ADAPTIVE HYBRIDIZED INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS FOR H(CURL)-ELLIPTIC PROBLEMS

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1 ADAPTIVE HYBRIDIZED INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS OR HCURL-ELLIPTIC PROBLEMS C. CARSTENSEN, R. H. W. HOPPE, N. SHARMA, AND T. WARBURTON Abstract. We develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin IPDG-H method for Hcurl- elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents equations. The method can be derived from a mixed formulation of the given boundary value problem and involves a Lagrange multiplier that is an approximation of the tangential traces of the primal variable on the interfaces of the underlying triangulation of the computational domain. It is shown that the IPDG-H technique can be equivalently formulated and thus implemented as a mortar method. The mesh adaptation is based on a residual-type a posteriori error estimator consisting of element and face residuals. Within a unified framework for adaptive finite element methods, we prove the reliability of the estimator up to a consistency error. The performance of the adaptive symmetric IPDG-H method is documented by numerical results for representative test examples in 2D. Key words. adaptive hybridized Interior Penalty Discontinuous Galerkin method, a posteriori error analysis, Hcurl-elliptic boundary value problems, semi-discrete eddy currents equations AMS subject classifications. 651, 65N3 1. Introduction. Discontinuous Galerkin DG methods are widely used algorithmic schemes for the numerical solution of partial differential equations PDE. or a comprehensive description, we refer to the survey article [24] and the references therein. As far as elliptic boundary value problems are concerned, DG methods can be derived from a primal-dual mixed formulation using local approximations of the primal and dual variables by polynomial scalar and vector-valued functions and appropriately designed numerical fluxes. Among the most popular schemes are Interior Penalty DG IPDG and Local DG LDG methods which have been analyzed by means of a priori estimates of the global discretization, e.g., in [3, 5, 23, 39]. or Hcurl-elliptic boundary value problems arising from a semi-discretization of the eddy currents equations, symmetric IPDG methods have been studied in [36]. The time-harmonic Maxwell equations have been addressed in [46]. On the other hand, the a posteriori error analysis and application of adaptive finite element methods EM for the efficient numerical solution of boundary and initialboundary value problems for PDE has reached some state of maturity as documented by a series of monographs. There exist several concepts including residual and hierarchical type estimators, error estimators that are based on local averaging, the so-called goal oriented dual weighted approach, and functional type error majorants cf. [2, 6, 7, 3, 44, 49] and the references therein. A posteriori error estimators for DG methods applied to second order elliptic boundary value problems have been developed and analyzed in [1, 11, 18, 38, 4, 47]. In particular, a convergence analysis of adaptive symmetric IPDG methods has been provided in [12, 34] and [41]. Residualand hierarchical-type a posteriori error estimator for Hcurl-elliptic problems have Dep. of Math., Humboldt Universität zu Berlin, D-199 Berlin, Germany Dep. of Comput. Sci. Eng., Yonsei University, Seoul , Korea Dep. of Math., Univ. of Houston, Houston, TX , U.S.A. Inst. of Math., Univ. of Augsburg, D Augsburg, Germany CAAM, Rice Univ., Houston, TX , U.S.A. The first author has been supported by the German National Science oundation DG within the Research Center MATHEON and by the WCU program through KOSE R The other authors acknowledge support by the NS grant DMS

