An a posteriori error indicator for discontinuous Galerkin...

IMA Journal of Numerical Analysis (2005) of 27doi: 10.1093/imanum/

An a posteriori error indicator for discontinuous Galerkin discretizationsof H(curl)–elliptic partial differential equations

PAUL HOUSTON∗School of Mathematical Sciences, University of Nottingham, University Park, Nottingham,

NG7 2RD, UK

ILARIA PERUGIA†Dipartimento di Matematica, Universita di Pavia, Via Ferrata 1, 27100 Pavia, Italy

DOMINIK SCHOTZAU‡Mathematics Department, University of British Columbia, 1984 Mathematics Road,

Vancouver, BC V6T 1Z2, Canada

IMA J. Numer. Anal., Vol. 27, pp. 122-150, 2007

We introduce a residual-based a posteriori error indicator for DG discretizations of H(curl;Ω )–ellipticboundary value problems that arise in eddy current models. We show that the indicator is both reliableand efficient with respect to the approximation error measured in terms of a natural energy norm. Wevalidate the performance of the indicator within an adaptive mesh refinement procedure and show itsasymptotic exactness for a range of test problems.

Keywords:Discontinuous Galerkin methods, a posteriori error analysis, eddy current problems

1. Introduction

The electric field-based eddy current model for the computation of quasistatic electromagnetic fieldsconsists of the following initial-boundary value problem: find the electric field E : Ω × (0,T ) → R

3

satisfying

∂t(σE)+∇× (µ−1∇×E) = ∂t js in Ω × (0,T),

n×E = 0 on Γ × (0,T ),

E|t=0 = E0 in Ω ,

cf. (3; 7). Here, Ω is an open bounded Lipschitz polyhedron in R3 with boundary Γ = ∂Ω and outward

normal unit vector n. For simplicity, we assume Ω to be simply-connected and Γ to be connected. Theright-hand side js = js(x, t) is a given external source field, with js(·, t)∈ L2(Ω)3 and ∇ · js(·, t)∈ L2(Ω),t ∈ (0,T ). The material coefficients σ and µ are the electric conductivity and the magnetic permeability,

∗[email protected]†[email protected]‡[email protected]

IMA Journal of Numerical Analysis c© Institute of Mathematics and its Applications 2005; all rights reserved.

2 of 27 P. HOUSTON, I. PERUGIA, AND D. SCHOTZAU

respectively; σ is a symmetric, uniformly positive semidefinite tensor, and µ is a symmetric, uniformlypositive definite tensor, both with bounded coefficients. Assuming σ is uniformly positive definite onthe whole of Ω and that an implicit time discretization is employed, at each time-step, one would needto solve a boundary value problem of the form

∇× (µ−1∇×u)+βu = j in Ω ,

n×u = 0 on Γ ,

where u is the approximation to E to be computed at the current time-step, j depends on js and theapproximation to E at the previous time-step, and β depends on σ and the size of the current time-step.In order to simplify the presentation, we assume from here on that µ and β are equal to the identitytensor, cf. the discussion presented in (5).

With this discussion in mind, the purpose of this paper is to introduce a residual-based a posteriorierror indicator for interior penalty discontinuous Galerkin (DG) approximations of the following modelproblem: find the vector unknown u (electric field) that satisfies

∇×∇×u+u = j in Ω , (1.1)

n×u = 0 on Γ , (1.2)

where, for generality, j is a given external source field in L2(Ω)3. By introducing the Sobolev space

H0(curl;Ω) := v ∈ L2(Ω)3 : ∇×v ∈ L2(Ω)3, n×v = 0 on Γ ,

the weak form of (1.1)–(1.2) is given by: find u ∈ H0(curl;Ω) such that

a(u,v) :=∫

Ω

((∇×u) · (∇×v)+u ·v

)dx =

∫

Ωj ·vdx (1.3)

for all v ∈ H0(curl;Ω). The system in (1.1)–(1.2) is the simplest prototype of a partial differentialequation that is elliptic over H0(curl;Ω).

The main motivation for using a discontinuous Galerkin approach for the numerical approximationof (1.1)–(1.2) is that DG methods, being based on discontinuous finite element spaces, can easily han-dle non-conforming meshes which contain hanging nodes and, in principle, local spaces of differentpolynomial orders; for the purposes of the current article, we shall only consider the h-version of theDG method. Moreover, the implementation of discontinuous elements can be based on standard shapefunctions, without the need to employ curl-conforming elemental mappings - a convenience that is par-ticularly advantageous for high-order elements and that is not straightforwardly shared by standard edgeor face elements commonly used in computational electromagnetics (see (8; 32; 1) and the referencestherein for hp-adaptive edge element methods). A further benefit of DG methods is that inhomoge-neous Dirichlet boundary conditions can easily be incorporated within the scheme, without the need toexplicitly evaluate edge- and face-element interpolation operators.

This paper represents a continuation of the series of articles (31; 30; 16; 17; 20; 19; 14; 15) con-cerned with the development of DG finite element methods for the numerical approximation of thetime-harmonic Maxwell equations. Indeed, here we have considered the design and analysis of noncon-forming DG methods for both low and high frequency approximations of these equations. In the lowfrequency regime, both regularized, cf. (30; 16), and mixed, cf. (17; 20; 19), formulations have beenproposed. For the high frequency regime, mixed and so-called direct formulations have been proposed

AN A POSTERIORI ERROR INDICATOR FOR DG DISCRETIZATIONS OF H(curl)–ELLIPTIC PDES 3 of 27

and analyzed; see (31; 15) and (14), respectively. For a recent review of this work, we refer to thearticle (18).

With the exception of (19), the aforementioned articles have only considered the derivation of a pri-ori error bounds. In this article we introduce an a posteriori error indicator for the DG approximationto (1.1)–(1.2) and prove both upper and lower bounds for the actual error measured in terms of a naturalenergy norm, thereby, demonstrating that the indicator is both reliable and efficient. We remark thatupper bounds of this type were derived in (19) for the mixed DG approximation of the low frequencyMaxwell’s equations. However, the bounds presented there are somewhat unsatisfactory in the sensethat the error estimators also depend on certain embedding parameters of the computational domain.The proof of these results relied on employing suitable Helmholtz decompositions of the error, togetherwith the conservation properties of the underlying DG method; cf. (6), where this approach was firstdeveloped in the context of DG methods employed for the numerical approximation of diffusion prob-lems. In this article we pursue a different technique, inspired by the recent articles (21; 22; 23); seealso (26) for related work. Here, the proof of the upper bound is based on rewriting the method in a non-consistent manner using lifting operators in the spirit of (4), see also (29), together with approximationresults which allow us to find a conforming finite element function which is close to any discontinuousone. Approximation results of this latter type have been developed for the h–version of the DG methodin the case when the underlying conforming finite element space is a subspace of H 1

0 (Ω) in the arti-cles (17; 26); the extension of this result to the hp–DG method was presented in the recent article (23).In this paper we exploit an analogous result derived in (14) for the h–version of the DG method whenthe underlying conforming finite element space is a subspace of H0(curl;Ω). With this approximationresult, the proof of the upper a posteriori error bound now rests on estimating the error between theanalytical solution u and a conforming approximant uc

h. To this end, we exploit the decomposition ofH0(curl;Ω) derived in (13, Lemma 2.4), together with approximation results for the standard Clementinterpolant, as well as for the quasi-interpolation operator constructed in (5, Section 5). The proofs ofthe lower (efficiency) a posteriori error bounds follow from the standard bubble technique introducedin (33) and (34).

