. Extensions to more general
situations are discussed in section 4.
2.1. The LDG method. To introduce our LDG method, we consider a general
discontinuous Galerkin (DG) method of which the LDG method is a particular but
important case. We consider a general triangulation T with hanging nodes whose
elements K are of various shapes.
To obtain the weak formulation with which our DG method is defined, we multiply
equations (1.2) and (1.3) by arbitrary, smooth test functions r and v, respectively,
and integrate by parts over the element K 2 T to obtain
Z
Z
ur
Z
Z
Z
Z
Note that the above equations are well defined for any functions (q; u) and (r; v) in
Next, we seek to approximate the exact solution (q; u) with functions (q in the
finite element space MN \Theta VN ae M \Theta V , where
and the local finite element space S(K) typically consists of polynomials. Note that
for a given element K, the restrictions to K of uN and of each of the components of q N
belong to the same local space; this renders the coding of these methods considerably
simpler than that of the standard mixed methods, especially for high-order polynomial
An analysis of the LDG method for elliptic problems 5
local spaces. In order to ensure the existence of the approximate solution of the DG
method, we require the following local and quite mild condition:
Z
Other than these properties, there is complete freedom in the choice of the local spaces
since no inter-element continuity is required at all.
The approximate solution (q defined by using the above weak formula-
tion, that is, by imposing that for all K 2 T , for all
Z
Z
Z
Z
Z
Z
where the numerical fluxes b uN and b q N have to be suitably defined in order to ensure
the stability of the method and in order to enhance its accuracy.
As pointed out in the introduction, we can see that the numerical fluxes b uN and b q N
are nothing but discrete approximations to the traces of u and q on the boundary of
the elements. To define these numerical fluxes, let us first introduce some notation.
be two adjacent elements of T ; let x be an arbitrary point of the set
which is assumed to have a non-zero (d \Gamma 1)-dimensional measure,
and let n + and n \Gamma be the corresponding outward unit normals at that point. Let
(q; u) be a function smooth inside each element K \Sigma and let us denote by (q
the traces of (q; u) on e from the interior of K \Sigma . Then, we define the mean values
ff\Deltagg and jumps [[\Delta]] at x 2 e as follows:
Note that the jump in u is a vector and the jump in q is a scalar which only involves
the normal component of q.
We are now ready to introduce the expressions that define the numerical fluxes in
(2.2) and (2.3). If the set e is inside the
domain\Omega\Gamma we take
b u
ffugg
where the auxiliary parameters C 11 , C 12 and C 22 depend on x 2 e and are still at
our disposal. The boundary conditions are imposed through a suitable definition of
(b q; b u), namely,
g N on \Gamma N ;
and
b u :=
where the superscript + denotes quantities related to the element the edge we are
considering belongs to, and We remark that the definition of (b q; b u) on
6 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
the boundary
@\Omega is still of the form (2.4) if the exterior traces are taken to
be
and C 12 is chosen such that C 12
Let us stress once more that the numerical fluxes we just defined are nothing but a
particular case of the so-called approximate Riemann solvers widely used in numerical
schemes for non-linear hyperbolic conservation laws.
This completes the definition of our DG method. The LDG method is obtained when
C 22 j 0; in this case, the function q N can be locally solved in terms of uN and hence
eliminated from the equations, as can be easily seen from (2.3). This local solvability
gives its name to the LDG method.
That this DG method actually defines a unique approximate solution depends in a
crucial way on the coefficients C 11 and C 22 . Indeed, we have the following result.
Proposition 2.1 (Well posedness of the DG method). Consider the DG method
defined by the weak formulation (2.2) and (2.3), and by the numerical fluxes in (2.4)
and (2.5). If the coefficients C 11 are positive and the coefficients C 22 are non-negative,
the DG method defines a unique approximate solution (q
Notice that the above result, which we prove in the next subsection, is independent
of the auxiliary vector parameter C 12 . The choice C
symmetry and stability of the DG method. Finally, let us point out that the role
of the auxiliary parameters C 11 and C 22 is to enhance the stability and hence the
accuracy of the method.
2.2. The classical mixed setting. The study of our DG method is greatly
facilitated if we recast its formulation in a classical mixed finite element setting. To
do that, we need to introduce some notation. We denote by E i the union of all interior
faces of the triangulation T , by ED the union of faces on \Gamma D , and by EN the union of
we assume that \Gamma
Now, we sum equations (2.2) and (2.3) over all elements and obtain, after some
simple manipulations, that the DG approximation (q N ; uN ) is the unique solution of
