COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING
Commun. Numer. Meth. Engng 2002; 18:69–75
Prepared using cnmauth.cls [Version: 2000/03/22 v1.0]
Local discontinuous Galerkin methods for elliptic problems
P. Castillo1
1
B. Cockburn1,‡
I. Perugia2,§
D. Schötzau1
School of Mathematics, University of Minnesota, 127 Vincent Hall, Minneapolis, MN 55455, USA
2 Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy
Commun. Numer. Meth. Engng., Vol. 18, pp. 69–75, 2002
SUMMARY
In this paper, we review the development of local discontinuous Galerkin methods for elliptic problems.
We explain the derivation of these methods and present the corresponding error estimates; we also
mention how to couple them with standard conforming finite element methods. Numerical examples are
displayed which confirm the theoretical results and show that the coupling works very well. Copyright
c 2002 John Wiley & Sons, Ltd.
key words:
Finite element methods, discontinuous Galerkin methods, elliptic problems.
1. Introduction
Over the last years, discontinuous Galerkin (DG) methods have been successfully applied to
a variety of problems where convection phenomena play an important role (see [10] and the
references therein for state-of-the-art surveys on DG methods). However, the need to extend
these methods to problems in which also diffusion must be taken into account has recently
created a renewed interest in DG techniques for elliptic problems; see, e.g., [2, 12, 14]. In this
paper, we review the development of one of those methods, namely, the local discontinuous
Galerkin (LDG) method. The LDG method was introduced in [11] in the context of convectiondiffusion systems, as generalization of the method proposed in [3] for the compressible NavierStokes equations. The LDG method was then further developed and analyzed in [5, 6]; for
purely elliptic problems, it has been recently investigated in [4] and [7]. The advantage of
applying the LDG method to elliptic problems relies on the ease with which it handles hanging
nodes, elements of general shapes, and local spaces of different types; these properties render
the LDG method ideally suited for hp-adaptivity and allow it to be easily coupled with other
‡ Supported
§ This
in part by the NSF (DMS-9807491) and by the University of Minnesota Supercomputing Institute.
work was carried out when the author was a Visiting Professor at the University of Minnesota.
c 2002 John Wiley & Sons, Ltd.
Copyright Received
Revised
70
LDG METHODS FOR ELLIPTIC PROBLEMS
methods. Indeed, the coupling of LDG methods with conforming finite elements (and the
performance of the resulting method in the presence of meshes with hanging nodes) is analyzed
and numerically tested in [13]; the motivation for this coupling comes from an application in
the framework of rotating electrical machines, see [1]. An extension of the LDG method to
the Stokes system, where the discretization of the divergence-free condition on the velocity is
the main difficulty, was proposed and analyzed in [8]. Extensions to Maxwell’s equations are
currently under way.
2. The LDG discretization
To illustrate the definition of the LDG method, consider the classical model elliptic problem:
−∆u = f in Ω,
u = gD on ΓD ,
∇u · n = g N · n on ΓN .
d
Here, Ω is a bounded domain in R , d = 2, 3, and n is the outward unit normal to its boundary
∂Ω = ΓD ∪ ΓN . We begin by introducing the auxiliary variable q = ∇u, and by rewriting our
model problem as follows:
q = ∇u,
−∇ · q = f,
in Ω.
Let now T be a triangulation of Ω into elements {K}; it may have hanging nodes and elements
of various shapes. The LDG method determines an approximation (q h , uh ) to (q, u) belonging
to the local finite element space Q(K) × U(K), typically consisting of polynomials, for each
K ∈ T . This approximate solution is obtained by imposing that, for all (r, v) ∈ Q(K) × U(K),
Z
Z
Z
q h · r dx = −
uh ∇ · r dx +
u
bh r · nK ds,
∂K
Z
ZK
ZK
bh · nK ds.
vq
f v dx +
q h · ∇v dx =
K
K
∂K
bh are the so-called numerical fluxes
Here, nK is the outward unit normal to K and u
bh and q
which are discrete approximations to the traces of u and q on the boundary of the elements.
The name “numerical fluxes” is borrowed from the high-resolution schemes for non-linear
hyperbolic conservation laws; in that framework, they are also called approximate Riemann
bh are needed in order to achieve a stable and accurate
solvers. Both numerical fluxes u
bh and q
scheme. To define these numerical fluxes, we have to introduce some notation. Let K + and K −
be two adjacent elements of T ; let x be an arbitrary point of the set e = ∂K + ∩ ∂K − (which
is not necessarily an entire edge of an element in T ) and let n+ and n− be the corresponding
outward unit normals at that point. Let (s, w) be a function smooth inside each element K ±
and denote by (s± , w± ) the traces of (s, w) on e from the interior of K ± . Then, for a scalar
function w and a vector-valued function s, we define the mean values {{·}} and jumps [[·]] at
x ∈ e as
{{w}} := (w+ + w− )/2,
[[w]] := w+ n+ + w− n− ,
{{s}} := (s+ + s− )/2,
[[s]] := s+ · n+ + s− · n− .
bh and u
If e is inside the domain, the numerical fluxes q
bh are defined by
bh := {{qh }} − α[[uh ]] − β[[q h ]],
q
c 2002 John Wiley & Sons, Ltd.
