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

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1 IMA Journal of Numerical Analysis 005) Page of 7 doi: 0.093/imanum/ An a posteriori error indicator for discontinuous Galerkin discretizations of Hcurl) elliptic partial differential equations PAUL HOUSTON School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 RD, U ILARIA PERUGIA Dipartimento di Matematica, Università di Pavia, Via errata, 700 Pavia, Italy DOMINI SCHÖTZAU Mathematics Department, University of British Columbia, 984 Mathematics Road, Vancouver, BC V6T Z, Canada IMA J. Numer. Anal., Vol. 7, pp. -50, 007 We introduce a residual-based a posteriori error indicator for DG discretizations of Hcurl; ) elliptic boundary value problems that arise in eddy current models. We show that the indicator is both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm. We validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems. eywords: Discontinuous Galerkin methods, a posteriori error analysis, eddy current problems. Introduction The electric field-based eddy current model for the computation of quasistatic electromagnetic fields consists of the following initial-boundary value problem: find the electric field E : 0,T ) R 3 satisfying t σe) + µ E) = t j s n E = 0 E t=0 = E 0 in, in 0,T), on Γ 0,T), cf. 3; 7). Here, is an open bounded Lipschitz polyhedron in R 3 with boundary Γ = and outward normal unit vector n. or simplicity, we assume to be simply-connected and Γ to be connected. The right-hand side j s = j s x,t) is a given external source field, with j s,t) L ) 3 and j s,t) L ), t 0,T ). The material coefficients σ and µ are the electric conductivity and the magnetic permeability, Paul.Houston@nottingham.ac.uk ilaria.perugia@unipv.it schoetzau@math.ubc.ca IMA Journal of Numerical Analysis c Institute of Mathematics and its Applications 005; all rights reserved.

2 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU respectively; σ is a symmetric, uniformly positive semidefinite tensor, and µ is a symmetric, uniformly positive definite tensor, both with bounded coefficients. Assuming σ is uniformly positive definite on the whole of and that an implicit time discretization is employed, at each time-step, one would need to solve a boundary value problem of the form µ u) + βu = j in, n u = 0 on Γ, where u is the approximation to E to be computed at the current time-step, j depends on j s and the approximation 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 identity tensor, cf. the discussion presented in 5). With this discussion in mind, the purpose of this paper is to introduce a residual-based a posteriori error indicator for interior penalty discontinuous Galerkin DG) approximations of the following model problem: find the vector unknown u electric field) that satisfies u + u = j in,.) n u = 0 on Γ,.) where, for generality, j is a given external source field in L ) 3. By introducing the Sobolev space H 0 curl;) := {v L ) 3 : v L ) 3, n v = 0 on Γ }, the weak form of.).) is given by: find u H 0 curl;) such that ) au,v) := u) v)+ u v dx = j vdx.3) for all v H 0 curl;). The system in.).) is the simplest prototype of a partial differential equation that is elliptic over H 0 curl;). The main motivation for using a discontinuous Galerkin approach for the numerical approximation of.).) is that DG methods, being based on discontinuous finite element spaces, can easily handle non-conforming meshes which contain hanging nodes and, in principle, local spaces of different polynomial orders; for the purposes of the current article, we shall only consider the h-version of the DG method. Moreover, the implementation of discontinuous elements can be based on standard shape functions, without the need to employ curl-conforming elemental mappings - a convenience that is particularly advantageous for high-order elements and that is not straightforwardly shared by standard edge or face elements commonly used in computational electromagnetics see 8; 3; ) and the references therein for hp-adaptive edge element methods). A further benefit of DG methods is that inhomogeneous Dirichlet boundary conditions can easily be incorporated within the scheme, without the need to explicitly evaluate edge- and face-element interpolation operators. This paper represents a continuation of the series of articles 3; 30; 6; 7; 0; 9; 4; 5) concerned with the development of DG finite element methods for the numerical approximation of the time-harmonic Maxwell equations. Indeed, here we have considered the design and analysis of nonconforming DG methods for both low and high frequency approximations of these equations. In the low frequency regime, both regularized, cf. 30; 6), and mixed, cf. 7; 0; 9), formulations have been proposed. or the high frequency regime, mixed and so-called direct formulations have been proposed

3 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES 3 of 7 and analyzed; see 3; 5) and 4), respectively. or a recent review of this work, we refer to the article 8). With the exception of 9), the aforementioned articles have only considered the derivation of a priori error bounds. In this article we introduce an a posteriori error indicator for the DG approximation to.).) and prove both upper and lower bounds for the actual error measured in terms of a natural energy norm, thereby, demonstrating that the indicator is both reliable and efficient. We remark that upper bounds of this type were derived in 9) for the mixed DG approximation of the low frequency Maxwell s equations. However, the bounds presented there are somewhat unsatisfactory in the sense that 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, together with the conservation properties of the underlying DG method; cf. 6), where this approach was first developed in the context of DG methods employed for the numerical approximation of diffusion problems. In this article we pursue a different technique, inspired by the recent articles ; ; 3); see also 6) for related work. Here, the proof of the upper bound is based on rewriting the method in a nonconsistent manner using lifting operators in the spirit of 4), see also 9), together with approximation results which allow us to find a conforming finite element function which is close to any discontinuous one. Approximation results of this latter type have been developed for the h version of the DG method in the case when the underlying conforming finite element space is a subspace of H0 ) in the articles 7; 6); the extension of this result to the hp DG method was presented in the recent article 3). In this paper we exploit an analogous result derived in 4) for the h version of the DG method when the underlying conforming finite element space is a subspace of H 0 curl;). With this approximation result, the proof of the upper a posteriori error bound now rests on estimating the error between the analytical solution u and a conforming approximant u c h. To this end, we exploit the decomposition of H 0 curl;) derived in 3, Lemma.4), together with approximation results for the standard Clément interpolant, as well as for the quasi-interpolation operator constructed in 5, Section 5). The proofs of the lower efficiency) a posteriori error bounds follow from the standard bubble technique introduced in 33) and 34). inally, we note that the techniques presented here can be readily extended to elliptic problems in H 0 curl;) with smooth coefficients. However, the extension to problems with non-smooth coefficients remains an open issue. The reason for this is that for non-smooth coefficients the decomposition in 3, Lemma.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 R d, d, we write H t D) to denote the usual Sobolev space of real-valued functions with regularity exponent t and norm t,d ; for t = 0, we write L D) instead of H 0 D). The space H t D) d consists of vector fields whose components belong to H t D); it is endowed with the usual product norm which we denote, for simplicity, also by t,d. or D R 3, we write Hcurl;D) and Hdiv;D) to denote the spaces of vector fields u L D) 3 with u L D) 3 and u L D), respectively, endowed with their corresponding graph norms curl and div, respectively. inally, we denote by H0 D) and H 0curl;D) the subspaces of H D) and Hcurl;D), respectively, of functions with zero trace and zero tangential trace, respectively.. Discontinuous Galerkin discretization In this section, we consider the interior penalty DG discretization of.).). To this end, we first introduce the following notation.

