Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator

Comput. Methods Appl. Mech. Engrg. 194 (2005) 499–510

www.elsevier.com/locate/cma

Energy norm a posteriori error estimation formixed discontinuous Galerkin approximations of the

Maxwell operator

Paul Houston a,*,1, Ilaria Perugia b, Dominik Schotzau c,2

a Department of Mathematics, University of Leicester, Leicester LE1 7RH, UKb Dipartimento di Matematica, Universita di Pavia, Via Ferrata 1, 27100 Pavia, Italy

c Department of Mathematics, University of British Columbia, 121-1984 Mathematics Road, Vancouver V6T 1Z2, Canada

Received 30 July 2003; accepted 27 February 2004

Abstract

In this paper we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approxi-

mations of the Maxwell operator. In particular, by employing suitable Helmholtz decompositions of the error, together

with the conservation properties of the underlying method, computable upper bounds on the error, measured in terms

of a natural (mesh-dependent) energy norm, are derived. Numerical experiments testing the performance of our a poste-

riori error bounds for problems with both smooth and singular analytical solutions are presented.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Discontinuous Galerkin methods; A posteriori error estimation; Mixed methods; Maxwell operator

1. Introduction

In recent work we have been concerned with the development of mixed discontinuous Galerkin (DG,

for short) finite element methods for the numerical approximation of the Maxwell operator in the time

harmonic regime; here, the underlying system of partial differential equations consists of a curl–curl

0045-7825/$ - see front matter � 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.cma.2004.02.025

* Corresponding author.

E-mail addresses: [email protected] (P. Houston), [email protected] (I. Perugia), [email protected]

(D. Schotzau).1 Funded by the EPSRC (Grant GR/R76615).2 Partially supported by the Swiss NSF (Project 21-068126.02).

mailto:[email protected]



500 P. Houston et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 499–510

operator subject to a divergence-free constraint. Indeed, in the series of papers [5–7] we have introduced,

analyzed and numerically tested non-conforming mixed finite element methods based on a discontinuous

Galerkin approach, where the underlying analytical solution is approximated by completely discontinuous

piecewise polynomials. Optimal a priori error bounds for both stabilized equal-order and non-stabilized

mixed-order methods have been derived in an appropriate mesh-dependent DG energy norm. One of thekey advantages of this approach in comparison with standard conforming finite element methods (see [8],

and the references cited therein) lies in the greater flexibility available for designing the underlying finite

element space. Indeed, within a DG approach, non-matching grids containing hanging nodes and non-

uniform, even anisotropic, polynomial approximation degrees can easily be handled. Thereby, DG meth-

ods are naturally suited for the application within adaptive refinement algorithms where the mesh and/or

polynomial degree are locally refined in order to yield reliable and efficient control of the discretization

error.

In this article, we initiate the development of the a posteriori error estimation and adaptive mesh designfor the non-stabilized mixed DG method introduced in [6]. In particular, computable upper bounds on the

error, measured in terms of a natural (mesh-dependent) DG energy norm, will be derived. Inspired by the

recent article [3], the proof of this error bound is based on employing suitable Helmholtz decompositions of

the error, together with the conservation properties of the underlying DG method. The performance of this

error bound within an adaptive local mesh refinement procedure will be demonstrated for problems with

both smooth and singular analytical solutions.

2. Model problem

We first introduce some notation: given a bounded domain D in Rd , d P 1, we write Ht(D) to denote the

usual Sobolev space of real-valued functions with regularity exponent t and norm k Ækt,D. For t = 0, we writeL2(D) in lieu of H

0(D). The space Ht(D)d consists of vector fields whose components belong to Ht(D); it is

endowed with the usual product norm which we denote, for simplicity, also by k Ækt,D. The dual spaces ofH1/2(oD) and H1/2(oD)d are denoted by H�1/2(oD) and H�1/2(oD)d, respectively, where oD denotes the

boundary of D; in both cases, we write k Æk�1/2,oD to denote the corresponding dual norm. Additionally,we write hÆ, ÆioD to denote the duality pairings in H�1/2(oD) · H1/2(oD) and H�1/2(oD)d · H1/2(oD)d, as well

as for the inner products in L2(oD) and L2(oD)d. For D � R3, we write H(curl;D) and H(div;D) to denote

the spaces of vector fields u 2 L2(D)3 with $ · u 2 L2(D)