Copyright Prepared using cnmauth.cls
u
bh := {{uh }} + β · [[uh ]],
Commun. Numer. Meth. Engng 2002; 18:69–75
71
LDG METHODS FOR ELLIPTIC PROBLEMS
with a scalar parameter α and a vector-valued parameter β to be properly chosen, whereas on
the boundary, we take
(
(
+ +
−
q+
gD on ΓD ,
h − α(uh n + gD n ) on ΓD ,
bh :=
q
and
u
bh :=
gN
on ΓN ,
u+
on ΓN .
h
This completes the definition of the LDG method. A few important points concerning this
method need to be briefly discussed:
• The aim of the parameters α and β in the definition of the numerical fluxes is to render the
formulation stable and to enhance its accuracy. For the method to be well defined, we must
have α > 0, whereas β can be arbitrary.
• The LDG method defines a unique solution under very mild compatibility conditions on the
local spaces Q(K) and U(K); in fact, it is enough to have ∇U(K) ⊆ Q(K) (see [4]). The
inter-element continuity of the approximate solution as well as the Dirichlet and Neumann
boundary conditions are enforced in a weak sense only through the numerical fluxes.
• The LDG method can be considered to be a mixed finite element method. However, the fact
that the numerical flux u
bh is independent of q h allows us to actually eliminate the variable q h
from the equations in an element-by-element fashion. This local solvability gives the name to
the LDG method; it is not shared, by classical or stabilized mixed finite element methods.
• Extensions of the LDG method to more general elliptic problems which include variable
diffusion tensors as well as lower order terms can be done in a straightforward way by applying
the techniques developed in [6].
3. Error estimates
A complete error analysis of the LDG method has been carried out in the case where the
local spaces are taken as standard polynomial spaces with possibly different approximation
degrees, Q(K) = P ℓ (K)d , U(K) = P k (K), with k ≥ 1 and ℓ = k or ℓ = k − 1. We discuss
the cases ℓ = k and ℓ = k − 1 separately, giving rise to equal-order elements and mixed-order
elements, respectively. Our theoretical results for the local spaces above, are summarized in
Table I below; h denotes the biggest diameter of the elements of the triangulation T .
Table I. Orders of convergence of the L2 -errors in q and u for smooth solutions and k ≥ 1.
α
Q(K)
U(K)
L2 -error in q
L2 -error in u
O(1)
P k (K)d
P k (K)
k
k + 1/2
P k (K)
k
k+1
O(1/h)
O(1)
O(1/h)
k
P (K)
d
P (K)
k − 1/2
k
P k−1 (K)d
P k (K)
k
k+1
P
k−1
d
(K)
k
Equal-order elements. In this case, it has been proved in [4] that the orders of convergence
of the L2 -norms of the errors in q and u are k and k + 1, respectively, provided that the
parameters α and β are taken to be of order O(h−1 ) and O(1), respectively. These orders
c 2002 John Wiley & Sons, Ltd.
Copyright Prepared using cnmauth.cls
Commun. Numer. Meth. Engng 2002; 18:69–75
72
LDG METHODS FOR ELLIPTIC PROBLEMS
were actually observed numerically in the experiments reported in [4]. For α of order O(1),
the
√ theoretical orders of convergence k and k + 1/2 were obtained in [4], resulting in a loss of
h in the approximation of u. However, no degradation in the order of convergence for u was
actually observed on unstructured triangular meshes in the numerical results reported in [4]
and [13]. Furthermore, it has been shown in [7] that on Cartesian grids, with a special choice
of the numerical fluxes (for which α is of order O(1) and β is such that |β · nK | = 21 ) and for
equal-order elements with Qk -polynomials, the LDG method super-converges and the orders
of convergence k + 21 in q and k + 1 in u are obtained. A similar phenomenon has not been
observed on unstructured grids.
Mixed-order elements. LDG methods with lower approximation degree for q h (i.e., with
ℓ = k − 1) have been first analyzed in [8] in the context of the Stokes problem. These results
immediately carry over to the model problem considered here: with α of order O(1/h) and β
of order O(1), the same orders of convergence k in q and k + 1 in u as for equal-order elements
are achieved. In this case, the error estimates are optimal from both an approximation point of
view and in terms of the smoothness requirements on the exact solution. However, numerical
tests in [8] and [13] indicate that the use of equal-order elements is not less efficient.