Finally, we note that the techniques presented here can be readily extended to elliptic problems inH0(curl;Ω) with smooth coefficients. However, the extension to problems with non-smooth coefficientsremains an open issue. The reason for this is that for non-smooth coefficients the decomposition in (13,Lemma 2.4) is no longer applicable; the same problem also arises in the analysis of conforming methods,cf. (5).

Before we proceed, we first introduce some notation: given a bounded domain D in Rd , d > 1, we

write Ht(D) to denote the usual Sobolev space of real-valued functions with regularity exponent t andnorm ‖ · ‖t,D; for t = 0, we write L2(D) instead of H0(D). The space Ht(D)d consists of vector fieldswhose components belong to H t(D); it is endowed with the usual product norm which we denote, forsimplicity, also by ‖·‖t,D. For D ⊂R

3, we write H(curl;D) and H(div;D) to denote the spaces of vectorfields u ∈ L2(D)3 with ∇×u ∈ L2(D)3 and ∇ ·u ∈ L2(D), respectively, endowed with their correspond-ing graph norms ‖ · ‖curl and ‖ · ‖div, respectively. Finally, we denote by H1

0 (D) and H0(curl;D) thesubspaces of H1(D) and H(curl;D), respectively, of functions with zero trace and zero tangential trace,respectively.

2. Discontinuous Galerkin discretization

In this section, we consider the interior penalty DG discretization of (1.1)–(1.2). To this end, we firstintroduce the following notation.


Let Th be a conforming, shape-regular and affine tetrahedral partition of Ω of granularity h =maxK∈Th hK , where the local mesh size hK is defined to be the diameter of the element K ∈ Th. Wedenote by Fh the set of all faces of elements in Th. For an approximation order ` > 1, we introduce thefollowing finite element space

Vh := v ∈ L2(Ω)3 : v|K ∈ P`(K)3, K ∈ Th, (2.1)

where P`(K) denotes the space of polynomials of total degree at most ` on K. With this notation, weconsider the following interior penalty DG method: find uh ∈ Vh such that

ah(uh,v) =

∫

Ωj ·vdx (2.2)

for all v ∈ Vh. The discrete form ah(·, ·) is given by

ah(u,v) := ∑K∈Th

∫

K

((∇×u) · (∇×v)+u ·v

)dx− ∑

F∈Fh

∫

F[[u]]T · ∇×vds

− ∑F∈Fh

∫

F[[v]]T · ∇×uds+ ∑

F∈Fh

∫

Fa[[u]]T · [[v]]T ds.

(2.3)

For a piecewise smooth function v, on interior faces, we write [[v]]T and v to denote the tangentialjump and mean value of the vector field v, respectively. On boundary faces we set [[v]]T = n× v andv= v; cf. (16) or (14). The function a penalizes the tangential jumps; it is referred to as the interiorpenalty stabilization function. On a given face F it is defined by

a|F := αh−1F , (2.4)

with hF denoting the diameter of face F and α being a positive constant that is independent of the meshsize. It is well-known that (2.2) is uniquely solvable, provided that α > αmin, for a threshold valueαmin that only depends on the shape-regularity of the mesh and the approximation order `, cf. (17), forexample.

REMARK 2.1 The well-posedness of (2.2) is guaranteed provided that α > Cinv, where Cinv is theconstant arising in the inverse estimate

‖w‖20,∂ K 6 Cinvh−1

K ‖w‖20,K ∀w ∈ P

`(K),

see, e.g., (30); see also (4). The constant Cinv depends only on the polynomial degree ` and the shape-regularity of the mesh (see, e.g., (24)). Indeed for square elements, by solving a local generalizedeigenvalue problem, it can be shown that Cinv = C′

inv`2, where C′

inv = (1+3/`+2/`2) 6 6 for all ` > 1;thereby, in this case we may set α = C′

inv`2, or indeed α = 6`2. In general, for other element types, and

indeed, for hybrid meshes, on the basis of the articles (16; 23), in Section 6 we actually set α = CIP`2,

with CIP = 10, the value of the constant CIP being chosen empirically, based on extensive numericalexperimentation; see the discussion at beginning of Section 6 for further details.

REMARK 2.2 In order to incorporate the inhomogeneous Dirichlet boundary condition n×u = g on Γ ,g in L2(Γ )3, within the above formulation, it is sufficient to simply add the following terms to thefunctional on the right-hand side of (2.2)

− ∑F∈FB

h

∫

Fg · (∇×v)ds+ ∑

F∈FBh

∫

Fag · (n×v)ds,

where FBh denotes the set of all faces of elements in Th which lie on the boundary Γ of Ω .


3. A posteriori error indicator

In this section, we present a reliable and efficient indicator for the error of the DG approximation mea-sured in terms of a natural energy norm.

3.1 Local a posteriori error indicators

We begin by introducing local error indicators. To do so, we introduce on each element K ∈ Th theindicator ηK which is given by the sum of the following five terms

η2K = η2

RK+η2

DK+η2

TK+η2

NK+η2

JK. (3.1)

The first term ηRK measures the residual of the underlying system of partial differential equations andis defined by

η2RK

= h2K‖jh−∇×∇×uh−uh‖2

0,K, (3.2)

where jh ∈ Vh denotes an appropriate approximation to j; we refer to Remark 3.2 concerning the choiceof jh. The second term ηDK measures the error in the divergence and is given by

η2DK

= h2K‖∇ · (jh −uh)‖2

0,K . (3.3)

The third term ηTK is the usual face residual related to the tangential jump of ∇×uh:

η2TK

=12 ∑

F∈∂ K\ΓhK‖[[∇×uh]]T‖2

0,F . (3.4)

The fourth term ηNK measures the normal jump [[jh−uh]]N of jh −uh over interior faces, that is,

η2NK

=12 ∑

F∈∂ K\ΓhK‖[[jh−uh]]N‖2

0,F . (3.5)

The last term ηJK measures the tangential jumps of the approximate solution uh and is defined by

η2JK

=12 ∑

F∈∂ K\Γ‖a 1

2 [[uh]]T‖20,F + ∑

F∈∂ K∩Γ‖a 1

2 (n×uh)‖20,F . (3.6)

3.2 Reliability

We now state and discuss the fact that the error indicator

ERR :=

(

∑K∈Th

η2K

) 12

(3.7)

provides a reliable upper bound on the approximation error with respect to an energy-type norm. To doso, we introduce the space V(h) := H0(curl;Ω)+Vh, and define the following norm:

‖v‖2DG := ‖v‖2

0,Ω + ∑K∈Th

‖∇×v‖20,K + ∑

F∈Fh

‖a 12 [[v]]T‖2

0,F .