the following variational problem: find (q VN such that
\Gammab(v;
for all (r; v) 2 MN \Theta VN . Here, the bilinear forms a, b and c are given by
a(q; r) :=
Z\Omega
Z
C 22 ds
Z
K2T
Z
ur
Z
ds \Gamma
Z
c(u; v) :=
Z
ds
Z
ED
C 11 uv ds:
The linear forms F , G are defined by
F (r) :=
Z
ED
Z
C 22 (g N \Delta n)(r \Delta n) ds;
Z\Omega
Z
ED
ds
Z
An analysis of the LDG method for elliptic problems 7
Note that these two linear forms contain all the data of the problem. In particular,
they contain both the Dirichlet and Neumann data, which is not the case for the
classical mixed finite element methods.
Equations (2.6) and (2.7) can be rewritten in a more compact form as follows:
by setting
We end this section by proving Proposition 2.1.
Proof of Proposition 2.1. Due to the linearity and finite dimensionality of the problem,
it is enough to show that the only solution to the equations (2.6) and (2.7) with
taking
adding the two equations, we get
which implies q
As a consequence, equation (2.6) becomes
after integration by parts, the form b(\Delta; \Delta) can be rewritten as
K2T
Z
Z
ds
Z
ED
we get that
K2T
Z
Hence, owing to (2.1), on every K 2 T and since uN is a continuous function
equal to zero on the Dirichlet boundary, we get that uN j 0. This completes the proof
of Proposition 2.1. 2
2.3. A priori error estimates. In this section we state and discuss our a priori
error bounds for the DG method. As pointed out at the beginning of this section,
we restrict our analysis to
domains\Omega such that, for smooth data, the solution u of
problem (1.1) belongs to H
We also assume that when f is in L
2(\Omega\Gamma and the
boundary data are zero, we have the elliptic regularity result k u
Grisvard [28] or [29].
We assume that every element K of the triangulation T is affine equivalent, see [16,
Section 2.3], to one of several reference elements in an arbitrary but fixed set; this
allows us to use elements of various shapes with possibly curved boundaries. For each
by hK the diameter of K and by ae K the diameter of the biggest ball
included in K; we set, as usual, h := maxK2T hK . The triangulations we consider can
8 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
have hanging nodes but have to be regular , that is, there exists a positive constant oe
such that
ae K
see [16, Section 3.1]. Moreover, we let the maximum number of neighbors of a given
element K be arbitrary but fixed. To formally state this property, we need to introduce
the set hK; K 0 i defined as follows:
interior of @K " @K 0 otherwise:
Thus, we assume that there exists a positive constant 1 such that, for each element
These three hypotheses allow for quite general triangulations and are not restrictive
in practice.
The only assumptions we use for the local space S(K) are that it contains the space
of polynomials of degree at most k on K and satisfies (2.1).
Next, we introduce a semi-norm that appears in a natural way in the analysis of
these methods. We denote by H k (D), D being a domain in R d , the Sobolev spaces of
integer orders, and by k \Delta k k;D and j \Delta j k;D the usual norms and semi-norms in H k (D)
and H k (D) d ; we omit the dependence on the domain in the norms whenever
We define j (q; u) j 2
A := A(q; u; q; u), that is,
where
\Theta 2 (q; u) :=
Z
C 22 ds
Z
ds
Z
ds
Z
ED
We assume that the stabilization coefficients C 11 and C 22 defining the numerical fluxes
in (2.4) and (2.5) are defined as follows:
ih ff
-h fi
independent of the mesh-size and jC 12 j
of order one. Our main result will be written in terms of the parameters - ? and - ?
defined as follows:
An analysis of the LDG method for elliptic problems 9
We are now ready to state our main result.
Theorem 2.2. Let (q; u) be the solution of (1.2)-(1.5) and let (q N ; uN ) be the approximate
solution given by the DG method (2.2) and (2.3). We assume the hypotheses
on the local spaces and on the form of the stabilization parameters described
above. The triangulations are assumed to satisfy the hypothesis (2.11); if ff 6= 0 or
fi 6= 0, we also assume that hypothesis (2.12) is satisfied. Then we have that, for
where C solely depends on oe, ffi (not when
Let us briefly discuss the above result:
ffl We begin by noting that the orders of convergence depend on the size of the stabilization
parameters C 11 and C 22 only through the quantities - ? and - ? . This fact
has several important consequences:
ffi The same orders of convergence are obtained with either C 22 = 0 or C 22 of
order h. This means that there is no loss in the orders of convergence if
instead of penalizing the jumps of the normal component of q N with a C 22
or order h, no penalization at all (the LDG method) is used.
ffi The same orders of convergence are obtained with either the LDG method
of order one or C 11 of order h \Gamma1 and C 22 of order one.
In general, the same orders of convergence are obtained by taking (ff;
or by taking (ff;
ffi The most remarkable cases occur when \Gammaff; fi 2 f0; 1g since it is for those values
that - ? and - ? achieve their maximum and minimum. The corresponding
orders of convergence are displayed in Table 2.1 for k - 1.