2 been studied in [8, 9, 1, 2, 37]. A convergence analysis for residual estimators has been developed in [19] for 2D and in [35] for 3D problems. rom a computational point of view, DG methods suffer from a relatively huge amount of globally coupled degrees of freedom DO compared to standard EM. Hybridization is a technique that gives rise to a significant reduction of the globally coupled DO. It has been introduced for mixed EM in [31] and further studied in [4, 13, 15, 25, 26]. Adaptive mixed hybrid methods on the basis of reliable a posteriori error estimators have been considered in [14, 45] and [5]. or DG methods, a survey of hybridized DG DG-H methods has been provided in [26], whereas a unified analysis has been developed in [28]. However, adaptive DG-H methods have not yet been investigated. In this paper, we will derive and analyze a residual-type a posteriori error estimator for hybridized symmetric IPDG IPDG-H methods applied to Hcurl-elliptic boundary value problems in 3D. The analysis will be carried out within a unified framework provided for adaptive finite element approximations in [17, 18, 2, 21, 22]. The paper is organized as follows: In Section 2, we introduce some basic notation and present the class of Hcurl-elliptic boundary value problems to be approximated by symmetric IPDG-H methods. Section 3 deals with the development of symmetric IPDG-H methods based on a mixed formulation of the elliptic boundary value problems. We establish its relationship with mortar techniques which allows the implementation as a mortar method. In section 4, we present the residual-type a posteriori error estimator and prove its reliability. inally, in section 5, we provide a detailed documentation of numerical results to illustrate the performance of the symmetric IPDG-H methods. 2. Basic Notations. Let Ω R 3 be a simply connected polyhedral domain with boundary Γ = Ω such that Γ = Γ D Γ N, Γ D Γ N =. We denote by DΩ the space of all infinitely often differentiable functions with compact support in Ω and by D Ω its dual space referring to <, > as the dual pairing between D Ω and DΩ. We further adopt standard notation from Lebesgue and Sobolev space theory. In particular, for a subset D Ω, we refer to L 2 D and L 2 D as the Hilbert spaces of scalar and vector-valued square integrable functions with inner products,,d and associated norms,d, respectively. urther, we denote by H 1 D the Sobolev space of square integrable functions with square integrable weak derivatives equipped with the inner product, 1,D and norm 1,D. or Σ D, we refer to H 1/2 Σ as the space of traces v Σ of functions v H 1 D on Σ. We set H,Σ 1 D := {v H1 Ω v Σ = } and refer to H Σ D as the associated dual space. or a simply connected polyhedral domain Ω with boundary Γ = Ω which can be split into J relatively open faces Γ 1,..., Γ J with Γ = J j=1 Γ j, we refer to Hcurl; Ω as the Hilbert space Hcurl; Ω := {u L 2 Ω curl u L 2 Ω}, equipped with the inner product u, v curl,ω := u, v,ω curl u, curl v,ω and the associated norm curl,ω. We further refer to Hcurl ; Ω as the subspace of irrotational vector fields. The space Hdiv; Ω is defined by Hdiv; Ω := { q L 2 Ω div q L 2 Ω} which is a Hilbert space with respect to the inner product u, v div,ω := u, v,ω div u, div v,ω and the associated norm div,ω. or vector fields u C Ω 3 := {u Ω u C R 3 }, the normal component trace reads η n u Γj := n Γj u Γj, j = 1,..., J with the exterior unit normal vector n Γj on Γ j. The normal component trace mapping can be extended by continuity to a surjective, continuous linear mapping η n : Hdiv; Ω H /2 Γ cf. [32]; Thm We define H div; Ω as the subspace of vector fields with vanishing normal components on 2

3 Γ. In order to study the traces of vector fields q Hcurl; Ω, following [16], we introduce the spaces L 2 tγ := {u L 2 Ω η n u = }, H 1/2 Γ := {u L 2 tγ u Γj H 1/2 Γ j for all j = 1,..., J}. 6 t j t k n j n k t jk Γ j Γ k ig Two adjacent faces Γ j, Γ k with common edge E jk or Γ j, Γ k Γ with j k and E jk := Γ j Γ k E h, the set of edges, we denote by t j and t k the tangential unit vectors along Γ j and Γ k and by t jk the unit vector parallel to E jk such that Γ j is spanned by t j, t jk and Γ k by t k, t jk cf. igure 2.1. Let J := {j, k {1,..., J} 2 Γ j Γ k = E jk E h } and define H 1/2 Γ := {u H 1/2 Γ t jk u j Ejk = t jk u k Ejk for j, k J }, H 1/2 Γ := {u H1/2 Γ t j u j Ejk = t k u k Ejk for j, k J }. We refer to H /2 Γ and H /2 Γ as the dual spaces of H 1/2 Γ and H 1/2 Γ with L 2 tγ as the pivot space. or u D Ω 3 we further define the tangential trace mapping γ t Γj := u n Γj Γj, j = 1,..., J, and the tangential components trace π t Γj := n Γj u n Γj Γj, j = 1,..., J. Moreover, for a smooth function u D Ω we define the tangential gradient operator Γ = grad Γ as the tangential components trace of the gradient operator, i.e., Γ u Γj := Γj u = π t,j u = n Γj u n Γj, j = 1,..., J, which leads to a continuous linear mapping Γ : H 3/2 Γ H 1/2 Γ cf. [16]. The tangential divergence operator div Γ : H /2 Γ H 3/2 Γ is defined, with the respective dual pairings,, as the adjoint operator of Γ, i.e., div Γ u, v = u, Γ v, v H 3/2 Γ, u H /2 Γ. inally, for u C Ω we define the tangential curl operator curl τ as the tangential trace of the gradient operator curl τ u Γj = curl Γj u = γ t,j u = u n j for j = 1,..., J. 2.1 The vectorial tangential curl operator is a linear continuous mapping The scalar tangential curl operator curl τ : H 3/2 Γ H 1/2 Γ. curl τ : H /2 Γ H 3/2 Γ is defined as the adjoint of the vectorial tangential curl operator via curl τ, i.e., < curl τ u, v > = < u, curl Γ v > for all v H 3/2 Γ and u H /2 Γ. 3