We remark that ‖·‖DG represents the natural energy norm arising from the definition of the DG bilinearform ah(·, ·) given in (2.3). While this norm does indeed depend on the interior penalty stabilizationparameter α , cf. (2.4), the value of α is fixed, for a given polynomial degree, and is independent of themesh size h.


THEOREM 3.1 Let u be the analytical solution of (1.1)–(1.2), with j ∈ L2(Ω)3, and uh ∈ Vh its DGapproximation obtained by (2.2) with α > αmin. Let the local error indicators be defined by (3.1)–(3.6).Then, the following upper bound for the error holds

‖u−uh‖DG 6 CEST

(

∑K∈Th

η2K

)1/2

+CAPPA (j− jh), (3.8)

where CEST and CAPP are positive constants; here, CEST is independent of the mesh size and CAPP isindependent of the mesh size and α . If α > max1,αmin then CEST may also be bounded independentlyof α . Further, A (j− jh) is the data approximation term given by

A (j− jh) = ‖j− jh‖0,Ω .

REMARK 3.1 From the proof of Theorem 3.1, it follows that

CEST =√

2Cdec maxCc,Cq max1,α− 12 C

− 12

lift +(1+Ccont)(2α−1Cconf +1

) 12

andCAPP = 2Cdec maxCc,Cq,

where Cdec is the constant arising in the stability bounds stated in (4.8), Cq and Cc are the interpolationconstants in (4.9) and (4.10), respectively, α is the constant arising in the definition of the interior penaltystabilization function (2.4), Clift is the constant in (4.3), Ccont is the continuity constants in Lemma 4.2and Cconf is the constant arising in the approximation result stated in Proposition 4.1.

REMARK 3.2 We note that Theorem 3.1 holds for any jh ∈ Vh. To ensure that the data approximationterm A (j− jh) does not dominate the overall a posteriori error bound stated in (3.8), jh should be chosenin such a manner so that (asymptotically) A (j− jh) tends to zero at, at least, the same rate as the firstterm on the right-hand side of (3.8) (and thereby also at, at least, the same rate as ‖u− uh‖DG, cf.Theorem 3.3 in the following section) as the mesh is refined for a given polynomial degree `. This canbe achieved, for example, by choosing jh to be the L2-projection of j onto the space Vh.

REMARK 3.3 In the case where the external source field j belongs to H(div;Ω) (for example, in manyeddy current problems, j is actually solenoidal), then an upper bound for the error analogous to the onestated in Theorem 3.1 may be derived with the data approximation term defined by

A (j− jh)2 = ∑

K∈Th

h2K(‖j− jh‖2

0,K +‖∇ · (j− jh)‖20,K).

In this case the normal jump error indicator ηNK is defined as

η2NK

=12 ∑

F∈∂ K\ΓhK‖[[uh]]N‖2

0,F ,

which now reflects the fact that the normal components of j across the element faces are continuous.

REMARK 3.4 The case of imposing the inhomogeneous boundary condition n×u = g on Γ may alsobe incorporated within the above error bound. Indeed, in the case when g is the tangential trace of a


function belonging to Vh ∩H1(Ω)3, the error indicator is simply modified by redefining the tangentialjump indicator ηJK as follows:

η2JK

=12 ∑

F∈∂ K\Γ‖a 1

2 [[uh]]T‖20,F + ∑

F∈∂ K∩Γ‖a 1

2 (n×uh −g)‖20,F.

For general g ∈ L2(Γ )3, a data error term which takes into account the error in the finite element ap-proximation of the boundary datum must be included within the a posteriori error bound.

3.3 Efficiency

Next, we discuss the efficiency of all the terms that constitute the local error indicators ηK .

PROPOSITION 3.2 Let u be the analytical solution of (1.1)–(1.2) and uh ∈ Vh its DG approximationobtained by (2.2) with α > αmin. Let the local error indicators be defined by (3.1)–(3.6). Then, thefollowing local bounds hold.

(i) For any element K ∈ Th, we have

ηRK 6 C (‖∇× (u−uh)‖0,K +hK‖u−uh‖0,K +hK‖j− jh‖0,K) ,

with a constant C > 0 that is independent of the mesh size and α .

(ii) For any element K ∈ Th, we have

ηDK 6 C (‖u−uh‖0,K +‖j− jh‖0,K) ,

with a constant C > 0 that is independent of the mesh size and α .

(iii) For any interior face F shared by two elements K and K ′, we have

h12F‖[[∇×uh]]T‖0,F 6 C ∑

K∈δF

(‖∇× (u−uh)‖0,K +hK‖u−uh‖0,K +hK‖j− jh‖0,K

),

with a constant C > 0 that is independent of the mesh size and α ; here, we have set δF = K,K ′.

(iv) For any interior face F shared by two elements K and K ′, we have

h12F‖[[jh−uh]]N‖0,F 6 C ∑

K∈δF

(‖u−uh‖0,K +‖j− jh‖0,K

), (3.9)

with a constant C > 0 that is independent of the mesh size and α ; again, δF = K,K ′.

(v) For interior faces F , we have

‖a 12 [[uh]]T‖0,F = ‖a 1

2 [[u−uh]]T‖0,F ,

and for boundary faces F ,

‖a 12 (n×uh)‖0,F = ‖a 1

2 (n× (u−uh))‖0,F .


The proof of Proposition 3.2 is given in Section 5. An immediate consequence of Proposition 3.2 isthe following result.

THEOREM 3.3 Let u be the analytical solution of (1.1)–(1.2), with j ∈ L2(Ω)3, and uh ∈ Vh its DGapproximation obtained by (2.2) with α > αmin. Let the local error indicators be defined by (3.1)–(3.6).Then our proposed error indicator ERR in (3.7) is efficient in the sense that

ERR 6 CEFF

(‖u−uh‖DG +A (j− jh)

),

where CEFF is a positive constant independent of the mesh size and α .

REMARK 3.5 In the case when j∈H(div;Ω), analogous efficiency bounds may also be derived. Indeed,bounds (i), (iii), and (v) in Proposition 3.2 remain unchanged; on the right-hand side of the bounds in(ii) and (iv), the term ‖j− jh‖0,K may be replaced by hK‖∇ · (j− jh)‖0,K . In the case of the latter bound,the term on the left-hand side of the inequality in (3.9) must also be modified in light of the changein the definition of the normal jump indicator ηNK outlined in Remark 3.3; by this we mean that the

term h12F‖[[jh −uh]]N‖0,F must be replaced by h

12F‖[[uh]]N‖0,F . Thereby, with the modified definition of

the data approximation term stated in Remark 3.3, an analogous efficiency bound to the one stated inTheorem 3.3 holds.

REMARK 3.6 We point out that the constants CEST and CAPP arising in Theorem 3.1, and CEFF definedin Theorem 3.3, are all independent of the constant α arising in the definition of the interior penaltyparameter a, which indicates that the resulting bounds are relatively insensitive to changes in α . Indeed,numerical experiments presented in the articles (22; 23) confirm this assertion in the context of bothnearly–incompressible linear elasticity and Poisson’s equation, respectively.