Table
Orders of convergence for
ffl In the case 1 - k - s, that is, when the degree of the polynomial approximation
is less than needed to fit the smoothness of the exact solution, we see in Table 2.2
that the best orders of convergence for
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
respectively, are obtained for both C 11 and C 22 of order one. When C 22
is taken to be of order h or equal to zero, the stability of the method is weakened
and, as a consequence, a loss in the orders of convergence of 1=2 takes place. If
now C 11 is taken to be of order h \Gamma1 , the full order of convergence in the error of
the potential is recovered. The numerical experiments of section 5 show that these
orders of convergence are actually achieved. However, the expected loss in the orders
of convergence when C 11 is taken of order one is not observed, which shows that
in practice the LDG method is essentially insensitive to the size of the stabilization
parameter C 11 .
ffl The influence of the choice of the coefficients C 12 on the accuracy has not been
explored in this paper; we only assume those to be of order one. In [22] it is shown
that the LDG method, with a suitable choice of the coefficients C 12 , still gives the
orders of convergence of k
respectively, if Cartesian grids and tensor product polynomials of degree k in each
variable are used.

Table
Orders of convergence for
ffl For the case k - s + 1, that is, when the degree of the polynomial approximation
is more than needed to fit the smoothness of the exact solution, we see in Table 2.3
that the LDG method performs at least as well as all the other methods; it performs
better if C 11 is of order h \Gamma1 .