4 The range spaces of the tangential trace mapping γ t and the tangential components trace mapping π t on Hcurl; Ω can be characterized by means of the spaces H /2 div Γ, Γ := { λ H /2 Γ div Γ λ H /2 Γ }, H /2 curl Γ, Γ := { λ H /2 Γ curl Γ λ H /2 Γ }, which are dual to each other with respect to the pivot space L 2 tγ. We refer to /2,divΓ,Γ and /2,curlΓ,Γ as the respective norms and denote by, /2,Γ the dual pairing see, e.g., [16] for details. It can be shown that the tangential trace mapping is a continuous linear mapping γ t : Hcurl; Ω H /2 div Γ, Γ, whereas the tangential components trace mapping is a continuous linear mapping π t : Hcurl; Ω H /2 curl Γ, Γ. The previous results imply that the tangential divergence of the tangential trace and the scalar tangential curl of the tangential components trace coincide: or u Hcurl; Ω it holds div Γ u n = curl Γ n u n = n curl u. We define H curl; Ω as the subspace of Hcurl; Ω with vanishing tangential traces on Γ. Given a polyhedral domain Ω R 3 with boundary Γ = Ω such that Γ = Γ D Γ N, Γ D Γ N =, we denote by T H Ω a shape-regular simplicial triangulation of Ω that aligns with Γ D and Γ N. We refer to H Ω as the set of interior faces = T T, T ± T H Ω, and to H Σ as the set of faces located on the boundary Σ Γ, while H Ω := H Ω H Γ is the set of all faces. urther, E H Σ stands for the set of edges on Σ. We denote by h T and h the diameter of an element T T H Ω and a face H Ω, respectively. or two quantities A, B R, we use the notation A B, if there exists a constant C R, independent of the mesh size of the triangulation T H Ω, such that A CB. We refer to Nd 1 Ω; T H Ω := {v H Hcurl; Ω v H T ND 1 T, T T H Ω} as the curl-conforming edge element space, where ND 1 T stands for the lowest order edge element of Nédélec s first family [43], and to Nd 1,Γ D Ω; T H Ω := {v H Nd 1 Ω; T H Ω γ t v H = on Γ D } as its subspace of vanishing tangential trace components on Γ D. or vector fields v H Hcurl; T, we denote by curl,h,ω the meshdependent norm 1/2 v H curl,h,ω := v H 2,T curl v H 2,T. Moreover, for such vector fields we set v ± H := v H T± along = T T H Ω and define { v {v H } := H v H /2, HΩ, v H, H Γ { v [v H ] := H v H, HΩ, H Γ 4

5 as the averages and jumps of v H across the interior faces of the triangulation. or scalar functions v H L 2 Ω, the averages {v H } and jumps [v H ] are defined analogously. The class of Hcurl-elliptic boundary value problems to be approximated by IPDG-H methods is of the form curl µ curl u σ u = f in Ω, 2.2a γ t u = g 1 on Γ D, 2.2b π t µ curl u = g 2 on Γ N. 2.2c We assume that f L 2 Ω, g 1 L 2 Γ D, and g 2 Hcurl Γ N ; Γ N. We further suppose that µ is a symmetric, uniformly positive definite matrix-valued function µ = µx, x Ω, and that σ is a scalar nonnegative function σ = σx, x Ω, that are elementwise constant with respect to a given coarse simplicial triangulation T H Ω of the computational domain. We note that the subsequent analysis also applies to Hcurl-elliptic problems in 2D as given by curl µ curl u σ u = f in Ω, 2.3a t ΓD u = g 1 on Γ D, 2.3b µ curl u = g 2 on Γ N, 2.3c where curl u = u 2 / x 1 u 1 / x 2 for u = u 1, u 2 T, whereas curl u = u/ x 2, u/ x 1 T for a scalar function u. Moreover, t ΓD stands for the tangential unit vector on the Dirichlet part Γ D of the boundary. The data f, g 1 and g 2 have to be chosen accordingly. We will develop the IPDG-H method and perform the a posteriori error analysis only in the 3D case. The necessary modifications for 2D problems are straightforward. 3. Hybridized IPDG Methods. A mixed formulation of 2.2a-2.2c can be derived by introducing p := µ curl u as an additional variable. Setting V := {v Hcurl; Ω γ t u = g 1 on Γ D }, Q := L 2 Ω, 3.1 V := {v Hcurl; Ω γ t u = on Γ D }, it amounts to the computation of u, p V Q with ap, q bu, q = l 1 q for all q Q, 3.2a bv, p cu, v = l 2 v for all v V. 3.2b 5