4 4 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU Let T h be a conforming, shape-regular and affine tetrahedral partition of of granularity h = max Th h, where the local mesh size h is defined to be the diameter of the element T h. We denote by h the set of all faces of elements in T h. or an approximation order l, we introduce the following finite element space V h := {v L ) 3 : v P l ) 3, T h },.) where P l ) denotes the space of polynomials of total degree at most l on. With this notation, we consider the following interior penalty DG method: find u h V h such that a h u h,v) = j vdx.) for all v V h. The discrete form a h, ) is given by ) a h u,v) := u) v)+ u v dx [[u]] T {{ v}}ds T h h h [[v]] T {{ u}}ds + a[[u]] T [[v]] T ds. h or a piecewise smooth function v, on interior faces, we write [[v]] T and {{v}} to denote the tangential jump and mean value of the vector field v, respectively. On boundary faces we set [[v]] T = n v and {{v}} = v; cf. 6) or 4). The function a penalizes the tangential jumps; it is referred to as the interior penalty stabilization function. On a given face it is defined by.3) a := αh,.4) with h denoting the diameter of face and α being a positive constant that is independent of the mesh size. It is well-known that.) is uniquely solvable, provided that α α min, for a threshold value α min that only depends on the shape-regularity of the mesh and the approximation order l, cf. 7), for example. REMAR. The well-posedness of.) is guaranteed provided that α C inv, where C inv is the constant arising in the inverse estimate w 0, C invh w 0, w P l ), see, e.g., 30); see also 4). The constant C inv depends only on the polynomial degree l and the shaperegularity of the mesh see, e.g., 4)). Indeed for square elements, by solving a local generalized eigenvalue problem, it can be shown that C inv = C inv l, where C inv = + 3/l + /l ) 6 for all l ; thereby, in this case we may set α = C inv l, or indeed α = 6l. In general, for other element types, and indeed, for hybrid meshes, on the basis of the articles 6; 3), in Section 6 we actually set α = C IP l, with C IP = 0, the value of the constant C IP being chosen empirically, based on extensive numerical experimentation; see the discussion at beginning of Section 6 for further details. REMAR. In order to incorporate the inhomogeneous Dirichlet boundary condition n u = g on Γ, g in L Γ ) 3, within the above formulation, it is sufficient to simply add the following terms to the functional on the right-hand side of.) B h g v)ds + B h ag n v)ds, where B h denotes the set of all faces of elements in T h which lie on the boundary Γ of.

5 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES 5 of 7 3. A posteriori error indicator In this section, we present a reliable and efficient indicator for the error of the DG approximation measured in terms of a natural energy norm. 3. Local a posteriori error indicators We begin by introducing local error indicators. To do so, we introduce on each element T h the indicator η which is given by the sum of the following five terms η = η R + η D + η T + η N + η J. 3.) The first term η R measures the residual of the underlying system of partial differential equations and is defined by η R = h j h u h u h 0,, 3.) where j h V h denotes an appropriate approximation to j; we refer to Remark 3. concerning the choice of j h. The second term η D measures the error in the divergence and is given by η D = h j h u h ) 0,. 3.3) The third term η T is the usual face residual related to the tangential jump of u h : η T = \Γ h [[ u h ]] T 0,. 3.4) The fourth term η N measures the normal jump [[j h u h ]] N of j h u h over interior faces, that is, η N = \Γ h [[j h u h ]] N 0,. 3.5) The last term η J measures the tangential jumps of the approximate solution u h and is defined by 3. Reliability η J = \Γ a [[uh ]] T 0, + Γ a n uh ) 0,. 3.6) We now state and discuss the fact that the error indicator ) ERR := η 3.7) provides a reliable upper bound on the approximation error with respect to an energy-type norm. To do so, we introduce the space Vh) := H 0 curl;) + V h, and define the following norm: v DG := v 0, + v 0, + a [[v]]t 0,. h We remark that DG represents the natural energy norm arising from the definition of the DG bilinear form a h, ) given in.3). While this norm does indeed depend on the interior penalty stabilization parameter α, cf..4), the value of α is fixed, for a given polynomial degree, and is independent of the mesh size h.