Table
Orders of convergence for
An analysis of the LDG method for elliptic problems 11
ffl In the case the DG method converges provided C 22 6= 0; in particular, for
constant coefficients C 11 and C 22 , we obtain estimates of order one for
. This is one of the few finite element methods for second-order
elliptic problems that actually converges for piecewise-constant approximations.
When C 22 = 0, that is, for the LDG method, our numerical results, which we do not
report in this paper, show that there is no positive order of convergence in this case,
as predicted by Theorem 2.2.
Finally, let us point out that the hypothesis (2.12) is not necessary when
2.4. The idea of the proof. The proof of Theorem 2.2 will be carried out in
section 3. The purpose of this section is to display as clearly as possible the basic
ingredients and the main steps of our error analysis. As usual, we express the error
as the following sum:
where \Pi and \Pi are projections from M and V onto the finite element spaces MN
and VN , respectively.
a. The basic ingredients. The basic ingredients of our error analysis are two. The
first one is, as it is classical in finite element error analysis, the so-called Galerkin
orthogonality property, namely,
This property is a straightforward consequence of the consistency of the numerical
fluxes.
The second ingredient is a couple of inequalities that reflect the approximation properties
of the projections \Pi and \Pi, namely,
for any (q; u); (\Phi;
for any (r; v) 2 MN \Theta VN and (q; u) 2 H
As we show next, all the error estimates we are interested in can be obtained solely
in terms of functionals KA and KB .
b. The estimate of the error in the A-semi-norm. We have the following result.
Lemma 2.3. We have
A (q; u; q; u) +KB (q; u):
Proof. j (\Delta; \Delta) j A is a semi-norm and, hence,
Since
by the definition of A, (2.9);
by assumption (2.19);
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
we have that
and so,
The estimate now follows from a simple application of the assumption (2.18). This
completes the proof.
c. Estimate of the error in u in non-positive order norms. To obtain an
estimate of k e u k \Gammat;D , where t is a natural number and D is a sub-domain of \Omega\Gamma
we only have to obtain an estimate of the error in the approximation of the linear
functional  denotes the L 2 -inner product, by  (u N ) since
In this paper, we are only interested in the case but we give here the general
argument to stress the fact that it is essentially the same for all natural numbers
t. Error estimates in negative order norms are very important, as we point out in
section 4 of this paper.
To obtain our estimate, we need to introduce the solution ' of the so-called adjoint
problem, namely,
@n
Lemma 2.4. Let t be a natural number. Then, we have
KA (q; u; \Phi; ')
with ' denoting the solution of (2.21)-(2.23) and
Proof. Since ' is the solution of the adjoint equation, it is easy to verify that if we
set
for all (s; w) 2 M \Theta V ; indeed, note that problem (1.1) can be rewritten as in (2.8).
Taking (s;
by the definition of A, (2.9);
, by the assumption (2.19) and the estimate (2.20), we
obtain
An analysis of the LDG method for elliptic problems 13
and hence,
The estimate now follows from a simple application of assumption (2.18), and from
the definition of a non-positive order norm. This completes the proof.
d. Conclusion. Thus, in order to prove our a priori estimates, all we need to do
is to obtain the functionals KA and KB ; this will be carried out in the next section.
Then, Theorem 2.2 will immediately follow after a simple application of Lemmas 2.3
and 2.4.
3. Proofs. In this section, we prove our main results. We proceed as follows.
First, we obtain the functional KA for general projection operators \Pi and \Pi. To
obtain the functional KB , the projections \Pi and \Pi are taken to be the standard L 2 -
projections, just as done by Cockburn and Shu [25] in their study of the LDG method
for transient convection-diffusion problems.
3.1. Preliminaries. The following two lemmas contain all the information we
actually use about our finite elements. The first one is a standard approximation result
for any linear continuous operator \Pi from H r+1 (K) onto S(K) satisfying
any w 2 P k (K); it can be easily obtained by using the techniques of [16]. The second
one is a standard inverse inequality.
Lemma 3.1. Let w \Pi be a linear continuous operator from
H r+1 (K) onto S(K) such that
for some constant C that solely depends on oe in inequality (2.11), k, d and r.
Lemma 3.2. There exists a positive constant C inv that solely depends on oe in inequality
(2.11), k and d, such that for all s 2 S(K) d we have
for all K 2 T .
We are now ready to prove our main result.
3.2. The functional KA . In this subsection we determine the functional KA
in (2.18), up to a multiplicative constant independent of the mesh-size. We start by
giving an expression for KA which is valid for coefficients C 11 and C 22 that vary from
face to face, for any regularity of the solution. Then we write KA for
the particular choice (2.15), (2.16) of C 11 and C 22 in Theorem 2.2.
Let \Pi and \Pi be arbitrary projections onto VN and MN , respectively, satisfying
(component-wisely) the assumptions in Lemma 3.1.
Lemma 3.3. Assume (q; u) 2 H s+1
the approximation property (2.18) holds true with
KA (q; u; \Phi; ') =X
14 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
where
K2T
K2T
K2T
22 h 2 minfs;kg+1
K2T
22 h 2 minft;kg+1
K2T
K2T
K2T
K2T
K2T
K2T
ii := supfC ii 2. The
positive constant C is independent of the mesh-size but depends on the approximation
constants in Lemma 3.1 and on the coefficients C 12 .
Furthermore, in the case where (\Phi;
KA (q; u; q;
Proof. We set, for convenience, - q :=
We start by writing
and then proceed by estimating each of the forms on the right-hand side separately.
The form a(\Delta; \Delta) can be written as
K2T
Z
C 22 (- q \Delta n)(- \Phi \Delta n) ds
Z
@Kn@\Omega C 22 (- q \Delta n)(- \Phi \Delta n) ds
Z
@Kn@\Omega C 22 (- out
ds
where the superscript 'out' denotes quantities taken on @K n
@\Omega from outside K. By
repeated applications of the Cauchy-Schwarz's inequality, we obtain that ja(- q ; - \Phi )j
is bounded by
K2T
22 - q \Delta nk 0;@K"\Gamma N kC2
22 - \Phi \Delta nk 0;@K"\Gamma N
22 - q \Delta nk
22 - out
22 - \Phi \Delta nk
K2T
K2T
An analysis of the LDG method for elliptic problems 15
K2T
22 k- q \Delta nk 2
K2T
22 k- \Phi \Delta nk 2
Now, a straightforward application of Lemma 3.1 yields
To deal with the second term, we first note that
K2T
r-
Z
ds
Z
ds
and obtain, after repeated applications of the Cauchy-Schwarz's inequality with suitably
chosen weights, that jb(- u ; - \Phi )j is bounded by
K2T
K2T
k-
K2T
K2T
k-
K2T
Once again, a straightforward application of Lemma 3.1 gives that
For the third term, we use the same arguments to get
Finally, proceeding as above, we get
K2T
K2T
This proves the first assertion. The second one immediately follows by taking into
account that
and the proof of the lemma is complete.
The following result is a straightforward consequence of the estimates in Lemma 3.3.
Corollary 3.4. Let (q; u) 2 H s+1 2 \Theta H s+2
be the exact solution of
0, be the solution of the dual problem (2.21)-(2.23),
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
and \Gammar'. Assume that coefficients C 11 and C 22 satisfy (2.15), (2.16) . Then
there exist a constant C that solely depends on oe, i, - , k and d such that
KA (q; u; \Phi;
for k - 1. Moreover,
KA (q; u; q;
Proof. From Lemma 3.3, we get
KA (q; u; \Phi; ') =C
\Theta h minfs;kg+1
and
KA (q; u; q;
\Theta