3 and $ Æ u 2 L2(D), respectively, endowed with

their corresponding graph norms k Ækcurl and k Ækdiv. Finally, we denote by H 10ðDÞ, H0(curl;D) and

H0(div;D) the subspaces of H1(D), H(curl;D) and H(div;D), respectively, of functions with zero trace, tan-

gential trace and normal trace, respectively, and by H(curl0;D) and H(div0;D) the subspaces of H(curl;D)

and H(div;D), respectively, of functions with zero curl and divergence, respectively.

Given X � R3, a simply connected bounded Lipschitz polyhedron with connected boundary C, we con-sider the following problem: find (u,p) such that

r ðl�1r uÞ � erp ¼ f in X; ð1Þ

r � ðeuÞ ¼ 0 in X; ð2Þ

n u ¼ 0 and p ¼ 0 on C: ð3Þ
Here, n(x) denotes the unit outward normal vector to C at x 2 C. The coefficients l = l(x) > 0 ande = e(x) > 0 are the magnetic permeability and the electric permittivity of the medium, respectively; for sim-plicity, here we assume that both l and e are smooth functions. Throughout, we consider the physicallymost relevant case of a divergence free source field and assume that f 2 H(div0;X); i.e., f 2 L2(X)
3 and

P. Houston et al. / Comput. Methods Appl. Mech. Engrg. 194 (2005) 499–510 501

r � f ¼ 0 in X: ð4Þ
This assumption actually implies that p 0. Nevertheless, it is necessary to discretize the Lagrange multi-plier p in order to ensure the stability and well-posedness of the corresponding mixed approximation.
The following embedding results from [1] will play a crucial role in the forthcoming a posteriori analysis:

under the above assumptions on the domain X, there exists a regularity exponent s 2 (1/2,1], dependingonly on X, such that

H 0ðcurl;XÞ \ Hðdiv;XÞ ,!HsðXÞ3;Hðcurl;XÞ \ H 0ðdiv;XÞ ,!HsðXÞ3:

ð5Þ

The maximal value of s for which the above embeddings hold is closely related to the regularity properties

of the Laplacian in polyhedra and only depends on the opening angles at the corners and edges of the do-

main, cf. [1]. In particular, for a convex domain, we can choose s = 1, cf. Remark 4 below.The embeddings in (5) imply that, for smooth coefficients l and e, the analytical solution u possesses the

following regularity properties: eu 2 Hs(X)3 and l�1$ · u 2 Hs(X)3, with s denoting the exponent in (5).

3. Mixed discontinuous Galerkin finite element approximation

In this section we define the mixed DG approximation to (1)–(3) proposed and analyzed in the recent

paper [6]. To this end, we first introduce the following notation: we consider shape regular meshesTh thatpartition X into tetrahedra {K}. We denote by hK the diameter of K 2 Th and set h ¼ maxK2ThhK .We write FI

h and FBh to denote the union of all interior and boundary faces, respectively, and set

Fh ¼ FIh [FB

h . We define the local mesh size function h on Fh by setting hðxÞ ¼ minfhKþ ; hK�g, if x isin the interior of oK+ \ oK�, where K+ and K� are two adjacent elements; if x is in the interior of oK \ C,we set h(x) = hK. For f � FI

h , shared by two elements K+ and K� with outward unit normals n±, respec-

tively, we define, with obvious notation, the jumps across f by svbT = n+ · v+ + n� · v�, svbN = v+ Æ n+ +v� Æ n�, sqbN = q+n+ + q�n�, and the averages by v = (v+ + v�)/2 and q = (q+ + q�)/2. On f � FB

h ,

we set svbT = n · v, sqbN = qn and v = v.The generic discontinuous finite element space is given by P ‘ðThÞ :¼ fu 2 L2ðXÞ : ujK 2 P‘ðKÞ 8K 2