REMARK 3.7 We note that the emphasis of this article is the design of an a posteriori error indicator forthe h–version (only) of the interior penalty DG approximation of the H(curl)–elliptic problem (1.1)–(1.2). Thereby, it is natural that the constants CEST and CAPP arising in Theorem 3.1, and CEFF definedin Theorem 3.3, should all depend on the polynomial degree `; indeed, this is evidenced in Section6. The extension of the analysis to the hp–version of the finite element method, whereby all constantswould then indeed be independent of both h and ` forms part of our programme of future research. Weremark that one of the key technical difficulties in our setting is the derivation of the hp–version ofProposition 4.1; this has been undertaken in article (23) for subspaces of H1(Ω) only.

4. Proof of Theorem 3.1

In this section we carry out the proof of Theorem 3.1. To this end, in Section 4.1 we present an ap-proximation property for DG functions. Section 4.2 is devoted to augmenting the form ah to the spaceV(h). After these preliminary considerations, we conclude the proof of Theorem 3.1 in Sections 4.3, 4.4and 4.5.

4.1 Approximation by conforming finite element functions

One of the main ingredients in our error analysis is an approximation property that allows us to finda conforming finite element function close to any discontinuous one; see (20) and (14) where similarideas are used in the a priori error analysis of DG methods for Maxwell’s equations. We also referto (26) and (23) where results of that type have been instrumental in deriving a posteriori bounds for DGdiscretizations of diffusion problems.


To this end, we first define Vch to be the largest conforming space underlying Vh, that is,

Vch := Vh ∩H0(curl;Ω). (4.1)

In fact, Vch is the finite element space based on the second family of Nedelec’s elements of degree `;

see (28) or (27, Section 8.2).We will make use of the following approximation property that is similar to the results in (14).

PROPOSITION 4.1 Let v ∈ Vh. Then there is a function vc ∈ Vch such that

‖v−vc‖2DG 6

(2α−1Cconf +1

)∑

F∈Fh

‖a 12 [[v]]T‖2

0,F ,

with a constant Cconf > 0 that only depends on the shape-regularity of the mesh and the approximationorder.

Proof. This follows from the construction in (14, Appendix): for v ∈ Vh there is an conformingapproximation vc ∈ Vc

h such that

∑K∈Th

‖∇× (v−vch)‖2

0,K 6 Cconf ∑F∈Fh

h−1F ‖[[v]]T‖2

0,F

and‖v−vc‖2

0,Ω 6 Cconf ∑F∈Fh

hF‖[[v]]T‖20,F ,

with a constant Cconf > 0 that only depends on the shape-regularity of the mesh and the approximationorder `. The approximation result then follows readily from the definition of a in (2.4).

4.2 Auxiliary formulation

Following the a priori and a posteriori analyses of (29) and (23), respectively, we will augment thebilinear form ah in (2.2) to V(h)×V(h) in a non-consistent manner. To this end, we define for v ∈ V(h)the lifting L (v) ∈ Vh by

∫

ΩL (v) ·wdx = ∑

F∈Fh

∫

F[[v]]T · wds ∀w ∈ Vh. (4.2)

Then there is a constant Clift > 0 only depending on the shape-regularity of the mesh and the approxi-mation degree ` such that

‖L (v)‖20,Ω 6 α−1Clift ∑

F∈Fh

‖a 12 [[v]]T‖2

0,F . (4.3)

We then introduce the auxiliary bilinear form

ah(u,v) := ∑K∈Th

∫

K

((∇×u) · (∇×v)+u ·v

)dx− ∑

K∈Th

∫

KL (u) · (∇×v)dx

− ∑K∈Th

∫

KL (v) · (∇×u)dx+ ∑

F∈Fh

∫

Fa [[u]]T · [[v]]T ds.


Note that ah = ah on Vh×Vh and ah = a on H0(curl;Ω)×H0(curl;Ω); hence the form ah can be viewedas an extension of both a and ah to the space V(h). The discrete problem (2.2) can then be equivalentlystated as follows: find uh ∈ Vh such that

ah(uh,v) =∫

Ωj ·vdx ∀v ∈ Vh. (4.4)

Since ah = a on H0(curl;Ω)×H0(curl;Ω), the following stability result holds.

LEMMA 4.1 We haveah(v,v) = ‖v‖2

curl = ‖v‖2DG

for all v ∈ H0(curl;Ω).

Moreover, the bilinear form ah is continuous on V(h); we refer to (4; 16; 31) for details.

LEMMA 4.2 For any u,v ∈ V(h), we have

|ah(u,v)| 6 Ccont‖u‖DG‖v‖DG,

where Ccont = max2,1 + α−1Clift, α is the constant arising in the definition of the interior penaltystabilization function (2.4) and Clift is the constant arising in (4.3).

4.3 A preliminary upper bound

We decompose the error between the analytical solution u and the DG approximation uh ∈Vh as follows

u−uh = (u−uch)− (uh−uc

h),

where uch ∈ Vc

h is the conforming approximation of uh from Proposition 4.1.

LEMMA 4.3 The following upper bound for the error holds

‖u−uh‖DG 6 supw∈H0(curl;Ω)

R(w)+(1+Ccont)‖uh −uch‖DG,

where Ccont is the continuity constant from Lemma 4.2 and R(·) is defined by

R(w) = infwh∈Vh

∣∣∣∣∫

Ωj · (w−wh)dx− ah(uh,w−wh)

∣∣∣∣‖w‖DG

∀w ∈ H0(curl;Ω).

Proof. By application of the triangle inequality we obtain

‖u−uh‖DG 6 ‖u−uch‖DG +‖uh−uc

h‖DG. (4.5)

In order to bound ‖u− uch‖DG, we set w := u− uc

h ∈ H0(curl;Ω) and exploit the coercivity propertystated in Lemma 4.1; thereby, we get

‖w‖2DG = ah(w,w) = ah(u,w)− ah(uh,w)+ ah(uh −uc

h,w).

Since w ∈ H0(curl;Ω), we have

ah(u,w) = a(u,w) =∫

Ωj ·wdx.


From (4.4), we get

ah(uh,wh) =∫

Ωj ·wh dx ∀wh ∈ Vh.

This allows us to conclude that

‖w‖2DG =

∫

Ωj · (w−wh)dx− ah(uh,w−wh)+ ah(uh −uc

h,w)

for any wh ∈ Vh. The continuity of ah(·, ·) from Lemma 4.2 yields the bound

‖w‖2DG 6

∣∣∣∣∫

Ωj · (w−wh)dx− ah(uh,w−wh)

∣∣∣∣+Ccont‖uh −uch‖DG‖w‖DG

for any wh ∈ Vh. Therefore,

‖u−uch‖DG 6 R(u−uc

h)+Ccont‖uh −uch‖DG.

Observing that R(u−uch) 6 supw∈H0(curl;Ω) R(w) and referring to the triangle inequality in (4.5) com-

pletes the proof.