Note that the above results hold for arbitrary ff and fi. If now we restrict ourselves
to the case of Theorem 2.2, the result follows after simple algebraic manipulations.
3.3. The functional KB . In this subsection we determine the functional KB
satisfying (2.19), up to a multiplicative constant independent of the mesh-size. Here,
we take \Pi to be L 2 -projection and Again, we start by determining
expressions which are valid for varying coefficients C 11 and C 22 , and we conclude by
considering the particular case of Theorem 2.2. We proceed as follows. We show that
there exists a form j (\Delta; \Delta) j B , which is a semi-norm in both variables, such that for any
with C independent of the mesh-size. Then it is enough to determine KB such that
for any (q; u) 2 M \Theta V . In the following lemma we prove that (3.1) is satisfied by
defining the semi-norm j (\Delta; \Delta) j B as
Z
ED
ds
Z
ds
Z
C 22
where for each internal or Neumann boundary face e we set
C 22 (x) otherwise:
Note that only the function values along faces enter the j (\Delta; \Delta) j B semi-norm. As can
be inferred from the proof of Lemma 3.5 below, this is due to the particular choice of
\Pi and \Pi as L 2 -projections.
An analysis of the LDG method for elliptic problems 17
Lemma 3.5. Let \Pi and \Pi be the L
-projection and L
-projection onto VN
and MN , respectively, and j (\Delta; \Delta) j B be defined by (3.3). Then (3.1) holds true, with a
constant C that solely depends on oe, k and d.
Proof. Setting by the definition of the form
A in (2.9),
Using Cauchy-Schwarz's inequality and the fact that \Pi is the L
d -projection, we
obtain
'Z
C 22 ds
Z
ds
'Z
ds
Z
ds
Furthermore,
Z
ds
Z
ED
ds
Multiplying and dividing by C2
11 and then applying Cauchy-Schwarz's inequality, we
obtain
'Z
ds
Z
ED
ds
'Z
ds
Z
ds
Analogously,
Z
(ff- ds
Z
ds
'Z
ds
Z
ds
'Z
(ff- ds
Z
ds
The first factor can be estimated as follows:
Z
ds
Z
ds -
Z
C 22 ds
Z
ds
Z
ds
Z
ds
Z
ds
Z
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
. By the inverse
inequality in Lemma 3.2,
Z
ds
Z
ds -
K2T
Z
e
ds
K2T
K2T
@Kg. Thus, combining the above estimates, we get
Finally,
Z
ds
Z
ED
ds
'Z
ds
Z
ED
ds
'Z
ds
Z
ED
ds
To complete the proof, we simply have to gather the estimates of the terms T i ,
4, and apply once again the Cauchy-Schwarz's inequality.
The function KB can be easily defined by applying the estimates in Lemma 3.1 to
defined in (3.3).
Lemma 3.6. For any (q; u) 2 H s+1 2 \ThetaH s+2
the approximation property
(3.2) holds true with
K2T
/e
K2T
e
where e
and C is a
constant independent of the mesh-size and solely depending on the approximation and
inverse inequality constants (cf. Lemmas 3.1 and 3.2).
From this lemma, we immediately obtain the following result.
Corollary 3.7. Let (q; u) 2 H s+1 2 \Theta H s+2 , s - 0. Assume that the coefficients
C 11 and C 22 satisfy (2.15), (2.16). The triangulations are assumed to satisfy
the hypothesis (2.11); if ff 6= 0 or fi 6= 0, we also assume that hypothesis (2.12) is
Then there exists a constant C that solely depends on oe, ffi, i, - , k and d such that
the constant C is independent of ffi.
An analysis of the LDG method for elliptic problems 19
Proof. If we take the coefficients C 11 and C 22 as in Theorem 2.2, we get, after a simple
computation,
/e
and
where the parameter ffi is defined in (2.12), and b
Note that the left-hand sides of the above inequalities are trivially uniformly bounded
when otherwise, we must invoke the hypothesis (2.12) to ensure the
boundedness of these quantities. We emphasize that this is the only instance in which
this hypothesis is used.
Hence we obtain
\Theta

where C is independent of the mesh-size but depends on ffi and on the approximation
and inverse inequality constants, and
-h fi otherwise:
The result follows after simple algebraic manipulations.
3.4. The proof of Theorem 2.2. From Lemma 2.3 and Corollaries 3.4 and 3.7,
we get
and since minfPA ; the estimate
follows.
Next, consider the L 2 -norm of the error
=\Omega in
Lemma 2.4. From the elliptic regularity of the adjoint problem (2.21)-(2.23), we
have . The estimates of ku \Gamma uN k 0 directly follow
from substituting the expression of KA (q; u; \Phi; ') given by Corollary 3.4, and the expressions
of KB (q; u), KB (\Phi; ') given by Corollary 3.7 in (2.24), and bounding k\Phik 1
and k'k 2 by k-k 0 . Indeed, we get
and since minfQA j
follows with
This completes the proof of Theorem 2.2.
4. Extensions. In this section, we indicate how to the extend our main result
in several possible directions.
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
4.1. The case of polygonal domains. In the case of a non-convex polygonal
domain in two dimensions, our assumptions on the smoothness of the solution u of
our model problem (1.1) and on the elliptic regularity inequality are no longer true.
Indeed, if for instance the Neumann boundary is empty, the Dirichlet data is smooth
and f is in L 2
(\Omega\Gamma3 we have, see Grisvard [28], that u 2 H s+2
and ! is the maximum interior angle of @ Moreover, if the Dirichlet data is zero,
we have
see (1.7) in Schatz and Wahlbin [37] and the references therein. This is the elliptic
regularity result that we must use.
To prove our error estimates in this case, we proceed as follows. First, we note that
our main result Theorem 2.2 can be easily extended to this case; indeed, a simple
density argument shows that Lemmas 3.3 and 3.6 remain valid for s; t 2 (\Gamma1=2; 0).
Now we proceed as in subsection 3.4 and obtain the desired estimates by using the
above mentioned lemmas and the above described elliptic regularity inequality. The
estimate of the error in the j(\Delta; \Delta)j A -seminorm remains the same but the estimate of
the L 2 -norm of the potential has to be suitably modified.
For turns out that only for non-zero orders of
convergence for
respectively. The results for k - 1 are displayed in Table 4.1 for
smooth solutions
and in Table 4.2 for non-smooth solutions
simply write fl instead of