6 6 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU THEOREM 3. Let u be the analytical solution of.).), with j L ) 3, and u h V h its DG approximation obtained by.) with α α min. Let the local error indicators be defined by 3.) 3.6). Then, the following upper bound for the error holds ) / u u h DG C EST η +C APPA j j h ), 3.8) where C EST and C APP are positive constants; here, C EST is independent of the mesh size and C APP is independent of the mesh size and α. If α max{,α min } then C EST may also be bounded independently of α. urther, A j j h ) is the data approximation term given by A j j h ) = j j h 0,. REMAR 3. rom the proof of Theorem 3., it follows that and C EST = C dec max{c c,c q max{,α C lift }} + +C cont) α C conf + ) C APP = C dec max{c c,c q }, where C dec is the constant arising in the stability bounds stated in 4.8), C q and C c are the interpolation constants in 4.9) and 4.0), respectively, α is the constant arising in the definition of the interior penalty stabilization function.4), C lift is the constant in 4.3), C cont is the continuity constants in Lemma 4. and C conf is the constant arising in the approximation result stated in Proposition 4.. REMAR 3. We note that Theorem 3. holds for any j h V h. To ensure that the data approximation term A j j h ) does not dominate the overall a posteriori error bound stated in 3.8), j h should be chosen in such a manner so that asymptotically) A j j h ) tends to zero at, at least, the same rate as the first term on the right-hand side of 3.8) and thereby also at, at least, the same rate as u u h DG, cf. Theorem 3.3 in the following section) as the mesh is refined for a given polynomial degree l. This can be achieved, for example, by choosing j h to be the L -projection of j onto the space V h. REMAR 3.3 In the case where the external source field j belongs to Hdiv;) for example, in many eddy current problems, j is actually solenoidal), then an upper bound for the error analogous to the one stated in Theorem 3. may be derived with the data approximation term defined by A j j h ) = h j jh 0, + j j h ) ) 0,. In this case the normal jump error indicator η N is defined as η N = \Γ h [[u h ]] N 0,, which now reflects the fact that the normal components of j across the element faces are continuous. REMAR 3.4 The case of imposing the inhomogeneous boundary condition n u = g on Γ may also be incorporated within the above error bound. Indeed, in the case when g is the tangential trace of a

7 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES 7 of 7 function belonging to V h H ) 3, the error indicator is simply modified by redefining the tangential jump indicator η J as follows: η J = \Γ a [[u h ]] T 0, + Γ a n u h g) 0,. or general g L Γ ) 3, a data error term which takes into account the error in the finite element approximation 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 η. PROPOSITION 3. Let u be the analytical solution of.).) and u h V h its DG approximation obtained by.) with α α min. Let the local error indicators be defined by 3.) 3.6). Then, the following local bounds hold. i) or any element T h, we have η R C u u h ) 0, + h u u h 0, + h j j h 0, ), with a constant C > 0 that is independent of the mesh size and α. ii) or any element T h, we have η D C u u h 0, + j j h 0, ), with a constant C > 0 that is independent of the mesh size and α. iii) or any interior face shared by two elements and, we have h [[ u h ]] T 0, C δ u uh ) 0, + h u u h 0, + h j j h 0, ), with a constant C > 0 that is independent of the mesh size and α; here, we have set δ = {, }. iv) or any interior face shared by two elements and, we have ) h [[j h u h ]] N 0, C u u h 0, + j j h 0,, 3.9) δ with a constant C > 0 that is independent of the mesh size and α; again, δ = {, }. v) or interior faces, we have and for boundary faces, a [[u h ]] T 0, = a [[u u h ]] T 0,, a n u h ) 0, = a n u u h )) 0,.

8 8 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU The proof of Proposition 3. is given in Section 5. An immediate consequence of Proposition 3. is the following result. THEOREM 3.3 Let u be the analytical solution of.).), with j L ) 3, and u h V h its DG approximation obtained by.) with α α min. Let the local error indicators be defined by 3.) 3.6). Then our proposed error indicator ERR in 3.7) is efficient in the sense that ) ERR C E u u h DG + A j j h ), where C E is a positive constant independent of the mesh size and α. REMAR 3.5 In the case when j Hdiv;), analogous efficiency bounds may also be derived. Indeed, bounds i), iii), and v) in Proposition 3. remain unchanged; on the right-hand side of the bounds in ii) and iv), the term j j h 0, may be replaced by h j j h ) 0,. 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 change in the definition of the normal jump indicator η N outlined in Remark 3.3; by this we mean that the term h [[j h u h ]] N 0, must be replaced by h [[u h ]] N 0,. Thereby, with the modified definition of the data approximation term stated in Remark 3.3, an analogous efficiency bound to the one stated in Theorem 3.3 holds. REMAR 3.6 We point out that the constants C EST and C APP arising in Theorem 3., and C E defined in Theorem 3.3, are all independent of the constant α arising in the definition of the interior penalty parameter a, which indicates that the resulting bounds are relatively insensitive to changes in α. Indeed, numerical experiments presented in the articles ; 3) confirm this assertion in the context of both nearly incompressible linear elasticity and Poisson s equation, respectively. REMAR 3.7 We note that the emphasis of this article is the design of an a posteriori error indicator for the h version only) of the interior penalty DG approximation of the Hcurl) elliptic problem.).). Thereby, it is natural that the constants C EST and C APP arising in Theorem 3., and C E defined in Theorem 3.3, should all depend on the polynomial degree l; indeed, this is evidenced in Section 6. The extension of the analysis to the hp version of the finite element method, whereby all constants would then indeed be independent of both h and l forms part of our programme of future research. We remark that one of the key technical difficulties in our setting is the derivation of the hp version of Proposition 4.; this has been undertaken in article 3) for subspaces of H ) only. 4. Proof of Theorem 3. In this section we carry out the proof of Theorem 3.. To this end, in Section 4. we present an approximation property for DG functions. Section 4. is devoted to augmenting the form a h to the space Vh). After these preliminary considerations, we conclude the proof of Theorem 3. in Sections 4.3, 4.4 and Approximation by conforming finite element functions One of the main ingredients in our error analysis is an approximation property that allows us to find a conforming finite element function close to any discontinuous one; see 0) and 4) where similar ideas are used in the a priori error analysis of DG methods for Maxwell s equations. We also refer to 6) and 3) where results of that type have been instrumental in deriving a posteriori bounds for DG discretizations of diffusion problems.