Let us remark that, in Lemma 4.3, the term ‖uh −uch‖DG can be bounded in terms of the jumps of

the discrete solution, due to Proposition 4.1. Thereby, we deduce that

‖uh −uch‖2

DG 6(2α−1Cconf +1

)∑

F∈Fh

‖a 12 [[uh]]T‖2

0,F =(2α−1Cconf +1

)∑

K∈Th

η2JK

, (4.6)

where ηJK is the jump residual defined in (3.6). In the following section, we now derive a computableupper bound for R(w).

4.4 Bound of R(w)

To bound the term R(w), we will use the regular decomposition from (13, Lemma 2.4): any w ∈H0(curl;Ω) can be written as

w = w0 +∇ϕ , (4.7)

with w0 ∈ H0(curl;Ω)∩H1(Ω)3 and ϕ ∈ H10 (Ω). Furthermore, there is a stability constant Cdec > 0

only depending on Ω such that

‖w0‖1,Ω 6 Cdec‖w‖curl, ‖ϕ‖1,Ω 6 Cdec‖w‖curl. (4.8)

We will further make use of the quasi-interpolation operator constructed in (5, Section 5): for anyw ∈ H0(curl;Ω)∩H1(Ω)3, there is a low-order approximation wh ∈ Vc

h that satisfies

∑K∈Th

(‖∇× (w−wh)‖2

0,K +h−2K ‖w−wh‖2

0,K +h−1K ‖w−wh‖2

0,∂ K

)6 C2

q‖w‖21,Ω , (4.9)

with a constant Cq > 0 that only depends on the shape-regularity of the mesh.Finally, we will make use of the Clement interpolant (see, e.g., (9, Section I.A.3)): for any ϕ ∈

H10 (Ω), there is a piecewise linear approximation ϕh ∈ H1

0 (Ω) such that

∑K∈Th

(‖∇(ϕ −ϕh)‖2

0,K +h−2K ‖ϕ −ϕh‖2

0,K +h−1K ‖ϕ −ϕh‖2

0,∂ K

)6 C2

c‖ϕ‖21,Ω , (4.10)

with a constant Cc > 0 that only depends on the shape-regularity of the mesh.We are now ready to prove the following result.


LEMMA 4.4 For any w ∈ H0(curl;Ω), the following bound holds

R(w) 6√

2Cdec maxCc,Cq max1,α− 12 C

− 12

lift (

∑K∈Th

η2K

) 12

+2Cdec maxCc,CqA (j− jh).

Here, Cdec is the constant arising in the stability bounds stated in (4.8), Cq and Cc are the constants in(4.9) and (4.10), respectively, and Clift is the constant in (4.3).

Proof. For any arbitrary w ∈ H0(curl;Ω), recalling that ‖w‖DG = ‖w‖curl, we obviously have that

R(w) 6

∣∣∣∣∫

Ωj · (w−wc

h)dx− ah(uh,w−wch)

∣∣∣∣‖w‖curl

(4.11)

for any wch ∈ Vc

h. Using the result in (4.7), we decompose w as

w = w0 +∇ϕ ,

and choose wch ∈ Vc

h in (4.11) aswc

h = w0h +∇ϕh,

where w0h is the quasi-interpolant of w0 defined in (4.9) and ϕh is the Clement interpolant of ϕ from

(4.10). In this way we obtain∫

Ωj · (w−wc

h)dx− ah(uh,w−wch) ≡ T1 +T2,

with

T1 =

∫

Ωj · (w0 −w0

h)dx− ah(uh,w0 −w0h),

T2 =

∫

Ω(j−uh) ·∇(ϕ −ϕh)dx.

(4.12)

Let us next bound T1 and T2.Bound of T1: We have

T1 =

∫

Ω(jh −uh) · (w0−w0

h)dx− ∑K∈Th

∫

K(∇×uh) ·

(∇× (w0 −w0

h))

dx

+ ∑K∈Th

∫

KL (uh) ·

(∇× (w0−w0

h))

dx+ ∑K∈Th

∫

K(j− jh) · (w0 −w0

h)dx,

for any jh ∈ Vh, cf. Remark 3.2. Integrating by parts the second term on the right-hand side of the aboveidentity and employing the conformity of w0 −w0

h yields

− ∑K∈Th

∫

K(∇×uh) ·

(∇× (w0−w0

h))

dx

= − ∑K∈Th

∫

K(∇×∇×uh) · (w0 −w0

h)dx− ∑K∈Th

∫

∂ KnK × (∇×uh) · (w0 −w0

h)ds

= − ∑K∈Th

∫

K(∇×∇×uh) · (w0 −w0

h)dx− ∑K∈Th

∑F∈∂ K\Γ

12

∫

F[[∇×uh]]T · (w0 −w0

h)ds.


Here, we denote by nK the outward unit normal vector on ∂K. Therefore,

T1 = ∑K∈Th

∫

K(jh −∇×∇×uh−uh) · (w0 −w0

h)dx+ ∑K∈Th

∫

KL (uh) ·

(∇× (w0 −w0

h))

dx

− ∑K∈Th

∑F∈∂ K\Γ

12

∫

F[[∇×uh]]T · (w0 −w0

h)ds+ ∑K∈Th

∫

K(j− jh) · (w0 −w0

h)dx

≡ T11 +T12 +T13 +T14.

Obviously, we can bound T11 by

T11 6 ∑K∈Th

ηRK h−1K ‖w0 −w0

h‖0,K , (4.13)

where ηRK is the residual defined in (3.2).To bound T12 we use the Cauchy-Schwarz inequality and the stability of the lifting operator in (4.3);

this yields

T12 6

(

∑K∈Th

‖L (uh)‖20,K

) 12(

∑K∈Th

‖∇× (w0−w0h)‖2

0,K

) 12

6 α− 12 C

12lift

(

∑F∈Fh

‖a 12 [[uh]]T‖2

0,F

) 12(

∑K∈Th

‖∇× (w0−w0h)‖2

0,K

) 12

= α− 12 C

12lift

(

∑K∈Th

η2JK

) 12(

∑K∈Th

‖∇× (w0−w0h)‖2

0,K

) 12

,

(4.14)

where the jump residuals ηJK are defined in (3.6).For T13, application of the Cauchy-Schwarz inequality gives

T13 6 ∑K∈Th

(

∑F∈∂ K\Γ

12

hK‖[[∇×uh]]T‖20,F

) 12(

∑F∈∂ K

12

h−1K ‖w0 −w0

h‖20,F

) 12

6 ∑K∈Th

ηTK h− 1

2K ‖w0−w0

h‖0,∂ K ,

(4.15)

with the residual ηTK defined in (3.4). Similarly,

T14 6 ∑K∈Th

hK‖j− jh‖0,Kh−1K ‖w0 −w0

h‖0,K. (4.16)

Using the Cauchy-Schwarz inequality, taking into account the approximation bound in (4.9) and thebounds in (4.13), (4.14), (4.15), and (4.16), we conclude that

T1 6 Cq max1,α− 12 C

12lift(

∑K∈Th

(η2

RK+η2

JK+η2

TK

)) 1

2

‖w0‖1,Ω +CqA (j− jh)‖w0‖1,Ω . (4.17)


Bound of T2: Next, let us bound the term T2 in (4.12). Adding and subtracting jh, and integration byparts, we obtain

T2 =∫

Ω(j− jh) ·∇(ϕ −ϕh)dx+

∫

Ω(jh −uh) ·∇(ϕ −ϕh)dx

=

∫

Ω(j− jh) ·∇(ϕ −ϕh)dx− ∑

K∈Th

∫

K∇ · (jh −uh)(ϕ −ϕh)dx

+ ∑K∈Th

∫

∂ K(jh −uh) ·nK(ϕ −ϕh)ds

=

∫

Ω(j− jh) ·∇(ϕ −ϕh)dx− ∑

K∈Th

∫

K∇ · (jh −uh)(ϕ −ϕh)dx

+ ∑K∈Th

∑F∈∂ K\Γ

12

∫

F[[jh −uh]]N(ϕ −ϕh)ds

≡ T21 +T22 +T23.