Table
Orders of convergence for
4.2. Estimates of the error in negative-order norms. It is very well known
that the error in linear functionals can be estimated in terms of the error in negative-
order norms. Moreover, Bramble and Schatz [11] showed how to exploit the oscillatory
nature of finite element approximations, captured in estimates of the error in negative-
order norms, to enhance the quality of the approximation by using a simple post-processing
on regions in which the exact solution is very smooth and the mesh is
locally translation invariant.
An analysis of the LDG method for elliptic problems 21

Table
Orders of convergence for
estimates of negative-order norms can be easily obtained for our general DG
method by following the argument described in subsection 2.4 and the technicalities
displayed in section 3.
4.3. Curvilinear elements. The analysis in section 3 covers the case of triangulations
of curvilinear elements affine-equivalent to fixed curvilinear reference elements.
The aim of this subsection is to show how our main result can be extended to the more
general case where such an affine equivalence can not be established anymore. This
is, for instance, the case when the problem domain has a boundary with a generic
curvature.
There are two distinctive possibilities to do that. The first one is to keep the finite
element spaces described in the introduction; in this case, the local space S(K) could
be taken to be simply P k (K), for example. For our main result to hold in this case,
only Lemmas 3.1 and 3.2 would have to be proven for these elements and for the case
in which \Pi is the L 2 -projection.
The other possibility is to consider elements obtained through the so-called Piola
transformation [12, Section III.1.3]. This transformation associates the function (q; u)
defined on K to the function (b q; b u) defined on b
K by
where FK denotes the mapping from b
K to K. With the above notation, our finite
element spaces are given by
It is easy to verify that the following properties are satisfied on each element K of
22 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
our triangulation:
Z
Z
Z
r
Z
Z
Z
This implies that with this choice of finite element spaces, our main result holds if Lemmas
3.1 and 3.2 hold for the reference element b
K and for the standard L 2 -projection,
provided the mappings FK are sufficiently smooth; see [12] and the references therein.
Indeed, the proof of section 3 holds in this case if we use the projections \Pi and \Pi
defined by
d \Piq := b
\Pi is the L 2 -projection into the space S( b
K) and b
\Pi). The only
slight modification of the proof occurs in section 3.3 in the definition of j (\Delta; \Delta)j 2
to which we have to add the term k q k 2
This implies that an extra term in the upper
bound of the term T 1 in the proof of Lemma 3.5 appears which is easily dealt with.
modification of the proof is required at all.
4.4. General elliptic problems. The extension of our main result to more
general elliptic problems which include lower order terms can be done in a straight-forward
way by applying our techniques to the formulation used by Cockburn and
Dawson [18].
4.5. Exponential convergence of hp-approximations. In the analysis of
the DG methods considered in this paper, we have only derived error estimates with
respect to the mesh-size h and we have not exploited the dependence of our estimates
on the approximation order k. However, this can be done by modifying Lemmas
3.3 and 3.6 correspondingly; see also the work of Houston, Schwab and S-uli [38]
and the references there. In addition, by using the proper mesh design principles
and by obtaining suitable approximation error estimates in the elements abutting
at solution singularities, exponential convergence of the DG method can be proved.
See, for example, the recent work of Wihler and Schwab [40] who showed exponential
convergence for a model elliptic problem on a polygonal
domain\Omega for the DG method
of Baumann and Oden with interior penalties.
5. Numerical results for the LDG method. The purpose of this section is
to validate our a priori error estimates for the LDG method (i.e., C 22 = 0) and to
assess how the quality of its approximations depends on the size of the stabilization
parameters C 11 . Since C 22 = 0, the function q N can be expressed locally in terms of
uN and hence can be eliminated from the equations. In our examples we solve the
resulting linear system for uN by using the standard Conjugate Gradient algorithm;
in order to obtain as much precision as possible, the stopping criterion is such that the
absolute residual norm is less than 10 \Gamma12 . The approximation q N is then recovered
in a post-processing step by using the local expression of q N in terms of uN .
We present numerical results using sequences of structured as well as unstructured
triangular meshes fT i g, where the mesh-size parameter of T i+1 is half
the one of T i . The numerical orders of convergence of the errors are computed for
An analysis of the LDG method for elliptic problems 23
polynomials of degree 1 to 6 in the L 2 -norm and A-semi-norm. These orders are
defined as follows. If e(T i ) denotes the error on mesh T i (in the corresponding norm),
then the numerical order of convergence r i is
In all our computations, we take C 12 normal to the edges and of modulus 1=2. The
stabilization coefficient C 11 is chosen to be of order h \Gamma1 . We emphasize, however,
that for all our experiments no significant difference has been observed in the errors
of the approximations when C 11 is of order one. We also remark that results for
are not included either, since no positive orders of convergence have been obtained,
as predicted in Theorem 2.2.
5.1. Smooth solutions. In our first example, we investigate the order of convergence
for smooth solutions. We solve the model problem (1.1)
with homogeneous Dirichlet boundary conditions and empty Neumann boundary. The
right hand side f is chosen such that the exact solution is given by
cos
The sequence of structured meshes used in this example is created from consecutive
global refinement of an initial coarse structured mesh; at each refinement, every triangle
is divided into 4 similar triangles. The number of triangles of the meshes are
16, 64, 256, 1024 and 4096. Since our analysis is valid for arbitrary meshes, we also
perform some tests with a sequence of unstructured meshes. It consists of a set of
meshes such that the maximum edge length is less than a certain value. This value
is reduced by a factor of two, from one mesh to the next. In this way, if we take
two consecutive meshes, one is not the global refinement of the other. The number of
elements of the meshes are 22, 88, 312, 1368 and 5404.
We show the orders of convergence in the L 2 -norm of the error in the gradient
in the A-semi-norm of the error of (q; u) and in the L 2 -norm of the error in u in