9 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES 9 of 7 To this end, we first define V c h to be the largest conforming space underlying V h, that is, V c h := V h H 0 curl;). 4.) In fact, V c h is the finite element space based on the second family of Nédélec s elements of degree l; see 8) or 7, Section 8.). We will make use of the following approximation property that is similar to the results in 4). PROPOSITION 4. Let v V h. Then there is a function v c V c h such that v v c DG α C conf + ) a [[v]] T 0,, h with a constant C conf > 0 that only depends on the shape-regularity of the mesh and the approximation order. Proof. This follows from the construction in 4, Appendix): for v V h there is an conforming approximation v c V c h such that and v v c h ) 0, C conf h h [[v]] T 0, v v c 0, C conf h [[v]] T 0,, h with a constant C conf > 0 that only depends on the shape-regularity of the mesh and the approximation order l. The approximation result then follows readily from the definition of a in.4). 4. Auxiliary formulation ollowing the a priori and a posteriori analyses of 9) and 3), respectively, we will augment the bilinear form a h in.) to Vh) Vh) in a non-consistent manner. To this end, we define for v Vh) the lifting L v) V h by L v) wdx = [[v]] T {{w}}ds h w V h. 4.) Then there is a constant C lift > 0 only depending on the shape-regularity of the mesh and the approximation degree l such that We then introduce the auxiliary bilinear form ã h u,v) := T h L v) 0, α C lift a [[v]]t 0,. 4.3) h ) u) v) + u v dx L v) u)dx + a[[u]] T [[v]] T ds. h L u) v)dx

10 0 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU Note that ã h = a h on V h V h and ã h = a on H 0 curl;) H 0 curl;); hence the form ã h can be viewed as an extension of both a and a h to the space Vh). The discrete problem.) can then be equivalently stated as follows: find u h V h such that ã h u h,v) = j vdx v V h. 4.4) Since ã h = a on H 0 curl;) H 0 curl;), the following stability result holds. LEMMA 4. We have for all v H 0 curl;). ã h v,v) = v curl = v DG Moreover, the bilinear form ã h is continuous on Vh); we refer to 4; 6; 3) for details. LEMMA 4. or any u,v Vh), we have ã h u,v) C cont u DG v DG, where C cont = max{, + α C lift }, α is the constant arising in the definition of the interior penalty stabilization function.4) and C lift 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 u h V h as follows u u h = u u c h ) u h u c h ), where u c h Vc h is the conforming approximation of u h from Proposition 4.. LEMMA 4.3 The following upper bound for the error holds u u h DG sup w H 0 curl;) Rw) + +C cont ) u h u c h DG, where C cont is the continuity constant from Lemma 4. and R ) is defined by j w w h )dx ã h u h,w w h ) Rw) = inf w H 0 curl;). w h V h w DG Proof. By application of the triangle inequality we obtain u u h DG u u c h DG + u h u c h DG. 4.5) In order to bound u u c h DG, we set w := u u c h H 0curl;) and exploit the coercivity property stated in Lemma 4.; thereby, we get w DG = ã hw,w) = ã h u,w) ã h u h,w) + ã h u h u c h,w). Since w H 0 curl;), we have ã h u,w) = au,w) = j wdx.

11 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES of 7 rom 4.4), we get ã h u h,w h ) = j w h dx w h V h. This allows us to conclude that w DG = j w w h )dx ã h u h,w w h ) + ã h u h u c h,w) for any w h V h. The continuity of ã h, ) from Lemma 4. yields the bound w DG j w w h )dx ã h u h,w w h ) +C cont u h u c h DG w DG for any w h V h. Therefore, u u c h DG Ru u c h ) +C cont u h u c h DG. Observing that Ru u c h ) sup w H 0 curl;) Rw) and referring to the triangle inequality in 4.5) completes the proof. Let us remark that, in Lemma 4.3, the term u h u c h DG can be bounded in terms of the jumps of the discrete solution, due to Proposition 4.. Thereby, we deduce that u h u c h DG α C conf + ) a [[u h ]] T 0, = α C conf + ) ηj, 4.6) h where η J is the jump residual defined in 3.6). In the following section, we now derive a computable upper bound for Rw). 4.4 Bound of Rw) To bound the term Rw), we will use the regular decomposition from 3, Lemma.4): any w H 0 curl;) can be written as w = w 0 + ϕ, 4.7) with w 0 H 0 curl;) H ) 3 and ϕ H 0 ). urthermore, there is a stability constant C dec > 0 only depending on such that w 0, C dec w curl, ϕ, C dec w curl. 4.8) We will further make use of the quasi-interpolation operator constructed in 5, Section 5): for any w H 0 curl;) H ) 3, there is a low-order approximation w h V c h that satisfies ) w w h ) 0, + h w w h 0, + h w w h 0, Cq w,, 4.9) with a constant C q > 0 that only depends on the shape-regularity of the mesh. inally, we will make use of the Clément interpolant see, e.g., 9, Section I.A.3)): for any ϕ H0 ), there is a piecewise linear approximation ϕ h H0 ) such that ) ϕ ϕ h ) 0, + h ϕ ϕ h 0, + h ϕ ϕ h 0, Cc ϕ,, 4.0) with a constant C c > 0 that only depends on the shape-regularity of the mesh. We are now ready to prove the following result.