Coupling with conforming finite elements. When the LDG methods described above are
coupled with the standard conforming finite element method that uses polynomials of degree
k, the orders of convergence are identical to those reported in the above paragraphs; see [13].
4. Numerical experiments
Results on an L-shaped domain. First, we report the results of two numerical experiments
from [4]. We solve the model problem on an L-shaped domain, using a sequence of unstructured
triangular meshes created from consecutive global refinement. The initial coarse mesh consists
of 22 elements. The parameter α is chosen as 1/h, and β such that |β · nK | = 21 .
Table II. H 5 -solution on L-shaped domain: Orders of convergence of the L2 -errors in q and u.
k
1
2
3
4
5
h
0.8494
1.7966
2.6595
2.6559
2.7630
L2 -error in q
h/2
h/4
0.8581 0.9148
1.8441 1.9136
2.8369 2.9260
3.7667 3.8908
3.7978 3.8723
h/8
0.9530
1.9550
2.9644
3.9571
3.8912
h
2.0435
3.0471
4.0360
5.0226
5.9726
L2 -error in u
h/2
h/4
1.9542 1.9552
2.9694 2.9740
3.9693 3.9831
4.8793 4.9274
4.8779 4.8875
h/8
1.9714
2.9844
3.9916
4.9528
4.8739
In Table II, we show the orders of convergence in the L2 -errors of q and u for the LDG
method using equal-order elements of degree k = 1, . . . , 5, for a solution belonging to H 5 (Ω).
The numerical orders are obtained from the errors of two consecutive meshes. The expected
orders of min(4, k) and min(5, k + 1), respectively, are clearly visible.
In Table III, we report results for the exact solution expressed in polar coordinates by
u(r, θ) = rγ sin (γθ), with γ = 2/3, that exhibits a singularity at the reentrant corner of
the L-shaped domain. The results show that we get orders close to 23 and 34 . These are the
c 2002 John Wiley & Sons, Ltd.
Copyright Prepared using cnmauth.cls
Commun. Numer. Meth. Engng 2002; 18:69–75
73
LDG METHODS FOR ELLIPTIC PROBLEMS
same orders that are obtained with a standard conforming method, for which the orders of
convergence in the H 1 (Ω)- and L2 (Ω)-norm are 32 − ε and 34 − ε, respectively, for all ε > 0.
Table III. Singular solution on L-shaped domain: Orders of convergence of the L2 -errors in q and u.
k
1
2
3
4
5
h
0.7818
0.7794
0.7362
0.7139
0.7016
L2 -error in q
h/2
h/4
0.6298 0.6420
0.6662 0.6665
0.6665 0.6666
0.6666 0.6666
0.6666 0.6666
h/8
0.6513
0.6666
0.6666
0.6667
0.6667
h
1.6098
1.5610
1.5015
1.4715
1.4535
L2 -error in u
h/2
h/4
1.5694 1.5793
1.5383 1.5014
1.4810 1.4449
1.4543 1.4215
1.4383 1.4083
h/8
1.5760
1.4639
1.4137
1.3950
1.3849
Coupling with conforming finite elements. In the following numerical example, taken
from [13], we consider the situation where the LDG method is applied only in a sub-domain
ΩLDG ⊂ Ω and a conforming method is used in the remaining part Ωconf = Ω \ ΩLDG . The
corresponding coupled method, introduced in [1] and analyzed in [13], combines the ease with
which the LDG method handles hanging nodes with the lower computational cost of standard
conforming finite elements. This approach is also very promising in the context of multi-physics
or multi-material problems, where the practitioner might want to use a DG method only in
certain parts of the computational domain.
Figure 1. Meshes used in the numerical experiments: Non-nested grids with 256 and 1024 elements
and hanging nodes on the line y = 0. The LDG region ΩLDG is shadowed.
The coupling across the common interior interface Γ of ΩLDG and Ωconf is achieved as follows.
On the LDG side, the interface Γ is considered as a Dirichlet boundary with datum given by
the trace on Γ of the approximation from the conforming side, whereas on the conforming side
Γ is considered as a Neumann boundary, with datum given by the corresponding flux from the
LDG side; see [13] for details.
We solve the model problem on Ω = (−1, 1) × (−1, 1), using a sequence of non-nested grids
with decreasing mesh-sizes with a reduction factor 2. The grids are obtained by meshing
independently the two subregions (−1, 1) × (−1, 0) and (−1, 1) × (0, 1), giving rise to possibly
non-matching grids along the line y = 0. We define the LDG region ΩLDG on each mesh as
the union of all the elements having at least one vertex on the line y = 0. The domain ΩLDG
c 2002 John Wiley & Sons, Ltd.