For term T21, we haveT21 6 ‖j− jh‖0,Ω‖∇(ϕ −ϕh)‖0,Ω . (4.18)

Term T22 can be bounded by

T22 6 ∑K∈Th

hK‖∇ · (jh−uh)‖0,Kh−1K ‖ϕ −ϕh‖0,K

= ∑K∈Th

ηDK h−1K ‖ϕ −ϕh‖0,K ,

(4.19)

with the residual ηDK defined in (3.3).Finally, for term T23, we have

T23 6 ∑K∈Th

(

∑F∈∂ K\Γ

12

hK‖[[jh−uh]]N‖20,F

) 12(

∑F∈∂ K

12

h−1K ‖ϕ −ϕh‖2

0,F

) 12

6 ∑K∈Th

ηNK h− 1

2K ‖ϕ −ϕh‖0,∂ K ,

(4.20)

with the residual ηNK defined in (3.5).Application of the approximation result in (4.10), the bounds in (4.18), (4.19), and (4.20), and the

Cauchy-Schwarz inequality yields

T2 6 Cc

(

∑K∈Th

(η2

DK+η2

NK

)) 1

2

‖ϕ‖1,Ω +CcA (j− jh)‖ϕ‖1,Ω . (4.21)

Conclusion: from the bounds in (4.17) and (4.21) for T1 and T2, respectively, we deduce that

T1 +T2 6 maxCc,Cq max1,α− 12 C

− 12

lift (

∑K∈Th

η2K

) 12 (

‖w0‖21,Ω +‖ϕ‖2

1,Ω) 1

2

+√

2maxCc,CqA (j− jh)(‖w0‖2

1,Ω +‖ϕ‖21,Ω) 1

2 .

(4.22)

Combining this estimate with the stability bounds in (4.8), we obtain the result.


4.5 Proof of Theorem 3.1

The proof of Theorem 3.1 is now an immediate consequence of Lemma 4.3, Lemma 4.4 and the boundin (4.6). A careful inspection of the proof reveals that all the constants can be bounded independentlyof α provided α > 1.

5. Proof of Proposition 3.2

In this section we prove the efficiency bounds stated in Proposition 3.2; here, we employ the bubblefunction technique introduced in (33). To this end, for an element K of Th, we denote by bK thestandard polynomial bubble function on K, and for an interior face F shared by two elements K and K ′,we denote by bF the standard polynomial bubble function on F ; moreover, set δF := K,K ′. With thisnotation, the following bounds hold.

LEMMA 5.1 Let v be a polynomial function on K; then there exists a constant C > 0 independent of vand hK such that

‖bKv‖0,K 6 C‖v‖0,K, (5.1)

‖v‖0,K 6 C‖b12Kv‖0,K , (5.2)

‖∇(bKv)‖0,K 6 Ch−1K ‖v‖0,K. (5.3)

Moreover, let F be an interior face shared by two elements K and K ′, and let w be a polynomial functionon F ; then there exists a constant C > 0 independent of w and hF such that

‖w‖0,F 6 C‖b12F w‖0,F . (5.4)

Finally, there exists an extension Wb ∈ H10 ((K ∪K ′

)) of bF w such that Wb|F = bFw and

‖Wb‖0,K 6 Ch12F‖w‖0,F ∀K ∈ δF , (5.5)

‖∇Wb‖0,K 6 Ch− 1

2F ‖w‖0,F ∀K ∈ δF , (5.6)

with a constant C > 0 independent of w and hF .

Proof. The proof of (5.1), (5.2), (5.4) and (5.5) is given in (33, Lemma 4.1). The proof of (5.3) and (5.6)can be obtained by similar arguments; see (2, Theorems 2.2 and 2.4).

Analogous bounds can be easily obtained for vector-valued functions; in particular, we shall needthe following estimates.

LEMMA 5.2 Let v be a (vector-valued) polynomial function on K; then there exists a constant C > 0independent of v and hK such that

‖bKv‖0,K 6 C‖v‖0,K, (5.7)

‖v‖0,K 6 C‖b12Kv‖0,K, (5.8)

‖∇× (bKv)‖0,K 6 Ch−1K ‖v‖0,K. (5.9)

Moreover, let F be an interior face shared by two elements K and K ′, and let w be a (vector-valued)polynomial function on F; then there exists a constant C > 0 independent of w and hF such that

‖w‖0,F 6 C‖b12F w‖0,F . (5.10)


Finally, there exists an extension Wb ∈ H10 ((K ∪K ′

))3 of bFw such that Wb|F = bF w and

‖Wb‖0,K 6 Ch12F‖w‖0,F ∀K ∈ δF , (5.11)

‖∇×Wb‖0,K 6 Ch− 1

2F ‖w‖0,F ∀K ∈ δF , (5.12)

with a constant C > 0 independent of w and hF .

We now proceed with the proof of Proposition 3.2.Bound (i): Let K be an element of Th. For notational convenience, we introduce the following

polynomials on K:

vh = jh −∇×∇×uh−uh, vb = bK vh.

Using the fact that ∇×∇×u+u = j in L2(K)3, we obtain

‖b12Kvh‖2

0,K =

∫

K(jh −∇×∇×uh−uh) ·vb dx

=

∫

K(j−∇×∇×uh−uh) ·vb dx+

∫

K(jh − j) ·vb dx

=∫

K(∇×∇× (u−uh)+(u−uh)) ·vb dx+

∫

K(jh − j) ·vb dx

=

∫

K(∇× (u−uh)) · (∇×vb) dx+

∫

K(u−uh) ·vb dx+

∫

K(jh − j) ·vb dx,

where in the last step we have integrated by parts the curl-curl term; we note that, since vb is zero on theboundary of K, the boundary terms vanish. Using (5.8) and the Cauchy-Schwarz inequality, we obtain

‖vh‖20,K 6 C (‖∇× (u−uh)‖0,K‖∇×vb‖0,K +‖u−uh‖0,K‖vb‖0,K +‖j− jh‖0,K‖vb‖0,K) ,

and, owing to (5.9) and (5.7), we conclude that

‖vh‖0,K 6 C(h−1

K ‖∇× (u−uh)‖0,K +‖u−uh‖0,K +‖j− jh‖0,K),

with a constant C > 0 that is independent of the mesh size. Multiplying by hK and noting that ηRK =hK‖vh‖0,K yields the desired bound (i).