12 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU LEMMA 4.4 or any w H 0 curl;), the following bound holds Rw) C dec max{c c,c q max{,α C lift }} ) η + C dec max{c c,c q }A j j h ). Here, C dec is the constant arising in the stability bounds stated in 4.8), C q and C c are the constants in 4.9) and 4.0), respectively, and C lift is the constant in 4.3). Proof. or any arbitrary w H 0 curl;), recalling that w DG = w curl, we obviously have that j w w c h )dx ã hu h,w w c h ) Rw) 4.) w curl for any w c h Vc h. Using the result in 4.7), we decompose w as w = w 0 + ϕ, and choose w c h Vc h in 4.) as w c h = w0 h + ϕ h, where w 0 h is the quasi-interpolant of w0 defined in 4.9) and ϕ h is the Clément interpolant of ϕ from 4.0). In this way we obtain j w w c h )dx ã hu h,w w c h ) T + T, with T = j w 0 w 0 h )dx ã hu h,w 0 w 0 h ), T = j u h ) ϕ ϕ h )dx. Let us next bound T and T. Bound of T : We have T = j h u h ) w 0 w 0 h )dx u h ) w 0 w 0 h )) dx T h + L u h ) w 0 w 0 h )) dx + j j h ) w 0 w 0 h )dx, T h T h 4.) for any j h V h, cf. Remark 3.. Integrating by parts the second term on the right-hand side of the above identity and employing the conformity of w 0 w 0 h yields u h ) w 0 w 0 h )) dx T h = u h ) w 0 w 0 h )dx n u h ) w 0 w 0 h )ds T h T h = u h ) w 0 w 0 h )dx \Γ [[ u h ]] T w 0 w 0 h )ds.

13 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES 3 of 7 Here, we denote by n the outward unit normal vector on. Therefore, T = T h j h u h u h ) w 0 w 0 h )dx + \Γ T + T + T 3 + T 4. Obviously, we can bound T by T h [[ u h ]] T w 0 w 0 h )ds + L u h ) w 0 w 0 h )) dx j j h ) w 0 w 0 h )dx T η R h w0 w 0 h 0,, 4.3) where η R is the residual defined in 3.). To bound T we use the Cauchy-Schwarz inequality and the stability of the lifting operator in 4.3); this yields ) ) T L u h ) 0, w 0 w 0 h ) 0, ) α C lift a [[uh ]] T 0, h w 0 w 0 h ) 0, ) ), = α C lift η J w 0 w 0 h ) 0, ) 4.4) where the jump residuals η J are defined in 3.6). or T 3, application of the Cauchy-Schwarz inequality gives T 3 \Γ ) h [[ u h ]] T 0, η T h w0 w 0 h 0,, ) h w0 w 0 h 0, 4.5) with the residual η T defined in 3.4). Similarly, T 4 h j j h 0, h w0 w 0 h 0,. 4.6) Using the Cauchy-Schwarz inequality, taking into account the approximation bound in 4.9) and the bounds in 4.3), 4.4), 4.5), and 4.6), we conclude that T C q max{,α C lift } ) η R + ηj + ηt ) w 0, +C q A j j h ) w 0,. 4.7)

14 4 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU Bound of T : Next, let us bound the term T in 4.). Adding and subtracting j h, and integration by parts, we obtain T = j j h ) ϕ ϕ h )dx + j h u h ) ϕ ϕ h )dx = j j h ) ϕ ϕ h )dx j h u h )ϕ ϕ h )dx = or term T, we have + j h u h ) n ϕ ϕ h )ds j j h ) ϕ ϕ h )dx j h u h )ϕ ϕ h )dx + \Γ T + T + T 3. Term T can be bounded by [[j h u h ]] N ϕ ϕ h )ds T j j h 0, ϕ ϕ h ) 0,. 4.8) T h j h u h ) 0, h ϕ ϕ h 0, = η D h ϕ ϕ h 0,, with the residual η D defined in 3.3). inally, for term T 3, we have ) T 3 T h [[j h u h ]] N 0, h \Γ η N h ϕ ϕ h 0,, ) h ϕ ϕ h 0, 4.9) 4.0) with the residual η N defined in 3.5). Application of the approximation result in 4.0), the bounds in 4.8), 4.9), and 4.0), and the Cauchy-Schwarz inequality yields ) T C c η D + ηn ) ϕ, +C c A j j h ) ϕ,. 4.) Conclusion: from the bounds in 4.7) and 4.) for T and T, respectively, we deduce that ) T + T max{c c,c q max{,α C lift }} η w 0, + ) ϕ, 4.) + max{c c,c q }A j j h ) w 0, + ϕ,). Combining this estimate with the stability bounds in 4.8), we obtain the result.

15 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES 5 of Proof of Theorem 3. The proof of Theorem 3. is now an immediate consequence of Lemma 4.3, Lemma 4.4 and the bound in 4.6). A careful inspection of the proof reveals that all the constants can be bounded independently of α provided α. 5. Proof of Proposition 3. In this section we prove the efficiency bounds stated in Proposition 3.; here, we employ the bubble function technique introduced in 33). To this end, for an element of T h, we denote by b the standard polynomial bubble function on, and for an interior face shared by two elements and, we denote by b the standard polynomial bubble function on ; moreover, set δ := {, }. With this notation, the following bounds hold. LEMMA 5. Let v be a polynomial function on ; then there exists a constant C > 0 independent of v and h such that b v 0, C v 0,, 5.) v 0, C b v 0,, 5.) b v) 0, Ch v 0,. 5.3) Moreover, let be an interior face shared by two elements and, and let w be a polynomial function on ; then there exists a constant C > 0 independent of w and h such that w 0, C b w 0,. 5.4) inally, there exists an extension W b H 0 ) ) of b w such that W b = b w and with a constant C > 0 independent of w and h. W b 0, Ch w 0, δ, 5.5) W b 0, Ch w 0, δ, 5.6) Proof. The proof of 5.), 5.), 5.4) and 5.5) is given in 33, Lemma 4.). The proof of 5.3) and 5.6) can be obtained by similar arguments; see, Theorems. and.4). Analogous bounds can be easily obtained for vector-valued functions; in particular, we shall need the following estimates. LEMMA 5. Let v be a vector-valued) polynomial function on ; then there exists a constant C > 0 independent of v and h such that b v 0, C v 0,, 5.7) v 0, C b v 0,, 5.8) b v) 0, Ch v 0,. 5.9) Moreover, let be an interior face shared by two elements and, and let w be a vector-valued) polynomial function on ; then there exists a constant C > 0 independent of w and h such that w 0, C b w 0,. 5.0)