6 The bilinear forms a, b and c and the functionals l 1 Q, l 2 V are given by ap, q := µ p q dx, 3.3a Ω Ω bu, q := curl u q dx, cu, v := Ω σ u v dx, l 1 q :=, l 2 v := f v dx Ω 3.3b 3.3c 3.3d Γ N g 2 γ t v dτ. 3.3e The operator-theoretic framework involves the operator A : V Q V Q defined, for all u, p V Q and all v, q V Q by Au, pv, q := ap, q bu, q bv, p cu, v. 3.4 Then, the system 3.2a-3.2b can be written in compact form as where lv, q := l 1 q l 2 v for all v, q V Q. Au, p = l, 3.5 Theorem 3.1. Under the assumptions on the data of 2.2a-2.2c, A is a continuous, bijective linear operator. Hence, for any l 1, l 2 Q V, the system 3.2a-3.2b admits a unique solution u, p V Q which continuously depends on the data, namely u, p V Q l 1 Q l 2 V. 3.6 Proof. The mapping properties are straightforward. If g 1, there exists a unique u g1 V such that for all v V cf., e.g., [42] Au g1, v, µ curl v = Ω µ curl u g1 curl v σu g1 v dx =, and hence, we may restrict ourselves to the case of A : V Q V Q. Now, for any u, p V Q we have Au, p3u, 2p µ curl u = A3u, 2p µ curl uu, p = 2µ p 2 L 2 Ω 3σ u 2 L 2 Ω µ curl u 2 L 2 Ω. This implies the inf-sup condition and the remaining degeneracy condition which implies bijectivity. Given a simplicial triangulation T H Ω, DG methods are based on the approximation of the vector field u and p by elementwise polynomials, thus giving rise to the 6

7 finite dimensional function spaces V H := {v H L 2 Ω v H T Π k T, T T H Ω, 3.7a γ t v H = g H,1 on H Γ D }, Q H := {q H L 2 Ω q H T Π k T, T T H Ω}. 3.7b Here and in the sequel, g H,1 Π k, H Γ D is some approximation of g 1 and Π k T, T T H Ω, as well as Π k, H Ω, stand for the sets of vector-valued functions whose components are polynomials of degree at most k N. DG methods amount to the computation of p H, u H Q H V H with a H p H, q H b H u H, q H d H û H, q H = l 1 H q H for all q H Q H, b H v H, p H d H v H, ˆp H c H u H, v H = l 2 H v H for all v H V H. 3.8a 3.8b Here and throughout, û H, ˆp H are appropriate numerical flux functions and the meshdependent bilinear forms a H, b H, c H, and d H are defined by means of a H p H, q H := µp H q H dx, 3.9a The functionals l 1 H b H u H, q H := c H u H, v H := d H u H, q H := T T T H Ω and l2 H are given by curl u H q H dx, σu H v H dx, 3.9b 3.9c γ t u H, π t q H. 3.9d l 1 H q H :=, l 2 H v H := T f v H dx H Γ N g 2 γ t v H dτ. 3.1a 3.1b In case of symmetric Interior Penalty Discontinuous Galerkin IPDG methods, the numerical fluxes read γ t û H := π t ˆp H := { {γt u H }, H Ω, H Γ { {πt µ curl u H } α h [γ tu H ], H Ω, H Γ 3.11a 3.11b with a suitable penalty parameter α >. The choice q H := µ curl v H in 3.8a and 3.11a,3.11b allow the elimination of p H from 3.8a,3.8b. This results in the following standard form of the symmetric IPDG method: ind u H V H such that a IP u H, v H = l IP v H for all v H V H

8 Here and in the sequel, the bilinear form a IP and the functional l IP read a IP u H, v H := µ curl u H curl v H σu H v H dx H Ω α H Ω l IP v H := T {π t µ curl u H } [γ t v H ] {γ t u H } [π t µ curl v H ] h T [γ t u H ] [γ t v H ]dτ, f v H dx H Γ N g 2 γ t v H dτ. 3.13a dτ 3.13b The idea of hybridization is to enforce the continuity of the tangential component traces of p H across the interior edges of the triangulation by a piecewise polynomial Lagrange multiplier which is an approximation of the tangential traces of u. or this purpose, we introduce the multiplier space M H := {µ H L 2 H Ω µ H Π k, H Ω} Choosing a numerical flux function ˆp H, not necessarily the same as in 3.11b, the IPDG-H method is to find p H, u H, λ H Q H V H M H with a H p H, q h b H u H, q H d H λ H, q H = l 1 H q H for all q H Q H, 3.15a b H v H, p H d H v H, ˆp H c H u H, v H = l 2 H v H for all v H V H, 3.15b d H µ H, ˆp H = for all µ H M H. 3.15c In IPDG-H methods, the penalty parameter α is typically chosen elementwise, i.e., α T = α T, T T H Ω, so that on H Ω with = T T, T ± T H Ω, we have to distinguish between α := α T and α := α T. The advantage of hybridized methods is that the primal and dual variables u H and p H can be eliminated from 3.15a-3.15c which results in a global variational problem for the Lagrange multiplier λ H M H of the form a S H λ H, µ H = l S H µ H for all µ H M H Once λ H M H has been computed, the primal and dual variables can be computed by the solution of low-dimensional, local problems. To this end, following the unified framework from [28], we set { { uh on T \ Γ λ H = D, ḡ on T Γ H,1 = D { ḡ H,2 = on T \ Γ N g H,2 on T Γ N on T \ Γ D g H,1 on T Γ D, with an approximation g H,2 Π k, H Γ N, of g 2. We define S p f, S u f Π k T 2, S p λ H, S u λ H Π k T 2, S p g H,1, S u g H,1 Π k T 2, S p g H,2, S u g H,2 Π k T 2 8