Thg, ‘ P 0, where P‘ðKÞ is the space of polynomials of degree less than or equal ‘ on K. For ‘ P 1, we

use the finite element spaces Vh ¼ P ‘ðThÞ3 and Qh ¼ P ‘þ1ðThÞ.The mixed DG finite element approximation to (1)–(3) is then defined as follows: find (uh,ph) 2 Vh · Qh

such that

ahðuh; vÞ þ bhðv; phÞ ¼ fhðvÞ 8v 2 Vh; ð6Þ

bhðuh; qÞ � chðph; qÞ ¼ 0 8q 2 Qh: ð7Þ

Here, ah, bh, ch and fh are defined, respectively, by

ahðu; vÞ ¼Z

Xl�1rh u � rh vdxþ

ZFh

asutT � svtT ds�ZFh

ðsutT � l�1rh v

þ svtT � l�1rh u Þds;

bhðv; pÞ ¼ �Z

Xev � rhpdxþ

ZFh

ev � sptN ds;


chðp; qÞ ¼ZFh

csptN � sqtN ds;

fhðvÞ ¼Z

Xf � vdx;

where $h denotes the elementwise ‘‘nabla’’ operator.

Remark 1. In order to incorporate the inhomogeneous Dirichlet boundary conditions n · u = g on C, g inL2(C)

3, within the above formulation, it is sufficient to simply modify the definition of the functional fh as

follows:

fhðvÞ ¼Z

Xf � vdx�

ZFB

h

g � l�1rh vdsþZFB

h

ag � ðn vÞds:

The interior penalty parameters a and c are defined, respectively, on each face f in Fh, as follows:

a ¼ al�1h�1 and c ¼ ceh�1: ð8Þ

Here, a and c are positive real constants, which are independent of h and the coefficients l and e. To ensurestability of the underlying DG method (6) and (7) the parameter a, which is associated to the interior pen-alty discretization of the curl–curl operator, must be chosen sufficiently large, cf. [5,6].

The stability and a priori error analysis of the mixed DG approximation to (1)–(3) was developed in the

recent article [6]. To this end, writing k � k20;Fhto denote the quantity

Pf�Fh

k � k20;f , we introduce the DG-norm

jjjðv; qÞjjj2DG ¼ kvk2curl;h þ kqk21;h;

where

kvk2curl;h ¼ ke1=2vk20;X þ kl�1=2rh vk20;X þ kl�1=2h�1=2svtTk

20;Fh

;

kqk21;h ¼ ke1=2rhqk20;X þ ke1=2h�1=2sqtNk2

0;Fh:

With this notation, we quote the following error bound.

Theorem 3.1. Assuming (4) holds, so that p 0, and that the interior penalty parameters a and c are selected

as in (8), where a is a sufficiently large real constant. Then, the mixed DG approximation (uh,ph) obtained by(6) and (7) satisfies the following a priori error bound

jjjðu� uh; p � phÞjjjDG 6 Chminfs;‘g½keuks;X þ kl�1r uks;X�;

where C is a positive constant independent of the mesh size h, and s is the regularity exponent defined in (5).

Remark 2. In contrast to the continuous level where p 0, when assumption (4) holds, on the discretelevel, we generally have ph 5 0. Thus, it is necessary to control the size of kphk1,h in order to control theerror ku � uhkcurl,h which is of primary interest.

Remark 3. The a priori error bound in Theorem 3.1 can be extended to arbitrary source terms f 2 L2(X)3;

cf. [6] for details. Indeed, provided that p 2 Hs+1(X), the following error estimate holds:

jjjðu� uh; p � phÞjjjDG 6 Chminfs;‘g½keuks;X þ kl�1r uks;X þ kpksþ1;X�;

where C is a positive constant independent of the mesh size h.


We conclude this section by establishing a crucial property of the DG scheme (6) and (7). To this end, we

define, for any K 2 Th and piecewise smooth function v, the flux sK(v) = nK · (l�1$ · v) on oK, and its dis-

crete version

sKðvÞ ¼nK ð l�1r v � asvtT Þ on oK n C;

nK ðl�1r v� aðnK vÞÞ on oK \ C:

(

The expression sKðvÞ is referred to as the numerical flux, cf. [2], for example. With this notation, we have the

following local conservation property of the DG method (6) and (7).