Copyright Prepared using cnmauth.cls
Commun. Numer. Meth. Engng 2002; 18:69–75
74
LDG METHODS FOR ELLIPTIC PROBLEMS
contains all the hanging nodes and shrinks towards the line y = 0 as h goes to zero; the
second and third meshes are depicted in Figure 1. In Table IV, we display the L2 -errors in q
and u obtained when the LDG method with equal-order P 1 -elements is applied in ΩLDG , and
standard conforming P 1 -elements are used in Ωconf . The results show that the errors converge
with the optimal orders, as expected, and that the coupling of LDG and conforming methods
with a shrinking LDG region can be successfully carried out.
Table IV. Convergence for the coupled method and P 1 -elements; α = 1, β = O(1).
reduction
in mesh-size h
−
2.0
2.0
2.0
L2 -norm of q
error
reduction
4.7506e–1
−
2.5514e–1
1.8620
1.3239e–1
1.9272
6.7376e–2
1.9649
L2 -norm of u
error
reduction
6.5124e–2
−
1.7317e–2
3.7607
4.5057e–3
3.8434
1.1496e–3
3.9194
5. Conclusions
The results of an error analysis of the LDG method for elliptic problems have been reviewed and
numerical experiments have been presented, showing the sharpness of the theoretical estimates.
It has also been indicated how to couple LDG and conforming finite element methods, in order
to exploit the advantages of the LDG method with a reduced computational cost.
REFERENCES
1. P. Alotto, A. Bertoni, I. Perugia, and D. Schötzau, Discontinuous finite element methods for the
simulation of rotating electrical machines, COMPEL, 20 (2001), 448–462.
2. D. Arnold, F. Brezzi, B. Cockburn, and D. Marini, Unified analysis of discontinuous Galerkin methods
for elliptic problems, SIAM J. Num. Anal., to appear.
3. F. Bassi and S. Rebay, A high-order accurate discontinuous finite element method for the numerical
solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), pp. 267–279.
4. P. Castillo, B. Cockburn, I. Perugia, and D. Schötzau, An a priori error analysis of the local
discontinuous Galerkin method for elliptic problems, SIAM J. Num. Anal., 38 (2000), 1676–1706.
5. P. Castillo, B. Cockburn, D. Schötzau, and C. Schwab, An optimal a priori error estimate for the
hp-version of the local discontinuous Galerkin method for convection–diffusion problems, Math. Comp.,
to appear.
6. B. Cockburn and C. Dawson, Some extensions of the local discontinuous Galerkin method for
convection-diffusion equations in multidimensions, Proceedings of the X Conference on the Mathematics
of Finite Elements and Applications: MAFELAP X, J. Whitemann, ed., Elsevier, 2000, pp. 225–238.
7. B. Cockburn, G. Kanschat, I. Perugia, and D. Schötzau, Superconvergence of the local discontinuous
Galerkin method for elliptic problems on Cartesian grids, SIAM J. Num. Anal., 39 (2001), 264–285.
8. B. Cockburn, G. Kanschat, D. Schötzau, and C. Schwab, Local discontinuous Galerkin methods for
the Stokes system, SIAM J. Num. Anal., to appear.
9. B. Cockburn, G. Karniadakis, and C.-W. Shu, The development of discontinuous Galerkin methods, in
Discontinuous Galerkin Methods: Theory, Computation and Applications, B. Cockburn, G. Karniadakis,
and C.-W. Shu, eds., vol. 11 of Lect. Notes Comput. Sci. Eng., Springer Verlag, 2000, pp. 3–50.
10. B. Cockburn, G. Karniadakis, and C.-W. Shu, eds., Discontinuous Galerkin Methods. Theory,
Computation and Applications, vol. 11 of Lect. Notes Comput. Sci. Eng., Springer Verlag, 2000.
c 2002 John Wiley & Sons, Ltd.
Copyright Prepared using cnmauth.cls
Commun. Numer. Meth. Engng 2002; 18:69–75
LDG METHODS FOR ELLIPTIC PROBLEMS
75
11. B. Cockburn and C.-W. Shu, The local discontinuous Galerkin finite element method for convection–
diffusion systems, SIAM J. Num. Anal., 35 (1998), pp. 2440–2463.
12. J. Oden, I. Babuška, and C. Baumann, A discontinuous hp finite element method for diffusion problems,
J. Comput. Phys., 146 (1998), pp. 491–519.
13. I. Perugia and D. Schötzau, On the coupling of local discontinuous Galerkin and conforming finite
element methods, J. Sci. Comp., to appear.
14. B. Rivière, M. Wheeler and V. Girault, Improved energy estimates for interior penalty, constrained
and discontinuous Galerkin methods for elliptic problems. Part I, Computational Geosciences, 3 (1999),
337–360.
c 2002 John Wiley & Sons, Ltd.
Copyright Prepared using cnmauth.cls
Commun. Numer. Meth. Engng 2002; 18:69–75