16 6 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU inally, there exists an extension W b H 0 ) ) 3 of b w such that W b = b w and with a constant C > 0 independent of w and h. W b 0, Ch w 0, δ, 5.) W b 0, Ch w 0, δ, 5.) We now proceed with the proof of Proposition 3.. Bound i): Let be an element of T h. or notational convenience, we introduce the following polynomials on : v h = j h u h u h, v b = b v h. Using the fact that u + u = j in L ) 3, we obtain b v h 0, = = = = j h u h u h ) v b dx j u h u h ) v b dx + j h j) v b dx u u h ) + u u h )) v b dx + j h j) v b dx u u h )) v b ) dx + u u h ) v b dx + j h j) v b dx, where in the last step we have integrated by parts the curl-curl term; we note that, since v b is zero on the boundary of, the boundary terms vanish. Using 5.8) and the Cauchy-Schwarz inequality, we obtain v h 0, C u u h ) 0, v b 0, + u u h 0, v b 0, + j j h 0, v b 0, ), and, owing to 5.9) and 5.7), we conclude that v h 0, C h u u h) 0, + u u h 0, + j j h 0, ), with a constant C > 0 that is independent of the mesh size. Multiplying by h and noting that η R = h v h 0, yields the desired bound i). Bound ii): To bound η D, we set Using 5.), we obtain v h = j h u h ), v b = b v h. v h 0, C b v h 0, = C j h u h )v b dx, with a constant C > 0 that is independent of the mesh size. rom.), it is clear that j u Hdiv; ) and j u) = 0 in L ). Hence v h 0, C j h j) + u u h )) v b dx.

17 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES 7 of 7 Integrating by parts and employing the Cauchy-Schwarz inequality, together with 5.3), yields v h 0, C h j j h 0, + h u u h 0, ). Since h v h 0, = η D, we deduce bound ii). Bound iii): Let be an interior face shared by two elements and. On we define the functions w h = [[ u h ]] T, w b = b w h, and denote by W b the extension in H0 ) ) 3 of w b which satisfies 5.) and 5.). Since [[ u]] T = 0 on interior faces, we have b w h 0, = [[ u h ]] T w b ds = [[ u h u)]] T w b ds ) = u h u) W b dx u h u) W b )dx δ = j j h ) u u h ) u u h )) W b dx δ ) u h u)) W b ) dx j j h ) W b dx + u u h ) W b dx. Noting that j u u = 0 in L ) 3, δ, employing 5.0), the Cauchy-Schwarz inequality, together with 5.) and 5.), we obtain w h 0, C δ h j h u h u h 0, + h u u h) 0, +h j j h 0, + h u u h 0, ). Multiplying this by h and taking into account the shape regularity of the mesh and the bound for η R, we get h [[ u h ]] T 0, C δ u uh ) 0, + h u u h 0, + h j j h 0, ). This gives the bound iii). Bound iv): Similarly, we set w h = [[j h u h ]] N, w b = b w h, and proceed as before. The scalar function w b can be extended to a function W b H0 ) ) satisfying 5.5) and 5.6). Since [[j u]] N = 0 on interior faces and j u) = 0 in L ), we have b w h 0, = [[j h u h ]] N w b ds = [[j h u h j + u]] N w b ds ) = j h u h )W b dx + j h u h j + u) W b dx. δ

18 8 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU Employing 5.4) and the Cauchy-Schwarz inequality, together with 5.5) and 5.6), we obtain w h 0, C h j h u h ) 0, + h u u h 0, + h j j h 0, ). δ Multiplying this by h and taking into account the shape regularity of the mesh and the bound for η D, we finally get ) h [[j h u h ]] N 0, C u u h 0, + j j h 0,. δ 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 of the proposed a posteriori error indicator in 3.7) within an automatic adaptive refinement procedure. Here, we restrict ourselves to the two-dimensional analogue of.).) approximated on -irregular triangular meshes. Additionally, we note that throughout this section we select the interior penalty parameter α in.4) as follows: α = C IP l, with C IP = 0. The dependence of α on the polynomial degree l has been chosen in view of standard hp-version stability properties for discontinuous Galerkin methods, cf. Remark., and the references 6; 3). In contrast, the choice of the constant C IP is based purely on numerical experience; indeed, we have consistently employed the same value of C IP for a wide range of problems, including linear advection-diffusion equations, the Stokes equations, second-order quasi-linear elliptic partial differential equations, the compressible Navier-Stokes equations, and the time-harmonic Maxwell system, for example. In all cases, this choice of C IP is sufficiently large to guarantee stability of the underlying interior penalty DG method, without being so large as to adversely affect the conditioning of the resulting system of linear/nonlinear equations. The adaptive meshes are constructed on the basis of the local error indicators η, by employing the fixed fraction strategy, with refinement and derefinement fractions set to 5% and 0%, respectively. Here, the emphasis will be to demonstrate the asymptotic exactness of the proposed a posteriori error indicator on non-uniform adaptively refined meshes. By asymptotic exactness, we mean that the error indicator 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 C EST arising in Theorem 3. equal to one and ensure that the corresponding effectivity indices are roughly constant on all of the meshes employed. Here, the effectivity index is defined as the ratio of the a posteriori error indicator and the DG-norm of the actual error. or computational simplicity, in all of our numerical experiments, the data approximation terms arising in the a posteriori error bound 3.8) have been neglected. In general, to ensure the reliability of the error indicator, the constants C EST and C APP arising in Theorem 3. must be determined; this involves estimating each of the individual constants appearing within our error analysis, see Remark 3. for the definition of C EST and C APP. A typical approach is to numerically estimate each of these constants by

19 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES 9 of Error Indicator True Error Degrees of reedom a) Effectivity Index Mesh Number b) Effectivity Index 0 5 Error Indicator True Error Degrees of reedom c) Mesh Number d) Error Indicator True Error Degrees of reedom e) Effectivity Index Mesh Number f) IG.. Example. Computed energy norm errors and corresponding effectivity indices, respectively, for: a) & b) l = ; c) & d) l = ; e) & f) l = 3.