9 as the solutions of the local problems µ S p f curl S u f = in T, 3.17a curl S p f σ S u f = f in T, γ t S u f = on T, µ S p λ H curl S u λ H = in T, 3.17b curl S p λ H σ S u λ H = in T, γ t S u λ H = λ H on T, µ S p ḡ H,1 curl S u ḡ H,1 = in T, 3.17c curl S p ḡ H,1 σ S u ḡ H,1 = in T, γ t S u ḡ H,1 = ḡ H,1 on T, µ S p ḡ H,2 curl S u ḡ H,2 = in T, 3.17d curl S p ḡ H,2 σ S u ḡ H,2 = in T, π t S u ḡ H,2 = ḡ H,2 on T. The numerical flux π t ˆp H is given by means of local numerical fluxes π t ˆp H = Ŝpf Ŝpλ H Ŝpḡ H,1 Ŝpḡ H, In particular, for the IPDG-H method 3.15a-3.15c we choose Ŝ p f = { πt µ curl S u f α T h γ ts u f on H Ω Γ D, π t µ curl S u f α T h π tµ curl S u f on H Γ N, Ŝ p λ H = Ŝ p ḡ H,i = π t µ curl S u λ H α T h γ ts u λ H λ H on H Ω Γ D, π t µ curl S u λ H α T h π tµ curl S u λ H λ H on H Γ N, π t µ curl S u ḡ H,i α T h γ ts u ḡ H,i ḡ H,i on H Ω Γ D, π t µ curl S u ḡ H,i α T h π tµ curl S u ḡ H,i ḡ H,i on H Γ N. 3.19a 3.19b 3.19c or sufficiently large α T, T T H Ω, both the local problems 3.17a-3.17d and the global variational problem 3.16 have unique solutions which can be shown along the same lines of proof as in [28] for standard second order elliptic boundary value problems. If λ H M H solves 3.16, then defines the solution of 3.15a-3.15c. p H = S p f S p λ H S p ḡ H,1 S p ḡ H,2, 3.2 u H = S u f S u λ H S u ḡ H,1 S u ḡ H,2 Theorem 3.2. Assume that the numerical flux ˆp H is given by 3.18 and that p H, u H, λ H is the solution of 3.15a-3.15c. Then, the numerical flux ˆp H and the 9

10 multiplier λ H satisfy ᾱ α π t µ curl u H α π t µ curl u H π t ˆp H := α α h [γ tu H ] on H Ω, on H Γ D, on H Γ N, ᾱ α γ t u H α γ t u H λ H = h [π t µ curl u H ] on H Ω, α T h π t µ curl u H on H Γ D, α T h π t µ curl u H on H Γ N, 3.21a 3.21b where ᾱ := α α on = T T for T ± T H Ω. Proof. Let H Ω. If we use 3.19a-3.19c and 3.2 in 3.18, we obtain π t ˆp H = π t µ curl u H α T h Hence, observing 3.17b, it follows that [π t ˆp H ] = [π t µ curl u H ] α h γ tu H γ t u H γ t S u λ H on. α h γ tu H 3.22 α α h λ H. The specification 3.14 of the multiplier space M H and equation 3.15c imply [π t ˆp H ] =. This results in 3.21b due to On the other hand, π t ˆp ± H = π tp ± H α ±h γ t u ± H λ H We deduce 3.21a by inserting 3.21b into The proof of 3.21a,3.21b for H Γ D and H Γ N follows from similar arguments. The representation 3.21b of the Lagrange multiplier λ H shows that it provides an approximation of the tangential trace on the interfaces H Ω which reminds of mortar methods for Hcurl-elliptic problems cf., e.g., [2, 51]. Indeed, the IPDG- H method 3.15a-3.15c can be equivalently formulated as a mortar method. To see this, choose q H = µ curl u H in 3.15a and the numerical flux ˆp H in 3.15a according to 3.21b. Then, by elimination of p H, λ H := λ H ᾱ α γ t u H α γ t u H satisfies ã H u H, v H b H λ H, v H = l 2 H v H for all v H V H, 3.24 bh µ H, u H d H λ H, µ H = for all µ H M H. 1