Lemma 3.1. For any K 2 Th, if w is constant on K and zero outside K, then the following equality holds:

hsKðuÞ � sKðuhÞ;wioK ¼ 12hsphtN ; ewioK �

ZKrph � ewdx:

Proof. From the continuous problem, we immediately have hsK(u),wioK = �Kf Æ wdx. Further, from the def-inition of the method, sK and sK , we obtain hsKðuhÞ;wioK þ 1

2hsphtN ; ewioK �

RKrph � ewdx ¼

RKf � wdx;

subtracting these two expressions gives the desired result. h

4. A posteriori error estimation

In this section we establish an a posteriori estimator for the error measured in terms of the energy normjjj Æ jjjDG.

Theorem 4.1. We assume that (4) holds, so that p 0. Let (uh,ph) be the mixed discontinuous Galerkin

approximation obtained by (6) and (7). Then there is a constant C > 0 independent of the mesh size h, suchthat

jjjðu� uh; p � phÞjjjDG 6 CXK2Th

g2K

!1=2;

where the elemental error indicator gK is given by

g2K ¼ h2sK kf �r ðl�1r uhÞ þ erphk2

0;K þ h2s�1K ksKðuhÞ � sKðuhÞk20;oK þ h�1K kl�1=2suhtTk2

0;oK

þ hKke1=2suhtNk20;oKnC þ h2Kkr � ðeuhÞk20;K þ ke1=2rphk

20;K þ h�1K ke1=2sphtNk

20;oK ;

with s denoting the regularity parameter in (5).

Remark 4. Recall that the value of s 2 (1/2,1] only depends on the opening angles at the corners andedges of the domain and that we can choose s = 1 for a convex domain. These limited smoothness

properties of the function spaces for Maxwell�s equations are the reason that some of the terms inthe estimator gK are weighted differently with respect to the local mesh size than their counter-parts arising for diffusion problems. Finally, we note that for the two-dimensional analogue of

problem (1)–(3) considered in the numerical experiments presented in Section 5, the parameter s can

be chosen as s � min(1,p/x), where x is the maximum opening angle of the polygon X under

consideration.


Remark 5. To deal with the case of inhomogeneous boundary conditions n · u = g on C, the definition ofthe numerical flux sK must be suitably modified on elements K inTh such that oK \ C 5 ;. Then, if g is thetangential trace of a function belonging to Vh \ H1(X)3, the error indicator is simply modified according tothe new definition of the numerical fluxes on boundary elements, together with a corresponding modifica-

tion of the jump indicators h�1K kl�1=2suhtTk0;oK on oK \ C. For general g 2 L2(C)3, a data oscillation term

which takes into account the error in the finite element approximation of the boundary datum must be

included within the above error estimator.

Remark 6. To bound the term kl�1/2$h · (u � uh)k0,X, we closely follow the approach in [3] which wasrecently developed for diffusion problems. In particular, we will make use of a suitable Helmholtz decom-

position of the error and of the conservation property derived in Lemma 3.1. A similar Helmholtz decom-

position, combined with a duality argument, is then employed to bound the term ke1/2(u � uh)k0,X. Incontrast to diffusion problems, we note that, due to the limited regularity of u, it is not clear how to obtainsharper a posteriori bounds for the L2-norm of the error (or for functional error estimation), even when

standard duality techniques are employed.

Proof of Theorem 4.1. For simplicity, we carry out the proof for l = e = 1; the extension to general smoothcoefficients follows analogously. From the definition of the norm jjj Æ jjjDG, the continuity of the tangentialtrace of u at the inter-element boundaries, and the fact that p = 0, we have

jjjðu� uh; p � phÞjjj2DG ¼ ku� uhk20;X þ

XK2Th

kr ðu� uhÞk20;K þ kh�1=2suhtTk20;Fh

þXK2Th

krphk20;K þ kh�1=2sphtNk

20;Fh

:

Thereby, we only need to estimate kek20;X andP

K2Thkr ek20;K , where we have set, for convenience,

e = u � uh; to this end, we proceed as follows.