20 0 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU 0 0 u uh DG l= l= l= Degrees of reedom a) b) IG.. Example. a) Comparison of the DG-norm of the actual error for l =,,3; b) Computational mesh after 9 adaptive mesh refinements with l = and 6454 elements. either solving generalised eigenvalue problems, or simply evaluating the maximum of each constant computed over a given finite dimensional space, cf. ; 5); see also 35; ), for related work. All computations presented in this section have been performed using the MADNESS software package; see 0) for details. 6. Example In this first example, we let be the domain,) and set j = + π )cosπx )sinπx ), +π )cosπx )sinπx )) ; thereby, the analytical solution to.).), subject to an appropriate inhomogeneous boundary condition, is given by u = cosπx )sinπx ), cosπx )sinπx )). Here, we note that since j is solenoidal, we also have that u = 0 in. In ig. 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 l =,, 3. Here, we see that for each polynomial degree the a posteriori error indicator over-estimates the true error by a consistent factor, thereby confirming the asymptotic exactness of the proposed error indicator for this smooth problem. Additionally, we note that the effectivity indices increase as the polynomial degree is increased; indeed, for linear elements, the error indicator over-estimates the true error by a consistent factor between 6 7; for quadratic elements, this factor is between 0 ; finally, the effectivity indices for cubic elements lie in the range between 7 8. In ig. a) we compare the true error, measured in terms of the DG-norm, computed on the sequence of adaptively refined meshes for each of the polynomial degrees employed. As we would expect for this smooth problem, an increase in l leads to a considerable decrease in the DG-norm of the error for a fixed number of degrees of freedom. inally, in ig. b) we show the mesh generated using the proposed a posteriori error indicator after 9 adaptive refinement steps with linear elements. Here, we see that while the mesh has been

21 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES of 7 largely uniformly refined throughout the entire computational domain, additional refinement has been performed where the solution has local maxima and minima; cf. 9). Analogous meshes are also generated by our adaptive algorithm for l = and l = 3; for brevity, these results have been omitted. 6. Example In this second example, we consider the numerical approximation of a smooth a.e. non-solenoidal solution to the two-dimensional analogue of.).). To this end, we again select =,), but now with j = e x 3x cosx sinx ),e x cosx x sinx )) ; thereby, with suitable boundary conditions for u), the analytical solution to.).) is given by u = e x x cosx + sinx ),e x x sinx ). In ig. 3 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 l =,,3. As in the previous example, we see that for each polynomial degree the a posteriori error indicator over-estimates the true error by a consistent factor; though, as before, the computed effectivity indices increase slightly as the polynomial degree is enriched. A comparison of the actual DG-norm of the error for l =,, 3 is presented in ig. 4a); this again highlights the improvement in accuracy per degree of freedom when higher-order polynomials are employed for smooth problems. inally, ig. 4b) depicts the mesh generated using the proposed a posteriori error indicator after 8 adaptive refinement steps with linear elements. Here, we see that the mesh has been largely uniformly refined throughout the entire computational domain, with additional refinement being performed in the vicinity of the top and bottom right-hand corners of where the curvature of the analytical solution rapidly changes. Analogous meshes are also generated by our adaptive algorithm for l = and l = Example 3 In this final example, we select to be the non-convex) L shaped domain,) \ [0,),0] and set j and suitable non-homogeneous boundary conditions for u) so that the analytical solution u to the two-dimensional analogue of.).) is given, in terms of the polar coordinates r,ϑ), by ux,x ) = r β sinβϑ)), 6.) where β = /3; the analytical solution given by 6.) then contains a singularity at the re-entrant corner located at the origin of. In particular, we note that u lies in the Sobolev space H /3 ε ), ε > 0. We note that both u and the forcing function j are solenoidal in this example. As for the previous two examples, in ig. 5 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 l =,, 3. Once again, we observe that for each polynomial degree the a posteriori error indicator over-estimates the true error by a consistent factor and that the computed effectivity indices increase slightly as the polynomial degree is enriched. However, in contrast to the previous two smooth examples, for this test problem we observe from ig. 6a) that an increase in the polynomial degree leads to a slight degradation in the DG-norm of the actual error computed on each of the sequences of adaptively refined meshes.

22 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU Error Indicator True Error Degrees of reedom a) Effectivity Index Mesh Number b) Error Indicator True Error Degrees of reedom c) Effectivity Index Mesh Number d) Error Indicator True Error Degrees of reedom e) Effectivity Index Mesh Number f) IG. 3. Example. Computed energy norm errors and corresponding effectivity indices, respectively, for: a) & b) l = ; c) & d) l = ; e) & f) l = 3.