Tables

5.1, 5.2 and 5.3, respectively. For both types of meshes, we observe that the
optimal order of convergence predicted by our theory, see Table 2.2, is achieved. Note
that since machine precision is achieved for very fine grids and high polynomials, the
corresponding orders of convergence are meaningless and are replaced by a horizontal
line.
To give the reader a better idea of this phenomenon, in Figure 5.1, we display the
actual errors in the potential u whose orders of convergence appear in the left side
of

Table

5.3. Note how the very last part of the curve corresponding to polynomials
of degree bends as a consequence of having reached machine accuracy.
P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau

Table
Smooth solution; order of convergence of the L 2 error in the gradient q.
k order of convergence
Structured meshes Unstructured meshes
3 2.7216 2.9488 2.9924 3.0008 2.9303 2.4986 3.2825 2.9120
6 5.8878 5.9683 5.9820 - 6.4090 5.0744 6.4362 -

Table
Smooth solution; order of convergence of the A-semi-norm of the error in (q; u).
k order of convergence
Structured meshes Unstructured meshes
3 2.8380 2.9745 3.0018 3.0052 3.0120 2.5618 3.2984 2.9233

Table
Smooth solution; order of convergence of the L 2 error in the potential u.
k order of convergence
Structured meshes Unstructured meshes
6 7.0129 6.9889 6.8763 - 7.3003 6.
An analysis of the LDG method for elliptic problems 25
log||
Fig. 5.1. Smooth solution; the L 2 error in the potential u for the structured meshes.
26 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
5.2. An exact solution in H
5(\Omega\Gamma but not in H
We solve the
model problem (1.1) with exact Dirichlet boundary conditions in the convex domain
1). The right hand side is chosen such that the exact solution of
the problem is the function u ff defined by
ae cos
in (\Gamma1; 0) \Theta (\Gamma1; 1);
cos
This function belongs to H ff+ 1
but does not belong to H ff+ 1+"(\Omega\Gamma6 for all " ? 0.
In this test, ff = 4:5 and so u ff 2 H
5(\Omega\Gamma3 The predicted orders of convergence of the
-norm of the error in the gradient and that of the A-semi-norm of the error are both
5, and the predicted order of convergence of the L 2 -norm of the error in the potential
is 4; see

Tables

2.2 and 2.3. These are precisely the orders observed in Tables 5.4, 5.5
and 5.6, respectively. We use the sequence of structured meshes from the previous
test. Similar results not reported here are obtained using unstructured meshes.

Table
of convergence of the L 2 error in the gradient q.
k order of convergence
3 2.1363 2.8375 2.9531 2.9844
6 3.8556 3.9387 3.9710 3.9860

Table
of convergence of the A-semi-norm of the error in (q; u).
k order of convergence
3 2.5591 2.8822 2.9641 2.9882
6 3.9801 3.9664 3.9776 3.9876

Table
of convergence of the L 2 error in the potential u.
An analysis of the LDG method for elliptic problems 27
5.3. Smooth solution on an L-shaped domain. We solve the model problem
(1.1) in an L-shaped domain with Dirichlet boundary conditions. The exact solution
is the function u ff , described above, with 4:5. For this test we use a sequence
of unstructured meshes, created from a global refinement of an unstructured coarse
mesh. The number of elements of the meshes are 22, 88, 352, 1408 and 5632.
In

Tables

5.7, 5.8 and 5.9 below, we can see that we obtain the same order of convergence
as in the convex case even though the standard elliptic regularity result guarantees
an order of convergence for the L 2 -error of the potential smaller by
as indicated in Table 4.1.
A similar phenomenon takes place with the very smooth solution from the first test.