Bound (ii): To bound ηDK , we set

vh = ∇ · (jh −uh), vb = bK vh.

Using (5.2), we obtain

‖vh‖20,K 6 C‖b

12Kvh‖2

0,K = C∫

K∇ · (jh−uh)vb dx,

with a constant C > 0 that is independent of the mesh size. From (1.1), it is clear that j−u ∈ H(div;Ω)and ∇ · (j−u) = 0 in L2(K). Hence

‖vh‖20,K 6 C

∫

K∇ · ((jh − j)+(u−uh)) vb dx.


Integrating by parts and employing the Cauchy-Schwarz inequality, together with (5.3), yields

‖vh‖0,K 6 C(h−1

K ‖j− jh‖0,K +h−1K ‖u−uh‖0,K

).

Since hK‖vh‖0,K = ηDK , we deduce bound (ii).Bound (iii): Let F be an interior face shared by two elements K and K ′. On F we define the functions

wh = [[∇×uh]]T , wb = bF wh,

and denote by Wb the extension in H10 ((K ∪ K ′

))3 of wb which satisfies (5.11) and (5.12). Since[[∇×u]]T = 0 on interior faces, we have

‖b12F wh‖2

0,F =

∫

F[[∇×uh]]T ·wb ds

=∫

F[[∇× (uh −u)]]T ·wb ds

= ∑K∈δF

(∫

K∇×∇× (uh −u) ·Wb dx−

∫

K∇× (uh −u) · (∇×Wb)dx

)

= ∑K∈δF

(∫

K((j− jh)−∇×∇× (u−uh)− (u−uh)) ·Wb dx

−∫

K(∇× (uh −u)) · (∇×Wb) dx−

∫

K(j− jh) ·Wb dx+

∫

K(u−uh) ·Wb dx

).

Noting that j−∇×∇×u−u = 0 in L2(K)3, K ∈ δF , employing (5.10), the Cauchy-Schwarz inequality,together with (5.11) and (5.12), we obtain

‖wh‖0,F 6 C ∑K∈δF

(h

12F‖jh−∇×∇×uh−uh‖0,K +h

− 12

F ‖∇× (u−uh)‖0,K

+h12F‖j− jh‖0,K +h

12F‖u−uh‖0,K

).

Multiplying this by h12F and taking into account the shape regularity of the mesh and the bound for ηRK ,

we get

h12F‖[[∇×uh]]T‖0,F 6 C ∑

K∈δF

(‖∇× (u−uh)‖0,K +hK‖u−uh‖0,K +hK‖j− jh‖0,K

).

This gives the bound (iii).Bound (iv): Similarly, we set

wh = [[jh−uh]]N , wb = bF wh,

and proceed as before. The scalar function wb can be extended to a function Wb ∈ H10 ((K ∪ K ′

))satisfying (5.5) and (5.6). Since [[j−u]]N = 0 on interior faces and ∇ · (j−u) = 0 in L2(K), we have

‖b12F wh‖2

0,F =

∫

F[[jh −uh]]Nwb ds

=

∫

F[[jh −uh − j+u]]Nwb ds

= ∑K∈δF

(∫

K∇ · (jh−uh)Wb dx+

∫

K(jh −uh − j+u) ·∇Wb dx

).


Employing (5.4) and the Cauchy-Schwarz inequality, together with (5.5) and (5.6), we obtain

‖wh‖0,F 6 C ∑K∈δF

(h

12F‖∇ · (jh−uh)‖0,K +h

− 12

F ‖u−uh‖0,K +h− 1

2F ‖j− jh‖0,K

).

Multiplying this by h12F and taking into account the shape regularity of the mesh and the bound for ηDK ,

we finally get

h12F‖[[jh−uh]]N‖0,F 6 C ∑

K∈δF

(‖u−uh‖0,K +‖j− jh‖0,K

).

This shows (iv).Bound (v): This bound follows immediately from the fact that [[u]]T = 0 on interior faces and that

n×u = 0 on boundary faces.

6. Numerical experiments

In this section we present a series of numerical examples to illustrate the practical performance ofthe proposed a posteriori error indicator in (3.7) within an automatic adaptive refinement procedure.Here, we restrict ourselves to the two-dimensional analogue of (1.1)–(1.2) approximated on 1-irregulartriangular meshes. Additionally, we note that throughout this section we select the interior penaltyparameter α in (2.4) as follows:

α = CIP `2,

with CIP = 10. The dependence of α on the polynomial degree ` has been chosen in view of standardhp-version stability properties for discontinuous Galerkin methods, cf. Remark 2.1, and the references(16; 23). In contrast, the choice of the constant CIP is based purely on numerical experience; indeed,we have consistently employed the same value of CIP for a wide range of problems, including linearadvection-diffusion equations, the Stokes equations, second-order quasi-linear elliptic partial differen-tial equations, the compressible Navier-Stokes equations, and the time-harmonic Maxwell system, forexample. In all cases, this choice of CIP is sufficiently large to guarantee stability of the underlying inte-rior penalty DG method, without being so large as to adversely affect the conditioning of the resultingsystem of linear/nonlinear equations.

The adaptive meshes are constructed on the basis of the local error indicators ηK , by employing thefixed fraction strategy, with refinement and derefinement fractions set to 25% and 10%, respectively.Here, the emphasis will be to demonstrate the asymptotic exactness of the proposed a posteriori errorindicator on non-uniform adaptively refined meshes. By asymptotic exactness, we mean that the errorindicator tends to zero at the same rate as the energy norm of the true error, as the mesh is refined.Thereby, as in (6), we set the constant CEST arising in Theorem 3.1 equal to one and ensure that thecorresponding effectivity indices are roughly constant on all of the meshes employed. Here, the effec-tivity index is defined as the ratio of the a posteriori error indicator and the DG-norm of the actual error.For computational simplicity, in all of our numerical experiments, the data approximation terms arisingin the a posteriori error bound (3.8) have been neglected. In general, to ensure the reliability of theerror indicator, the constants CEST and CAPP arising in Theorem 3.1 must be determined; this involvesestimating each of the individual constants appearing within our error analysis, see Remark 3.1 for thedefinition of CEST and CAPP. A typical approach is to numerically estimate each of these constants by


103

104

105

10−1

100

101

Error IndicatorTrue ErrorPSfrag replacements

Degrees of Freedom0 2 4 6 8 10

0

1

2

3

4

5

6

7

8

9

PSfrag replacementsE

ffec

tivity

Inde

x

Mesh Number(a) (b)

103

104

105

10−3

10−2

10−1

100



0

5

10

15

PSfrag replacements

Eff

ectiv

ityIn

dex

Mesh Number(c) (d)

103

104

105

10−4

10−3

10−2

10−1

100



0

5

10

15

20

25

PSfrag replacements

Eff

ectiv

ityIn

dex

Mesh Number(e) (f)

FIG. 1. Example 1. Computed energy norm errors and corresponding effectivity indices, respectively, for: (a) & (b) ` = 1; (c) &(d) ` = 2; (e) & (f) ` = 3.