23 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES 3 of u uh DG l= l= l= Degrees of reedom a) b) IG. 4. Example. a) Comparison of the DG-norm of the actual error for l =,,3; b) Computational mesh after 8 adaptive mesh refinements with l = and 8774 elements. inally, in ig. 7 we show the mesh generated using the local error indicators after 9 adaptive refinement steps with linear elements. Here, we see that the mesh has been largely refined in the vicinity of the re-entrant corner located at the origin of the computational domain. Similar grids are also generated for l =,3. Analogous behaviour is also observed for a non-smooth problem with non-solenoidal j, and thereby 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 discretizations of Hcurl; ) elliptic boundary value problems. We have shown that the indicator is both reliable and efficient with respect to the DG energy norm. The analysis of the reliability bound relies on employing a non-consistent reformulation of the DG scheme, together with a decomposition result for the underlying discontinuous space. Numerical experiments presented in this article clearly demonstrate that the proposed a posteriori indicator converges to zero at the same asymptotic rate as the energy norm of the actual error on sequences of adaptively refined meshes. Acknowledgments PH was supported by the EPSRC Grant GR/R7665). DS was supported in part by the Natural Sciences and Engineering Research Council of Canada NSERC). REERENCES [] M. Ainsworth and J. Coyle. Hierarchic hp-edge element families for Maxwell s equations on hybrid quadrilateral/triangular meshes. Comput. Methods Appl. Mech. Engrg., 90: , 00. [] M. Ainsworth and J.T. Oden. A posteriori error estimation in finite element analysis. John Wiley & Sons, 000.

24 4 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU Effectivity Index Error Indicator True Error Degrees of reedom a) Mesh Number b) Error Indicator True Error Degrees of reedom c) Effectivity Index Mesh Number d) 0 0 Effectivity Index Error Indicator True Error Degrees of reedom e) Mesh Number f) IG. 5. Example 3. Computed energy norm errors and corresponding effectivity indices, respectively, for: a) & b) l = ; c) & d) l = ; e) & f) l = 3.

25 AN A POSTERIORI ERROR INDICATOR OR DG DISCRETIZATIONS O Hcurl) ELLIPTIC PDES 5 of 7 0 u uh DG l= l= l= Degrees of reedom IG. 6. Example 3. Comparison of the DG-norm of the actual error for l =,, 3. a) b) IG. 7. Example 3. a) Computational mesh after 9 adaptive mesh refinements with l = and 088 elements; b) Zoom of a) around the origin.

26 6 of 7 P. HOUSTON, I. PERUGIA, AND D. SCHÖTZAU [3] H. Ammari, A. Buffa, and J.C. Nédélec. A justification of eddy currents model for the Maxwell equations. SIAM J. Appl. Math., 60:805 83, 000. [4] D.N. Arnold,. Brezzi, B. Cockburn, and L.D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 39: , 00. [5] R. Beck, R. Hiptmair, R. Hoppe, and B. Wohlmuth. Residual based a-posteriori error estimators for eddy current computation. Modél. Math. Anal. Numér., 34:59 8, 000. [6] R. Becker, P. Hansbo, and M.G. Larson. Energy norm a posteriori error estimation for discontinuous Galerkin methods. Comput. Methods Appl. Mech. Engrg., 9:73 733, 003. [7] A. Bossavit. Électromagnétisme. Springer Verlag, 993. [8] L. Demkowicz and L. Vardapetyan. Modeling of electromagnetic absorption/scattering problems using hp adaptive finite elements. Comput. Methods Appl. Mech. Engrg., 5:03 4, 998. [9] V. Girault and P.Raviart. inite element methods for Navier-Stokes equations. Springer, Berlin Heidelberg New York, 986. [0] E. Hall and P. Houston. MADNESS: Multi dimensional ADaptive finite Element Solution Software. Users Manual Version.0. Technical report, University of Nottingham, in preparation). [] D.C. Handscomb. Errors of linear interpolation on a triangle. Technical Report NA95/09, Oxford University Computing Laboratory, 995. [] I. Harari and T. Hughes. What are C and h?: Inequalities for the analysis and design of finite element methods. Comput. Methods Appl. Mech. Engrg., 97:57 9, 99. [3] R. Hiptmair. inite elements in computational electromagnetism. Acta Numerica, :37 339, 00. [4] P. Houston, I. Perugia, A. Schneebeli, and D. Schötzau. Interior penalty method for the indefinite time-harmonic Maxwell equations. Numer. Math., 00:485 58, 005. [5] P. Houston, I. Perugia, A. Schneebeli, and D. Schötzau. Mixed discontinuous Galerkin approximation of the Maxwell operator: The indefinite case. Modél. Math. Anal. Numér., 394):77 754, 005. [6] P. Houston, I. Perugia, and D. Schötzau. hp-dgem for Maxwell s equations. In. Brezzi, A. Buffa, S. Corsaro, and A. Murli, editors, Numerical Mathematics and Advanced Applications ENUMATH 00, pages Springer, 003. [7] P. Houston, I. Perugia, and D. Schötzau. Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J. Numer. Anal., 4: , 004. [8] P. Houston, I. Perugia, and D. Schötzau. Recent developments in discontinuous Galerkin methods for the timeharmonic Maxwell s equations. International Compumag Society Newsletter, ):0 7, 004. [9] P. Houston, I. Perugia, and D. Schötzau. Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator. Comput. Methods Appl. Mech. Engrg., 94:499 50, 005. [0] P. Houston, I. Perugia, and D. Schötzau. Mixed discontinuous Galerkin approximation of the Maxwell operator: Non-stabilized formulation. J. Sci. Comp., :35 346, 005. [] P. Houston, D. Schötzau, and T. Wihler. Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem. J. Sci. Comp., ): , 005. [] P. Houston, D. Schötzau, and T. Wihler. An hp-adaptive mixed discontinuous Galerkin EM for nearly incompressible linear elasticity. Comput. Methods Appl. Mech. Engrg., to appear). [3] P. Houston, D. Schötzau, and T. P. Wihler. Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems. Math. Models Methods Appl. Sci., to appear). [4] P. Houston, C. Schwab, and E. Süli. Discontinuous hp-finite element methods for advection diffusion reaction problems. SIAM J. Numer. Anal., 40:33 63, 00. [5] P. Houston and E. Süli. Adaptive Lagrange Galerkin methods for unsteady convection-diffusion problems. Math. Comp., 7033):77 06, 00. [6] O.A. arakashian and. Pascal. A posteriori error estimation for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal., 4: , 003.