Table
5 -solution on L-shaped domain; order of convergence of the L 2 error in the gradient q.
k order of convergence
3 2.6595 2.8369 2.9260 2.9644
6 3.0742 3.9120 4.0307 4.1347

Table
5 -solution on L-shaped domain; order of convergence of the A-semi-norm of the error in (q; u).
k order of convergence
3 2.7984 2.8763 2.9379 2.9688
6 4.0916 3.9158 4.0313 4.1347

Table
5 -solution on L-shaped domain; order of convergence of the L 2 error in the potential u.
28 P. Castillo, B. Cockburn, I. Perugia and D. Sch-otzau
5.4. Non-smooth solution on an L-shaped domain. Finally, we present
numerical results for the classical L-shaped domain test with a singularity at the
reentrant corner. We consider the model problem (1.1) in an L-shaped domain with
zero right hand side and Dirichlet boundary conditions such that the exact solution
is given by
For conforming finite element methods, it has been shown that the orders of convergence
in the H 1 and L 2 norms are 2
respectively. The
numerical results for the LDG method on the sequence of unstructured meshes described
in the previous experiment are reported in Tables 5.10, 5.11 and 5.12. They
show that the rates of convergence predicted by Table 4.2 are achieved by the LDG
method. Observe that the same rates of convergence as in the conforming case are
achieved.

Table
Non-smooth solution on L-shaped domain; L 2 error in the gradient q.
k order of convergence

Table
Non-smooth solution on L-shaped domain; A-semi-norm of the error in (q; u).
k order of convergence

Table
Non-smooth solution on L-shaped domain; L 2 error in the potential u.
An analysis of the LDG method for elliptic problems 29
6. Concluding remarks. In this paper, we present the first a priori error analysis
for a general DG method that includes the LDG method and allows for triangulations
with hanging nodes and elements of several shapes.
We have proven that the orders of convergence of the approximations given by the
LDG method with the stabilization parameter C 11 of order h \Gamma1 are optimal; these
results have been confirmed by our numerical experiments which also indicate that
the quality of the approximation does not deteriorate when C 11 is taken to be of order
one. Theoretically, a loss of 1=2 in the orders of convergence can take place but this
phenomenon was not observed in the particular test problems we considered; as a
consequence, the sharpness of our error estimates in this case remains to be studied.
We have also theoretically shown that the effect of taking non-zero stabilization parameters
C 22 does not significantly improve the orders of convergence of the LDG
method. An exception is, of course, the piecewise constant case in which the LDG
method has an order of convergence of 0 whereas the DG method with C 11 and C 22
of order one do converge with orders of convergence of at least 1=2 and 1 in the error
of the gradient and potential, respectively.
In this paper, nothing has been said about how to chose the parameters C 12 . In
a forthcoming paper [22], it will be shown that, in the case of Cartesian grids and
tensor product polynomials, the orders of convergence of the LDG method can actually
increase if C 12 is suitably chosen.
Let us end by pointing out tha the implementation of codes for hp-adaptive versions
of the LDG method for general elliptic and transient convection-diffusion-reaction
problems is the subject of ongoing work.



--R

Discontinuous Galerkin methods for elliptic problems
An interior penalty finite element method with discontinuous elements
Nonconforming elements in the finite element method with penalty
Finite element methods for elliptic equations using nonconforming elements
Piecewise solenoidal vector fields and the Stokes problem


A discontinuous hp finite element method for the Navier-Stokes equations
A discontinuous hp finite element method for the Euler and the Navier-Stokes equations
A finite element method for domain decomposition with non-matching grids
Higher order local accuracy by averaging in the finite element method
Mixed and hybrid finite element methods
Discontinuous finite elements for diffusion problems
An optimal error estimate for the local discontinuous Galerkin method
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
The devising of discontinuous Galerkin methods for non-linear hyperbolic con- servatin laws
Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions
TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case
First international symposium on discontinuous Galerkin methods
TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems
Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids
TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework




Elliptic problems in nonsmooth domains

An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation
A discontinuous hp finite element method for diffusion problems
A conservative DGM for convection-diffusion and Navier-Stokes problems
A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation
A discontinuous Galerkin method applied to nonlinear parabolic equations
Improved energy estimates for interior penalty
Interior penalty preconditioners for mixed finite element approximations of elliptic problems
Maximum norm estimates in the finite element method in plane polygonal domains.

An elliptic collocation-finite element method with interior penalties
Exponential convergence of the hp-DGFEM for diffusion problems in two space dimensions

--TR

--CTR
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finite elements;elliptic problems;discontinuous Galerkin methods