11 Here and throughout the following, the bilinear forms ã H, b H and d H read ã H u H, v H := T Ω T Ω T Ω T Γ bh λ H, v H := d H λ H, µ H := T µ curl u H curl v H σu H v H dx ᾱ α T π t µ curl u H γ t v H dτ ᾱ α T α T h γ tu H γ t v H dτ ᾱ α T γ t u H π tµ curl v H dτ α T h π t µ curl u H π t µ curl v H dτ, H Ω H Ω λ H [π t µ curl v H ] dτ, ᾱ h λ H µ H dτ. The variational system 3.24 represents a symmetric saddle point problem which can be solved as in the standard mortar approach. Denoting by ÃH, B H, D H the matrices and by b H the vector associated with the bilinear forms and the right-hand side in the first equation of 3.24, the algebraic form of the saddle point problem is ÃH BH uh B T H D H λ H = bh Static condensation of u H results in the equivalent Schur complement system DH B T HÃ H B H λh = B T HÃ H b H A posteriori error analysis. The residual a posteriori error estimator for the symmetric IPDG-H method 3.15a-3.15c is given by η := H Γ N They consist of the element residuals ηt,1 2 ηt,2 2 ηt,3 2 η 2,3 η 2,4 1/2. H Ω η,1 2 η, η T,1 := µp H curl u H,T for all T T H Ω, 4.2a η T,2 := h T f curl p H σu H,T for all T T H Ω, 4.2b η T,3 := h T f σu H,T for all T T H Ω, 4.2c 11

12 and the face residuals η,1 := h 1/2 [π t p H ], for all H Ω, 4.3a η,2 := h 1/2 n [f σu H ], for all H Ω, 4.3b η,3 := h 1/2 g 2 π t p H, for all H Γ N, 4.3c η,4 := h 1/2 n f σu H, for all H Γ N. 4.3d The nonconformity of the symmetric IPDG-H method results in some consistency error ξ := 1/2 min u H ṽ H 2,T curl u H ṽ H 2,T 4.4 ṽ H V with the unique minimizer ũ H V of 4.4 and ξ 2 = u H ũ H 2,Ω curlu H ũ H 2,Ω. Theorem 4.1. Let p, u Q V and p H, u H, λ H Q H V H M H be the solutions of 3.5 and 3.15a-3.15c, let η and ξ be the residual error estimator and the consistency error of 4.1 and 4.4. Then, u, p u H, p H := p p H 2 Q u u H 2 curl,h,ω 1/2 η ξ. 4.5 We will provide the proof of Theorem 4.1 by a series of lemmas. We assume p H, ũ H Q V to be some approximation of the solution p, u Q V of the mixed problem 3.5 obtained by means of the solution p H, u H, λ H of the symmetric IPDG-H method 3.15a-3.15c. It is an immediate consequence of Theorem 3.1 that the error p p H, u ũ H satisfies p p H, u ũ H Q V Res 1 Q Res 2 V 4.6 with residuals Res 1 Q and Res 2 V, Res 1 q := l 1 q a p H, q bũ H, q for q Q, 4.7a Res 2 v := l 2 v bv, p H cũ H, v for v V. 4.7b Lemma 4.1. Let p H, u H, λ H Q H V H M H be the solution of 3.15a- 3.15c with the numerical flux ˆp H from The choice of p H = p H and of ũ H V as the unique minimizer of 4.4 imply Res 1 Q η 2 T,1 1/2 ξ. 4.8 Proof. With the L 2 -projection q H = P QH q of q Q onto Q H, we have q H,Ω q,ω and Res 1 q = Res 1 q P QH q Res 1 P QH q

13 In view of 4.7a and 3.3a,3.3b,3.3d, it follows that Res 1 q P QH q = T curl u H µ p H q P QH q dx T curl ũ H u H q P QH q dx. Straightforward estimation and q P QH q,ω q Q yield Res 1 q P QH q curl ũ H u H 2,T η 2 T,1 1/2 ξ q Q. curl u H µ p H 2,T 1/ /2 q P Q q 2,T Similar arguments for the last term in 4.9 and P QH q,ω q,ω reveal Res 1 P QH q 1/2 curl u H µ p H 2,T 1/2 PQH q,ω 4.11 curlû H u H 2,T 1/2 PQH q,ω η 2 T,1 1/2 ξ q Q. The combination of 4.1 and 4.11 concludes the proof. Lemma 4.2. or p H = p H and some approximation ũ H V H let the residual Res 2 of 4.7b satisfy Nd 1 ;Γ D Ω; T H Ω Ker Res Then, it holds Res 2 V ηt,2 2 ηt,3 2 η,1 2 η,2 2 H Ω 1/2 η,3 2 η,4 2 ξ. H Γ N 4.13 Proof. Given any v V, Theorem 1 in [48] shows that there exist v H Nd 1 ;Γ D Ω; T H Ω, ϕ H 1,Γ D Ω, and z H 1,Γ D Ω 3 such that v v H = ϕ z

14 and with appropriate patches ω T and ω ϕ,t h T v curl;ωt for T T H Ω, 4.15a ϕ,t v curl;ωt for T T H Ω, 4.15b h /2 ϕ, v curl;ω for H Ω Γ N, 4.15c z,t h T v curl;ω for T T H Ω, 4.15d h /2 γ t z, v,ω for H Ω Γ N. 4.15e It is a consequence of 4.12 and 4.14 that Res 2 v = Res 2 v v h = Res 2 ϕ Res 2 z The first term on the right-hand side in 4.16 reads Res 2 ϕ = T T σu H ϕ dx f ϕ dx An application of Green s formula gives T T f σu H ϕ dx = H Ω n [f σu H ] ϕ dτ H Γ N T g 2 γ t ϕ dτ 4.17 σũ H u H ϕ dx. H Γ N f σu H ϕ dx 4.18 n f σu H ϕ dτ. Since γ t ϕ = curl ϕ on H Γ N, a further application of Stokes formula yields H Γ N = H Γ N g 2 γ t ϕ dτ = E E H Γ N E curl g 2 ϕ dτ t E g 2 ϕ ds. H Γ N E E H Γ N E g 2 curl ϕ dτ 4.19 t E {g 2 } [ϕ] t E [g 2 ]{ϕ} ds Since g 2 Hcurl Γ N ; Γ N, we have curl g 2 =, H Γ N. Since ϕ H,Γ 1 D Ω, we have ϕ = on E E H Γ N. Consequently, 4.19 yields H Γ N g 2 γ t ϕ dτ =

15 With 4.18 and 4.2, 4.17 leads to Res 2 ϕ H Ω H Γ N h 2 T f σu H 2,T h n [f σu H ] 2, h n f σu H 2, ũ H u H 2,T This and 4.15a-4.15c imply Res 2 ϕ 1/2 1/2 η 2 T,3 H Γ N 1/2 H Ω 1/2 H Γ N ϕ 2,T 1/2. 1/2 η 2,4 h ϕ 2, H Ω 1/2 h 2 T ϕ 2,T 1/2 1/2 h ϕ 2, 1/2 ξ v curl;ω. η 2,2 1/ On the other hand, the second term on the right-hand side of 4.16 reads Res 2 z = f z dx g 2 γ t z dτ 4.22 T T T H Γ N p H curl z dx σũ H u H z dx. T σu H z dx Since [γ t z] = on H Ω, an application of Stokes theorem gives p H curl z dx = curl p H z dx T H Ω This and 4.22 lead to Res 2 z = H Ω T [π t p H ] γ t z dτ T H Γ N f curl p H σu H z dx [π t p H ] γ t z dτ T σũ H u H z dx. 15 H Γ N π t p H γ t z dτ. g 2 π t p H γ t z dτ

16 Hence, Res 2 z is bounded from above by Res 2 z H Ω H Γ N h 2 T f curl p H σu H 2,T h [π t p H ] 2, h g 2 π t p H 2, h 2 T ũ H u H 2,T This and 4.15d,4.15e result in Res 2 z 1/2 H Ω 1/2 1/2 1/2 η 2 T,2 H Γ N h γ tz 2, H Γ N 1/2 η 2,3 1/2 h 2 T z 2,T 1/2 h γ t z 2, h 2 T z 2,T 1/2. H Ω 1/2 ξ v curl;ω. The combination of 4.21 and 4.23 plus 4.16 concludes the proof. Lemma 4.3. or v H Nd p,γ D Ω; T H Ω it holds 1/2 η 2,1 1/ Res 2 v H = c H u H ũ H, v H Proof. We have Res 2 v H = l 2 H v H b H v H, p H c H ũ H, v H 4.25 = l 2 H v H b H v H, p H c H u H, v H c H u H ũ H, v H. Since v H Nd 1,Γ D Ω; T H Ω V H is an admissible test function in 3.15b, it follows that b H v H, p H c H u H, v H = l 2 H v H d H v H, ˆp H Since 3.21a, the last term vanishes The combination of concludes the proof. d H v H, ˆp H = Proof of Theorem 4.1. In view of Lemma 4.2 we define It follows that Res 2 := Res 2 c H u H ũ H, Res 2 V Res 2 V ξ

17 In view of 4.24, we have Nd 1,Γ D Ω; T H Ω Ker Res 2. Hence, Lemma 4.3 with Res 2 replaced by Res 2 yields Res 2 V η 2 T,2 η 2 T,3 H Γ N H Ω η 2,3 η 2,4 1/2. η,1 2 η, As in the case of the symmetric IPDG method cf., e.g., [2, 37], the consistency error admits the upper bound ξ H Ω η 2,5 1/2, η,5 := h /2 [γ t u H ], The combination of 4.8, and the triangle inequality u, p u H, p H u, p ũ H, p H u H ũ H curl,h,ω conclude the proof. 5. Numerical Results The Adaptive Cycle. The adaptive IPDG-H method is realized within an adaptive cycle with the basic steps SOLVE, ESTIMATE, MARK, and REINE. SOLVE stands for the numerical solution of the hybridized IPDG scheme with the mortar approach of section 3 implemented in the nudg code from [33] for the solution of The step ESTIMATE is devoted to the computation of the element residuals η T,i, 1 i 3, and the face residuals η,i, 1 i 4 cf. 4.2a-4.2c and 4.3a-4.3c as the basic constituents of the residual error estimator η cf Moreover, the consistency error ξ cf. 4.4 is estimated by the additional face residuals η,5 according to The following step MARK deals with the marking of elements and faces for refinement by a bulk criterion, also known as Dörfler marking [29]. In particular, given a universal constant < θ < 1, sets M T T H Ω {1, 2, 3} and M H Ω {1, 2, 3, 4, 5} of almost minimal cardinality are determined such that θ η 2 ηt,i 2 η,i T,i M T,i M The bulk criterion 5.1 is implemented by a greedy algorithm. or sufficiently small θ, it is expected that the bulk criterion may yield asymptotic optimal complexity cf., e.g., [12] in case of adaptive IPDG methods for standard second order elliptic boundary value problems. The final step REINE takes care of the practical realization of the adaptive refinement. Elements T T H Ω and faces H Ω such that T, i M T for some 1 i 3 and, i M for some 1 i 5 are refined by bisection Numerical Examples. or the illustration of the performance of the residual a posteriori error estimator we consider two examples of Hcurl-elliptic boundary value problems in 2D from 2.3a-2.3c. Both examples feature solutions in Hcurl; Ω with components in H s Ω for some < s < 1. The first one has an an 17

18 irrotational solution on an L-shaped domain with a singularity at the reentrant corner and the second one exhibits a solenoidal solution on a circle with a cut out wedge having a singularity at the origin. or both problems, the penalty parameters in the IPDG-H method have been chosen according to α ± := κk 1 2 /2 with κ = 1. Example 1: We consider the L-shaped domain Ω :=, 1 2 \[, 1] [, ] with Dirichlet boundary Γ D :=, 1, 1, Neumann boundary Γ N := Γ \ Γ D and data µ = σ = 1. The right-hand sides f, g 1, g 2 in 2.3a-2.3c are chosen such that u = gradr 2/3 sin 2 3 ϕ is the exact solution in polar coordinates. The solution is in Hcurl; Ω H 2/3 ε Ω for any ε > and exhibits a singularity at the reentrant corner. Mesh Level Mesh Level 8 Mesh Level ig Ex. 1: The initial mesh left and the meshes after 8 middle and 18 right adaptive refinement steps k = 4 and Θ =.1..5 k=1.5 k = 4 log u, p u H, p H θ=.1 θ=.3 3 θ=.5 θ =.7 uniform log DO log u, p u H, p H θ=.1 θ=.3 3 θ=.5 θ =.7 uniform log DO ig Ex. 1: The error u, p u H, p H versus the number of degrees of freedom on a logarithmic scale for various θ, k = 1 left and k = 4 right. igure 5.1 shows the initial mesh left and the meshes obtained after 8 middle and 18 right refinement steps of the adaptive algorithm in case k = 4 and θ =.1. We observe a pronounced refinement in a vicinity of the reentrant corner. igure 5.2 displays the global discretization error u, p u H, p H cf. 4.5 as a function of the number of degrees of freedom DO on a logarithmic scale for both uniform refinement and adaptive refinement in case k = 1 left and k = 4 right. The 18

19 results of the adaptive refinement are shown for various values of the constant θ in the bulk criterion 5.1. Both for k = 1 and k = 4 the benefits of adaptive versus uniform refinement can be clearly seen. In case k = 1, we observe a dependence of the convergence rate on the parameter θ which is much less pronounced in case k = 4. According to the theory for IPDG methods applied to standard second order elliptic boundary value problems cf. [12] and the numerical results in [34], we see that optimality is asymptotically achieved for small θ. Example 2: The domain Ω is the unit circle with a cut out wedge see igure 5.3. We assume µ = σ = 1 and Neumann boundary conditions on Γ = Ω. The data f and g 2 are chosen such that u = curlr 4/7 sin 4 7ϕ is the exact solution in polar coordinates. The solution is in Hcurl; Ω H 4/7 ε Ω for any ε > and exhibits a singularity at the origin. We use isoparametric elements for a proper resolution of the curved part of the boundary. Mesh Level Mesh Level 6 Mesh Level ig Ex. 2: The initial mesh left and the meshes after 6 middle and 14 right adaptive refinement steps k = 4 and Θ =.1..5 K=1 k = 4 log u, p u H, p H.5 θ=.1 2 θ=.3 θ=.5 θ=.7 uniform log DO log u, p u H, p H θ=.1 θ=.3 θ=.5 θ =.7 uniform log DO ig Ex. 2: The error u, p u H, p H versus the number of degrees of freedom on a logarithmic scale for various θ, k = 1 left and k = 4 right. As for the previous example, igures 5.3 and 5.4 display the history of the refinement process. We basically observe a similar behavior with asymptotically optimal convergence for small θ. However, in the pre-asymptotic regime, the decrease of the discretization error is less pronounced. The reason is that there are two main sources for the error: the singularity at the origin and the resolution of the curved boundary. Since the error is dominated by the singularity, the greedy algorithm realizing the 19

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