103

104

105

10−4

10−3

10−2

10−1

100

l=1l=2l=3PSfrag replacements

‖u−

u h‖ D

G

Degrees of Freedom(a) (b)

FIG. 2. Example 1. (a) Comparison of the DG-norm of the actual error for ` = 1,2,3; (b) Computational mesh after 9 adaptivemesh refinements with ` = 1 and 16454 elements.

either solving generalised eigenvalue problems, or simply evaluating the maximum of each constantcomputed over a given finite dimensional space, cf. (11; 25); see also (35; 12), for related work.

All computations presented in this section have been performed using the MADNESS software pack-age; see (10) for details.

6.1 Example 1

In this first example, we let Ω be the domain (−1,1)2 and set j = ((1 + 2π2)cos(πx1)sin(πx2),−(1+2π2)cos(πx2)sin(πx1))

>; thereby, the analytical solution to (1.1)–(1.2), subject to an appropriateinhomogeneous boundary condition, is given by

u = (cos(πx1)sin(πx2),−cos(πx2)sin(πx1))>.

Here, we note that since j is solenoidal, we also have that ∇ ·u = 0 in Ω .In Fig. 1 we plot the estimated and actual DG-norm of the error computed on the sequence of

meshes generated by our adaptive algorithm, together with their corresponding effectivity indices, for` = 1,2,3. Here, we see that for each polynomial degree the a posteriori error indicator over-estimatesthe true error by a consistent factor, thereby confirming the asymptotic exactness of the proposed errorindicator for this smooth problem. Additionally, we note that the effectivity indices increase as thepolynomial degree is increased; indeed, for linear elements, the error indicator over-estimates the trueerror by a consistent factor between 6–7; for quadratic elements, this factor is between 10–11; finally,the effectivity indices for cubic elements lie in the range between 17–18. In Fig. 2(a) we compare thetrue error, measured in terms of the DG-norm, computed on the sequence of adaptively refined meshesfor each of the polynomial degrees employed. As we would expect for this smooth problem, an increasein ` leads to a considerable decrease in the DG-norm of the error for a fixed number of degrees offreedom.

Finally, in Fig. 2(b) we show the mesh generated using the proposed a posteriori error indicatorafter 9 adaptive refinement steps with linear elements. Here, we see that while the mesh has been


largely uniformly refined throughout the entire computational domain, additional refinement has beenperformed where the solution has local maxima and minima; cf. (19). Analogous meshes are alsogenerated by our adaptive algorithm for ` = 2 and ` = 3; for brevity, these results have been omitted.

6.2 Example 2

In this second example, we consider the numerical approximation of a smooth a.e. non-solenoidalsolution to the two-dimensional analogue of (1.1)–(1.2). To this end, we again select Ω = (−1,1)2,but now with j = (ex1(3x2 cosx2 − sinx2),ex1(2cosx2 − x2 sinx2))

>; thereby, (with suitable boundaryconditions for u), the analytical solution to (1.1)–(1.2) is given by

u = (ex1(x2 cosx2 + sinx2),ex1 x2 sinx2)>.

In Fig. 3 we plot the estimated and actual DG-norm of the error computed on the sequence ofmeshes generated by our adaptive algorithm, together with their corresponding effectivity indices, for` = 1,2,3. As in the previous example, we see that for each polynomial degree the a posteriori errorindicator over-estimates the true error by a consistent factor; though, as before, the computed effectivityindices increase slightly as the polynomial degree is enriched. A comparison of the actual DG-norm ofthe error for ` = 1,2,3 is presented in Fig. 4(a); this again highlights the improvement in accuracy perdegree of freedom when higher-order polynomials are employed for smooth problems.

Finally, Fig. 4(b) depicts the mesh generated using the proposed a posteriori error indicator after 8adaptive refinement steps with linear elements. Here, we see that the mesh has been largely uniformlyrefined throughout the entire computational domain, with additional refinement being performed in thevicinity of the top and bottom right-hand corners of Ω where the curvature of the analytical solutionrapidly changes. Analogous meshes are also generated by our adaptive algorithm for ` = 2 and ` = 3.

6.3 Example 3

In this final example, we select Ω to be the (non-convex) L–shaped domain (−1,1)2 \ [0,1)× (−1,0]and set j (and suitable non-homogeneous boundary conditions for u) so that the analytical solution u tothe two-dimensional analogue of (1.1)–(1.2) is given, in terms of the polar coordinates (r,ϑ ), by

u(x1,x2) = ∇(rβ sin(βϑ )), (6.1)

where β = 2/3; the analytical solution given by (6.1) then contains a singularity at the re-entrant cornerlocated at the origin of Ω . In particular, we note that u lies in the Sobolev space H2/3−ε(Ω)2, ε > 0.We note that both u and the forcing function j are solenoidal in this example.

As for the previous two examples, in Fig. 5 we plot the estimated and actual DG-norm of the errorcomputed on the sequence of meshes generated by our adaptive algorithm, together with their corre-sponding effectivity indices, for ` = 1,2,3. Once again, we observe that for each polynomial degreethe a posteriori error indicator over-estimates the true error by a consistent factor and that the computedeffectivity indices increase slightly as the polynomial degree is enriched. However, in contrast to theprevious two smooth examples, for this test problem we observe from Fig. 6(a) that an increase in thepolynomial degree leads to a slight degradation in the DG-norm of the actual error computed on each ofthe sequences of adaptively refined meshes.


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FIG. 4. Example 2. (a) Comparison of the DG-norm of the actual error for ` = 1,2,3; (b) Computational mesh after 8 adaptivemesh refinements with ` = 1 and 8774 elements.

Finally, in Fig. 7 we show the mesh generated using the local error indicators after 9 adaptive refine-ment steps with linear elements. Here, we see that the mesh has been largely refined in the vicinity ofthe re-entrant corner located at the origin of the computational domain. Similar grids are also generatedfor ` = 2,3. Analogous behaviour is also observed for a non-smooth problem with non-solenoidal j, andthereby non-solenoidal analytical solution; for brevity, these results have been omitted.

7. Conclusions

In this article we have introduced a new residual-based a posteriori error indicator for DG discretizationsof H(curl;Ω)–elliptic boundary value problems. We have shown that the indicator is both reliable andefficient with respect to the DG energy norm. The analysis of the reliability bound relies on employinga non-consistent reformulation of the DG scheme, together with a decomposition result for the under-lying discontinuous space. Numerical experiments presented in this article clearly demonstrate that theproposed a posteriori indicator converges to zero at the same asymptotic rate as the energy norm of theactual error on sequences of adaptively refined meshes.

Acknowledgments

PH was supported by the EPSRC (Grant GR/R76615). DS was supported in part by the Natural Sciencesand Engineering Research Council of Canada (NSERC).

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102

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l=1l=2l=3

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FIG. 6. Example 3. Comparison of the DG-norm of the actual error for ` = 1,2,3.

(a) (b)

FIG. 7. Example 3. (a) Computational mesh after 9 adaptive mesh refinements with ` = 1 and 2088 elements; (b) Zoom of (a)around the origin.


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