Step 1. Estimate ofP

K2Thkr ek20;K . We follow the approach of [3] and write the L2-orthogonal

decomposition of $h · e 2 L2(X)3 as

rh e ¼ ru þr w; ð9Þ

with u 2 H 1ðXÞ=R and w 2 H0(curl;X) \ H(div0;X), with kwkcurl,X 6 Ck$ · wk0,X. This follows from thedecomposition L2(X)

3 = $H1(X) � H0(div0;X), and from [1, Theorem 3.17 and Corollary 3.19]. From

the embeddings in (5), we have that w belongs to Hs(X)3 and satisfies kwks,X 6 Ck$ · wk0,X. Therefore,the following stability estimate of the decomposition (9) holds:

kruk20 þ kwk2s;X 6 CXK2Th

kr ek20;K : ð10Þ

Employing this decomposition, we have that
XK2Th
kr ek20;K ¼XK2Th

ZKr e � rudxþ

XK2Th

ZKr e � r wdx Iþ II: ð11Þ

We first deal with term I: using the definition of e, the smoothness of u and u, integration by parts andthe fact that n · u = 0 on oK, we have

XK2Th

ZKr e � rudx ¼ �

XK2Th

hnK ðru nKÞ; nK uhioK :


Next, fixing v in Vh \ H0(curl;X), we observe that
XK2Th
hnK ðru nKÞ; nK vioK ¼ 0:

Thereby,
XK2Th
ZKr e � rudx ¼

XK2Th

hnK ðru nKÞ; nK ðv� uhÞioK

6

XK2Th

knK ruk�1=2;oKknK ðv� uhÞk1=2;oK :

Using the trace estimate

knK rk�1=2;oK 6 C½krk0;K þ hKkr rk0;K �; r 2 Hðcurl;KÞ;

together with (10), we obtain

XK2Th

ZKr e � rudx 6 Ckruk0;X

XK2Th

knK ðv� uhÞk21=2;oK

!1=2

6 Ckrh ek0;XXK2Th

knK ðv� uhÞk21=2;oK

!1=2:

Inverse estimates and the discrete trace inequality yield
XK2Th
knK ðv� uhÞk21=2;oK 6 CXK2Th

h�1K knK ðv� uhÞk20;oK 6 CXK2Th

h�2K kv� uhk20;K

6 CXK2Th

ðh�2K kv� uhk20;K þ kr ðv� uhÞk20;KÞ:

A careful inspection of the proof of Theorem 4.1 in [6] shows that

infv2Vh\H0ðcurl;XÞ

XK2Th

ðh�2K kv� uhk20;K þ kr ðv� uhÞk20;KÞ 6 Ckh�1=2suhtTk2

0;Fh:

As a consequence, we obtain

XK2Th

ZKr e � rudx 6 Ckrh ek0;X

XK2Th

h�1K ksuhtTk2

0;oK

!1=2: ð12Þ

In order to deal with term II, we first write wh to denote the L2-projection of w onto P 0ðThÞ, so that

kw� whk0;K þ h1=2K kw� whk0;oK 6 ChsKkwks;K 8K 2 Th: ð13Þ

Thereby, replacing w by w � wh in term II, integrating by parts and using the fact that u satisfies (1) with

p 0, we obtain
XK2Th
ZKr e � r wdx ¼

XK2Th

ZKðf �rr uhÞ � ðw� whÞdx

�XK2Th

hsKðeÞ;wioK þXK2Th

hsKðeÞ;whioK :

From the continuity properties of u and w, and the fact that the numerical fluxes sK are single-valued atinter-element boundaries, we have


XK2Th

hsKðeÞ;wioK ¼XK2Th

hsKðuhÞ � sKðuhÞ;wioK :

Employing Lemma 3.1, we can write
XK2Th
hsKðeÞ;whioK ¼XK2Th

hsKðuhÞ � sKðuhÞ;whioK þXK2Th

1

2hsphtN ;whioK �

XK2Th

ZKrph � wh dx:

Thus,
XK2Th
ZKr e � r wdx ¼

XK2Th

ZKðf �rr uh þrphÞ � ðw� whÞdx

�XK2Th

hsKðuhÞ � sKðuhÞ;w� whioK �XK2Th

1

2hsphtN ;w� whioK ;

where we also have added and subtractedP

K2Th

RKrph � wdx, used integration by parts and the fact that

$ Æ w = 0. Finally, using estimates (13) and (10), we obtain the following bound:
XK2Th
ZKr e � rwdx

6 CXK2Th

h2sK kf �rr uh þrphk20;K þ

XK2Th

h2s�1K ksKðuhÞ � sKðuhÞk20;oK þXK2Th

h2s�1K ksphtNk20;oK

!1=2

krh ek0;X: ð14Þ

Substituting (12) and (14) into (11) and dividing through by k$h · ek0,X, we conclude that
XK2Th
kr ek20;K

6 CXK2Th

h2sK kf �rr uh þrphk2

0;K þXK2Th

h2s�1K ksKðuhÞ � sKðuhÞk20;oK

þXK2Th

h�1K ksuhtTk2

0;oK þXK2Th

h2s�1K ksphtNk2

0;oK

!1=2:

Step 2. Estimate of kek0,X. We decompose e 2 L2(X)3 into e = $w + W, with w 2 H 1

0ðXÞ andW 2 H(div0;X). This decomposition is L2-orthogonal; hence

kek20;X ¼ krwk20;X þ kWk20;X:

Since $w is curl-free and has zero tangential component along the boundary, we have that $h · W = $h · e

and $ Æ W = 0.In order to bound kWk0,X, we consider the problem of finding z such that $ · $ · z = W and $ Æ z = 0 in

X, with n · z = 0 on C. We notice that, from (5), z 2 Hs(X)3, $ · z 2 Hs(X)3, and kzks,X + k$ · zks,X 6

CkWk0,X. By integration by parts and elementary manipulations, taking into account that $ · z 2H(curl;X), we have

kWk20;X ¼Z

XW � edx ¼

XK2Th

ZKrr z � edx ¼

XK2Th

ZKr z � r edxþ

XK2Th

hsKðzÞ; eioK

¼XK2Th

ZKr z � r edxþ

XK2Th

1

2hr z; suhtT ioKnC þ

XK2Th

hr z; nK uhioK\C:


From the stability estimate k$ · zks,X 6 CkWk0,X, the weighted Cauchy–Schwarz inequality, and thetrace inequality kr zk20;oK 6 Ch�1K kr zk2s;K (see [9]), we obtain, after division by kWk0,X, the following

kWk0;X 6 CXK2Th

kr ek20;K þXK2Th

h�1K ksuhtTk20;oK

!1=2:

The first term on the right-hand of the above inequality can then be bounded in terms of computable

quantities using the estimate obtained in Step 1. To deal with the term k$wk0,X, we first note that

krwk20;X ¼Z

Xrw � ðrw þ WÞdx ¼

ZXrw � edx ¼ �bhðe;wÞ:

Further, we let wh be the standard Clement interpolation of w that satisfies

kw � whk0;K þ hKkw � whk1;K þ h1=2K kw � whk0;oK 6 ChKkwk1;dK ; ð15Þ

where dK is the patch of elements adjacent to K (see [8, Section 5.6.1]). Since sw � whbN = 0, using Galerkinorthogonality, integration by parts and the fact that $ Æ u = 0, we can write

bhðe;wÞ ¼ bhðe;w � whÞ ¼ �XK2Th

ZKr � uhðw � whÞdxþ

XK2Th

1

2hw � wh; suhtNi0;oKnC:

Therefore, using the weighted Cauchy–Schwarz inequality and (15), after dividing by k$wk0,X, we obtain

krwk0;X 6 CXK2Th

h2Kkr � uhk20;K þXK2Th

hKksuhtNk20;oKnC

!1=2:

Conclusion of the proof. Combining the results obtained in Steps 1 and 2 gives the estimate stated in the

theorem. h

5. Numerical experiments

In this section we present a series of numerical examples to illustrate the practical performance of the

proposed a posteriori error estimator within an automatic adaptive refinement procedure. Here, we re-

strict ourselves to two-dimensional model problems with constant coefficients l e 1. Furthermore,in each of the examples shown below, we set the polynomial degree ‘ equal to 1 and the constants aand c arising in the interior penalty parameters, cf. (8), equal to 10 and 1, respectively. The adaptivemeshes are constructed by employing the fixed fraction strategy, with refinement and derefinement frac-

tions set to 25% and 10%, respectively. Here, the emphasis will be to demonstrate the asymptotic exact-

ness of the proposed a posteriori error indicator on non-uniform adaptively refined meshes. Thereby, as

in [3], we set the constant C arising in Theorem 4.1 equal to one and ensure that the corresponding effec-

tivity indices are roughly constant on all of the meshes employed; in general, to ensure the reliability of

the error estimator, C must be numerically determined for the underlying problem at hand, cf. [4], forexample.

5.1. Example 1

In this first example, we let X be the convex domain (�1,1)2 (i.e. so that s = 1 in (5)) and f =

(2cos(px) sin(py)p2,�2cos(py) sin(px)p2); thereby, the analytical solution to (1) is given by u = (cos(px)

0 2 4 6 8 100

1

2

3

4

5

6

7

8

9

10

Eff

ectiv

ity I

ndex

Mesh Number

(a) (b)

Fig. 1. Example 1. (a) Computational mesh after 8 adaptive refinements with 3210 elements; (b) effectivity indices.

(a) (b)

Fig. 2. Example 1. Numerical approximation to: (a) u1; (b) u2.


sin(py),�cos(py) sin(px)) and p = 0. In Fig. 1(a) we show the mesh generated using the proposed a

posteriori error indicator after 8 adaptive refinement steps. Here, we see that while the mesh has

been largely uniformly refined throughout the entire computational domain, additional refinementhas been performed where the solution has local maxima and minima; cf. Fig. 2 where we plot the

isolines of the numerical approximation uh computed on this mesh. Finally, in Fig. 1(b) we show

the history of the effectivity indices on each of the meshes generated by our adaptive algorithm. Here,

we observe that the error estimator over-estimates the true error by a consistent factor of between 5

and 6, thereby confirming the asymptotic exactness of the proposed error indicator for this smooth

problem.

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Eff

ectiv

ity I

ndex

Mesh Number

(a) (b)

Fig. 3. Example 2. (a) Computational mesh after 6 adaptive refinements with 1308 elements; (b) effectivity indices.


5.2. Example 2

In this final example, we select X to be the (non-convex) L–shaped domain with vertices (1,0), (1,1),(�1,1), (�1,�1), (0,�1) and (0,0). Furthermore, we set f = 0 and impose n · u = g on C, where g is chosenso that u(x,y) = $(r2/3 sin(2#/3)), in terms of the polar coordinates (r,#); thereby, on the basis of Remark 4,we set s = 2/3. In Fig. 3(a) we show the mesh generated using the local error indicators gK after 6 adaptiverefinement steps. Here, we see that the mesh has been largely refined in the vicinity of the re-entrant cornerlocated at the origin. In Fig. 3(b) we show the history of the effectivity indices on each of the meshes

generated by our adaptive algorithm. As in the previous example, we observe that the error estimator

over-estimates the true error by a consistent factor, thereby confirming the asymptotic exactness of the

proposed error indicator; however, here we see that for this non-smooth example, the effectivity indices

are smaller than the corresponding quantities computed for the smooth example considered in Section

5.1. Finally, in Fig. 4 we plot the isolines of the numerical approximation uh computed on the final mesh.

(a) (b)

Fig. 4. Example 2. Numerical approximation to: (a) u1; (b) u2.


6. Concluding remarks

In this article we have derived a residual–based energy norm a posteriori error bound for the mixed DG

approximation of the time-harmonic Maxwell operator. The analysis is based on employing Helmholtz

decompositions of the error, together with the conservation properties of the underlying method. Numer-ical experiments presented in this article clearly highlight the asymptotic exactness of the proposed estima-

tor on adaptively refined meshes.

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Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator