An hp-adaptive Mixed Discontinuous Galerkin FEM for Nearly Incompressible Linear Elasticity

Transcription

1 An hp-adaptive Mixed Discontinuous Galerkin FEM for Nearly Incompressible Linear Elasticity Paul Houston 1 Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK ( Paul.Houston@mcs.le.ac.uk) Dominik Schötzau Mathematics Department, University of British Columbia, Vancouver, BC V6T 1Z, Canada ( schoetzau@math.ubc.ca) Thomas P. Wihler School of Mathematics, University of Minnesota, Minneapolis MN, 5555, USA ( wihler@math.umn.edu) Abstract We develop the a posteriori error estimation of mixed hp version discontinuous Galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions. Computable upper and lower bounds on the error measured in terms of a natural (mesh dependent) energy norm are derived. The bounds are explicit in the local mesh sizes and approximation orders, and are independent of the locking parameter. A series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure. Key words: Discontinuous Galerkin methods, a posteriori error estimation, hp-adaptivity, linear elasticity, volume locking. 1 Supported by the EPSRC (Grant GR/R76615). Supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). Supported by the Swiss National Science Foundation, Project PBEZ-101. Preprint submitted to Elsevier Science 6 April 005

2 1 Introduction One of the main challenges in the design of finite element methods for linear elasticity problems is their robustness with respect to nearly incompressible materials. Indeed, it is well-known that the performance of standard loworder elements can significantly deteriorate if the compressibility parameter tends to a certain critical limit; this phenomenon is referred to as volume locking. In order to avoid this effect, several remedies have been proposed within the literature. For example, certain non-conforming and discontinuous Galerkin methods are free of volume locking; see, e.g., [,1,9,0], and the references cited therein. Another class of locking-free methods is obtained in a natural way by the use of mixed finite elements; here, we only mention [5] and the references cited therein. Mixed approaches are based on replacing the divergence of the displacement by an additional variable (Lagrange multiplier) that is discretized separately. The finite element approximation of linear elasticity and, more generally, partial differential equations of elliptic type, poses another major challenge: in polygonal or polyhedral domains arising in applications of engineering interest, the analytical solution of the underlying partial differential equations often exhibit strong singularities in the vicinity of corners or edges of the domain. Typically, these singularities can only be accurately resolved by employing meshes that are locally refined towards vertices, edges and/or faces of the computational domain, where these singularities are located. A more sophisticated, and in particular, more efficient extension of this approach is based on employing the hp-version of the finite element method: this combines locally refined meshes with variable approximation orders. It is well-known that this combination can achieve exponential rates of convergence, even in the presence of corner/edge singularities; for details, we refer the reader to [,11,1,7] and the references cited therein. For the practical realization of hp-discretizations, so-called discontinuous Galerkin (DG, for short) finite element methods provide an ideal computational framework. Indeed, in comparison to standard conforming finite element methods, DG schemes support the design of more general finite element spaces. For example, within a DG approach, non-matching grids containing hanging nodes and non-uniform, even anisotropic, approximation degree distributions can easily be handled. For recent surveys on the design and analysis of DG methods in a range of applications, we refer the reader to [1,6 8], and the references cited therein. In this paper, we propose and study a mixed hp-adaptive DG method for nearly incompressible linear elasticity problems in two dimensional polygonal domains. The method is based on mixed discontinuous elements, whose hp-

3 version stability and a priori convergence properties were previously studied in [5,6] in the context of the Stokes equations; see also [1] for an h-version approach. In the current article, we present an hp-version a posteriori error analysis for this DG method; in particular, we derive computable upper and lower energy norm a posteriori error bounds that are explicit in the local mesh sizes and approximation orders. We emphasize that, since our discretization is based on employing the mixed form of the equations of linear elasticity, we naturally obtain reliability and efficiency constants for the a posteriori bounds that are independent of the compressibility parameter. In particular, this implies that they do not deteriorate in the incompressible limit, which, in the mixed setting, corresponds to the Stokes problem. The a posteriori error analysis presented in this article is a continuation of our work in [15,16], where energy norm a posteriori error estimation of mixed h-version DG approximations for the Stokes equations and hp-version DG methods for Poisson s equation, respectively, was recently developed. Here, the proof of the upper bound on the energy norm of the error is based on rewriting the underlying DG method in a non-consistent manner using the lifting operators from [1,,5], and employing a norm equivalence result for hp-version DG spaces; see [16, Section 5]. This crucial result has been obtained by establishing an approximation property in the spirit of the h-version results in [1,0]. The lower (efficiency) bounds are derived based on employing the techniques developed in [] for conforming hp version finite element methods. As in [], reliability and efficiency of our error bounds cannot be established uniformly with respect to the polynomial degree, since the proof of efficiency relies on employing inverse estimates which are suboptimal in the spectral order. Finally, we note that, for the Stokes problem, the results of this paper have been previously announced in the conference article [17]. The outline of this article is as follows. In Section, we introduce the hp DG method for the numerical approximation of linear elasticity problems in mixed form. In Section, our a posteriori error bounds are presented and discussed; here, both upper and lower energy norm bounds will be considered. The proofs of these results will be presented in Section. In Section 5, we present a series of numerical experiments to illustrate the performance of the proposed error estimators within an automatic hp mesh refinement algorithm. Finally, in Section 6 we summarize the work presented in this paper and draw some conclusions. Throughout this article, we use the following notation: for an interval D R or a bounded Lipschitz domain D R, let L (D) be the Lebesgue space of square integrable functions, endowed with the usual norm 0,D. The standard Sobolev space of functions with integer regularity exponent s > 0 is denoted by H s (D). We write s,d and s,d for the corresponding norms and semi-norms, respectively. As usual, we define H 1 0(D) as the subspace of func-

4 tions in H 1 (D) with zero trace on D. Furthermore, for a function space X(D), let X(D) and X(D) be the spaces of all vector and tensor fields whose components belong to X(D), respectively. Without further specification, these spaces are equipped with the usual product norms (which, for simplicity, are denoted similarly as the norm in X(D)). For vectors v, w R, and matrices σ, τ R, we use the standard notation ( v) ij = j v i, ( σ) i = j=1 j σ ij, and σ : τ = i,j=1 σ ij τ ij. Furthermore, let v w be the matrix whose ij-th component is v i w j. With this notation, we note that the following identity holds v σ w = i,j=1 v i σ ij w j = σ : (v w). Mixed hp-dg Method for Linear Elasticity In this section, we introduce the equations governing linear elasticity considered in this article and present a mixed hp-dg method for their discretization..1 Linear Elasticity Problems On a given polygonal domain R with boundary Γ =, we consider the linear elasticity problem: find a vector field (displacement) u = (u 1, u ) H 1 0() such that u 1 ( u) = f 1 ν in, (1) u = 0 on Γ. () Here, is the divergence operator, ν (0, 1 /) is the Poisson ratio, and f L () is an external force (scaled by (1+ν)/E, where E > 0 is Young s modulus). In order to write the equations (1) () in mixed form, we introduce the additional variable (Lagrange multiplier) We note that p L 0 (), where p = 1 1 ν u. L 0 () = {q L () : } q dx = 0.

5 Hence, (1) () is equivalent to finding (u, p) H0 1() L 0 () such that u + p = f in, u + (1 ν)p = 0 in, () u = 0 on Γ. The standard variational formulation of () is then given by: find (u, p) H0 1() L 0 () such that A(u, v) + B(v, p) = B(u, q) + C(p, q) = 0 f v dx, () for all (v, q) H0 1() L 0 (), where A(u, v) = u : v dx, B(v, q) = q v dx, and C(p, q) = (1 ν) pq dx. More compactly, () can be written as follows: find (u, p) H0() 1 L 0() such that a(u, p; v, q) = f v dx (v, q) H0 1 () L 0 (), (5) with a(u, p; v, q) = A(u, v) + B(v, p) B(u, q) + C(p, q). We note that, for ν (0, 1 /), the form a is coercive on H0() 1 L 0() and hence the solution (u, p) H0 1() L 0 () of () exists and is unique. Remark 1 It can be seen from (1) () that the following stability estimate holds u 1, ν u 0, C f 0,, with a constant C > 0 that is independent of the Poisson ration ν (0, 1 /). This immediately implies that, as ν 1 /, the constraint u 0 naturally arises, which corresponds to (nearly) incompressible materials. It is well-known (see []) that this incompressibility constraint may cause a loss of uniformity (with respect to ν) in the asymptotic convergence regime of finite element methods based on discretizing the primal variables in (1) (). This does not mean that those methods do not converge at all; however, it may happen that the convergence begins to take place at such high numbers of degrees of freedom that, in certain cases, the method is not feasible in practice. This lack of robustness of the FEM with respect to incompressible materials is referred to as volume locking. In this paper, we will present a mixed DG method 5

6 based on () that is able to overcome this effect in a natural manner. For considerations on locking-free DG methods based on employing primal variables, we refer to [1,9,0]. Remark We note that the case when ν = 1 / in () corresponds to the standard Stokes problem for incompressible fluid flow. Due to the continuous inf-sup condition inf sup 0 q L 0 () 0 v H0 1() q v dx K > 0, (6) v 0, q 0, where K is the inf-sup constant, depending only on, the mixed variational formulation () is still well-posed and has a unique solution (u, p) H 1 0 () L 0() for ν = 1 /; see [5,10] for details. The inf-sup condition (6) also ensures the robustness of the mixed problem () with respect to ν.. Meshes and Trace Operators In this section, we introduce the notation required for the definition of the hp-dg method. Meshes: Throughout, we assume that the domain can be subdivided into conforming, shape-regular affine meshes T h = {K} K Th consisting of triangles and/or parallelograms. For each K T h, we denote by n K the unit outward normal vector to the boundary K, and by h K the elemental diameter. Furthermore, we assign to each element K T h an approximation order k K 1. The local quantities h K and k K are stored in the vectors h = {h K } K Th and k = {k K } K Th, respectively. We denote by E I (T h ) the set of all interior edges of T h, by E B (T h ) the set of all boundary edges, and define E(T h ) = E I (T h ) E B (T h ). We assume that the local mesh sizes and approximation degrees are of bounded variation, i.e. there is a constant ϱ 1 such that ϱh K h K ϱ 1 h K, ϱk K k K ϱ 1 k K, (7) whenever K and K share a common edge. 6

7 Averages and Jumps: Next, we define average and jump operators. To this end, let K + and K be two adjacent elements of T h ; furthermore, let x be an arbitrary point of the interior edge = K + K E I (T h ). Moreover, for scalar-, vector-, and matrix-valued functions q, v, and τ, respectively, that are smooth inside each element K ±, we denote by (q ±, v ±, τ ± ) the traces of (q, v, τ) on taken from within the interior of K ±, respectively. Then, we define the following averages at x : {q } = 1 /(q + + q ), {v } = 1 /(v + + v ), {τ } = 1 /(τ + + τ ). Similarly, the jumps at x are given by [q ] = q + n K + + q n K, [v ] = v + n K + + v n K, [v ] = v + n K + + v n K, [τ ] = τ + n K + + τ n K. On boundary edges E B (T h ), we set {q } = q, {v } = v, {τ } = τ, as well as [q ] = qn, [v ] = v n, [v ] = v n, and [τ ] = τn. Here, n denotes the unit outward normal vector to Γ.. Mixed hp-discontinuous Galerkin Discretization We now define an hp-dg method for the approximation of the linear elasticity problem () in mixed form and discuss its well-posedness. Given a mesh T h on and a degree vector k = {k K }, k K 1, we approximate () by finite element functions (u h, p h ) V h Q h, where V h = { v L () : v K S k K (K), K T h }, Q h = { q L 0 () : q K S k K 1 (K), K T h }. (8) Here, for k 0, S k (K) denotes the space P k (K) of polynomials of total degree at most k on K, if K is a triangle, and the space Q k (K) of polynomials of degree at most k in each variable on K, if K is a quadrilateral. We consider the mixed hp-dg method: find (u h, p h ) V h Q h such that A h (u h, v) + B h (v, p h ) = B h (u h, q) + C h (p h, q) = 0 f v dx, for all (v, q) V h Q h. The forms A h, B h and C h are given, respectively, by (9) 7

8 A h (u, v) = h u : h v dx E(T h ) + c [u] : [v ] ds, B h (v, q) = E(T h ) C h (p, q) = (1 ν) q h v dx + pq dx. E(T h ) ( { h v } : [u] + { h u } : [v ] ) ds {q }[v ] ds, (10) Here, h and h denote the discrete gradient and divergence operators, respectively, defined elementwise. The function c L (E(T h )) is the so-called discontinuity stabilization function that is chosen as follows. Define the functions h L (E(T h )) and k L (E(T h )) by Then we set min{h K, h K }, x = K K E I (T h ), h(x) := h K, x = K Γ E B (T h ), max{k K, k K }, x = K K E I (T h ), k(x) := k K, x = K Γ E B (T h ). c = γh 1 k, (11) with a parameter γ > 0 that is independent of h and k. As in (5), the discrete formulation (9) is equivalent to finding (u h, p h ) V h Q h such that a h (u h, p h ; v, q) = f v dx (1) for all (v, q) V h Q h, where a h (u, p; v, q) = A h (u, v) + B h (v, p) B h (u, q) + C h (p, q). (1) Henceforth, we assume that γ is chosen sufficiently large; under this condition, the problem (9), cf., also, (1), is uniquely solvable. Proposition.1 Let ν (0, 1 /). Then, there is a constant γ min > 0 such that, for all γ γ min, the mixed hp-dg method (9), cf., also, (1), possesses a unique solution (u h, p h ) V h Q h. Proof. This readily follows from the coercivity of the forms A h (for γ sufficiently large) and C h. Remark In the limiting case ν = 1 /, the discrete problem (9) is also uniquely solvable. This follows from the discrete inf-sup conditions in [5] 8

9 and [1] that show that the form B h is stable uniformly in h. While on quadrilateral meshes the discrete inf-sup constant has been shown to decay at the most like 1/ max{k K } in the approximation orders k K, a similar hp-version result on triangular meshes still needs to be established. Remark The form A h corresponds to the so-called symmetric interior penalty discretization of the Laplace operator; for a detailed review of a wide class of DG methods for diffusion problems and the Stokes system, we refer to the articles [1,5], respectively. Remark 5 In the case of inhomogeneous Dirichlet boundary conditions, u = g on Γ, with a datum g satisfying the compatibility condition Γ g n ds = 0, the functional on the right-hand side of the first equation in (9) must be replaced by F h (v) = f v dx E B (T h ) (g n) : h v ds + E B (T h ) c g v ds. Additionally, the right-hand side of the second equation in (9) is set equal to G h (q) = E B (T h ) q g n ds. Locking-Free hp-a Posteriori Error Estimation In this section we present and discuss a locking-free reliable and efficient hp-a posteriori estimator for the error of the hp-dg method (9), measured in terms of the energy norm DG given by (v, q) DG = h v 0, + E(T h ) c [v ] ds + ( ν) q 0,. The proofs of the corresponding a posteriori error bounds, Theorems.1 and., will be given in Section..1 Weighted Error Indicators In order to study the dependence on the polynomial degree in the a posteriori error analysis for conforming hp-finite element methods, the use of weighted local error indicators η K was recently proposed in []. Following that approach, we derive a family of weighted estimators η α;k, K T h, α [0, 1], for the mixed hp-dg method proposed in this paper. We note that, as for 9

10 conforming hp-methods, simultaneous reliability and efficiency uniformly in the polynomial degrees cannot be achieved for any fixed α [0, 1]. On a reference element K, we define the weight function Φ K(x) = dist(x, K). 1 For an arbitrary element K T h, we set Φ K = c K Φ K FK, where F K : K K is the elemental transformation and c K is a scaling factor chosen such that K Φ K dx = meas(k). Similarly, on the reference interval Î = ( 1, 1), we define the weight function ΦÎ(x) = 1 x. For an interior edge, the weight Φ is then defined by Φ = c ΦÎ F 1, where F is the affine transformation that maps ( 1, 1) onto and c is chosen such that Φ ds = length(). As in [], for each element K T h and α [0, 1], we introduce the weighted local error indicator η α;k, which is given by Here, ( ) ηα;k = kk α η α;rk,1 + ηα;e K + η RK, + ηj K. (1) ηα;r K,1 = k K h K (Π hf + u h p h )Φ α / K 0,K, (15) ηr K, = u h + (1 ν)p h 0,K, (16) are residual terms corresponding to the first and second equations in (), respectively, and Π h f denotes the elementwise L -projection of f onto the space S k K 1 (K), K T h. Furthermore, η α;e K = 1 K\Γ is a weighted edge residual. Finally, the term η J K k 1/ h 1 / ([p h ] [ h u h ])Φ α / 0,, (17) = K c 1 / [u h ] 0, (18) measures the discontinuities (jumps) of the approximate displacement u h over element edges.. Reliability and Efficiency The aim of this section is to present the a posteriori error estimates for the mixed hp-dg method (9). The first result is concerned with establishing the reliability of the error indicators η α;k. Theorem.1 Let α [0, 1]. Furthermore, let (u, p) H0 1() L 0 () be the solution of the linear elasticity problem () and (u h, p h ) V h Q h its mixed 10

11 hp-dg approximation obtained by (9). Then, the following a posteriori error bound holds (u u h, p p h ) DG C EST η α;k K T h 1/ + C OSC OSC(f, K) K T h where the elemental error indicators η α;k are given by (1) and OSC(f, K) = h Kk K f Π h f 0,K 1/, is a data oscillation term. The constants C EST, C OSC of ν, h, k, γ, and α. > 0 are independent Remark 6 We stress that the constants C EST and C OSC in Theorem.1 do not depend on γ, provided that γ is chosen sufficiently large. In addition, we emphasize that these constants are completely independent of the Poisson ratio ν; in particular, they do not deteriorate as ν 1 /, thereby clearly indicating the robustness of the hp-dg method with respect to volume locking. In fact, our a posteriori analysis also applies to the case when ν = 1 /, which corresponds to the Stokes system for incompressible fluid flow (see also [17]); cf. Remark. Remark 7 In order to incorporate inhomogeneous boundary conditions u = g on Γ, the error indicators η α;k are simply adjusted by modifying the jump indicators c 1 / [u h ] 0, K on K Γ; cf. [16]. Next, we discuss the efficiency of the error indicators η α;k. Theorem. Let (u, p) H 1 0() L 0() be the analytical solution of () and (u h, p h ) V h Q h its mixed hp-dg approximation obtained by (9). Writing η α;k to denote the weighted error indicators defined in (1), we have the following bounds: (a) Let α [0, 1]. For any ε > 0, there is a constant C ε, independent of ν, h, k, γ, α, and K T h, such that η α;r K,1 C ε ( k (1 α) K ( (u uh ) 0,K + p p h 0,K) ) + k max(1+ε α,0) K kk h K f Π hf 0,K. (b) There is a constant C, independent of ν, h, k, γ, and K T h, such that η R K, C ( (u u h ) 0,K + p p h 0,K). (c) Let α [0, 1]. For any ε > 0, there is a constant C ε, independent of ν, 11

12 h, k, γ, α, and K T h, such that η α;e K C ε k max(1+ε α,0) ( ) K (k K h (u u h ) 0,δK + p p h 0,δK + k ε K k K h K f Π hf 0,δ K ), where δ K = {K T h : K and K share a common edge }. (d) The jump term satisfies the following equality η J K = K c 1 / [u u h ] 0,. Remark 8 From Theorem.1, we see that the best upper (reliable) a posteriori error bounds are obtained for α = 0, while the best efficiency bounds arise when α = 1. As for conforming hp-fem, cf. [], for example, simultaneous reliability and efficiency cannot be achieved for any fixed α [0, 1]. This is perhaps not surprising, since in the special case when singularities present in the underlying analytical solution arise on inter-element boundaries, the energy norm of the error may decay to zero at a superconvergent rate as the polynomial degree is increased; see, for example, [7], where hp approximation results in weighted Sobolev norms are developed. Proofs.1 Proof of Theorem.1 The proof of Theorem.1 will be outlined in the following sections; it is based on employing the recent hp-decomposition result for discontinuous spaces from [16], together with a non-consistent reformulation of the hp-dg method (9) using lifting operators (see, e.g., [1]). A similar approach has also been developed for the a posteriori error analysis of the h-version of the interior penalty DG method for the Stokes problem in the article [15]..1.1 Decomposition of hp-dg Spaces We split the DG space V h from (8) into a conforming part V c h := V h H 1 0(), 1

13 and a purely nonconforming part V h. Here, V h is defined as the orthogonal complement of V c h in V h with respect to the norm v 1,h = h v 0, + E(T h ) h 1 k [v ] ds. With this notation, the following norm equivalence result holds; see [16], for details. Proposition.1 The expression v E(T h ) k h 1 [v ] ds is a norm on V h. This norm is equivalent to the norm 1,h and there is a constant C P > 0, independent of h and k, such that for all v V h. v 1,h C P E(T h ) k h 1 [v ] ds 1/ 1/ C P v 1,h.1. Lifting Operators and Extended Forms In this section, we define a suitable extension of the forms A h and B h from (10) to the continuous level using the lifting operators introduced in [1]; see also [,5]. To this end, we define the space Furthermore, by using the auxiliary space V (h) = H 1 0() + V h. (19) Σ h = {τ L () : τ K S k K (K), K T h }, we introduce the lifting operator L : V (h) Σ h by L(v) : τ dx = E(T h ) In addition, we define M : V (h) Q h by M(v)q dx = E(T h ) [v ] : {τ } ds τ Σ h. [v ] {q } ds q Q h. 1

14 The above lifting operators are stable; see [,5] for details. More precisely, for any v V (h), the following bounds hold L(v) 0, C L M(v) 0, C L E(T h ) E(T h ) k h 1 [v ] ds, k h 1 [v ] ds, where the constant C L > 0 is independent of h and k. We are now in a position to introduce the following extended forms: Ã h (u, v) = ( h u : h v dx L(u) : h v + L(v) : h u ) dx c [u] : [v ] ds, + E(T h ) B h (v, q) = q h v dx + C h (p, q) = (1 ν) pq dx. Moreover, we define M(v)q dx, ã h (u, p; v, q) = Ãh(u, v) + B h (v, p) B h (u, q) + C h (p, q). (0) We emphasize that, in contrast to the form a h, defined in (1), the form ã h is well-defined on (V (h) L ()). Furthermore, we observe that ã h a h on (V h Q h ), ã h a on (H 1 0() L ()). (1).1. Stability Results In this section, we present some basic stability properties of the form ã h ; firstly, we have the following continuity result. Lemma.1 For any (u, p) V h L () and (v, q) H 1 0 () L (), we have ã h (u, p; v, q) C C (u, p) DG (v, q) DG, where C C = max(5, C L γ 1 ) 1 /, and C L is the constant from (0). Proof. We first note that for v H 1 0 (), we have L(v) [v ] 0 and M(v) 0, from the stability of the lifting operators (0). Employing these results, together with the Cauchy-Schwarz inequality, and the inequality v 0, v 0, v H 1 (), 1

15 we deduce that ã h (u, p; v, q) Ãh(u, v) + B h (v, p) + B h (u, q) + C h (p, q) h u 0, v 0, + L(u) 0, v 0, + p 0, v 0, + 1 q 0, h u 0, + M(u) 0, 1 q 0, + (1 ν) p 0, q 0, (5 h u 0, + C Lγ 1 E(T h ) ( v 0, + ( ν) q 0, c 1 / [u] 0, + ( ν) p 0, ) 1/ max(5, C L γ 1 ) 1 / (u, p) DG (v, q) DG, ) 1/ which completes the proof. Next, we prove a stability result for the form ã h restricted to H 1 0 () L 0 (). This result is a direct consequence of the definition of the extended forms Ãh, B h and C h and the inf-sup condition in (6). Lemma. There exists a positive stability constant C S such that, for any (u, p) H0 1() L 0 (), there is (v, q) H 0 1() L 0 () with ã h (u, p; v, q) (u, p) DG, (v, q) DG C S. () The constant C S is independent of ν, γ, k, and h, but depends on the inf-sup constant K in (6). Proof. We proceed as in [8, Lemma.1] (and the references therein). Let p L 0 (). Then, by (6), there exists w H 0 1() such that p w dx K p 0,, w 0, p 0,, where K > 0 is the inf-sup constant from (6). Hence, since L(u) L(w) 0 and M(u) M(w) 0, we obtain ã h (u, p; w, p) = u : w dx p w dx + p u dx + (1 ν) p 0, K K u 0, w 0, + K p 0, K p 0, K u 0, + (1 ν) p 0, K 1 u 0, K 1 u 0, + ( K + 1 ν ) p 0,. 15

16 Furthermore, the fact that u 0, u 0,, leads to ( K ã h (u, p; w, p) K 1 u 0, ) p ν 0, ( K K 1 u 0, )( + min, 1 ν) p 0,. Additionally, we have that ã h (u, p; u, p) u 0,. Thus, a pair (v, q) satisfying () may be found by choosing an appropriate linear combination of (u, p) and (w, p) (independent of ν)..1. A Posteriori Error Estimation We now proceed to complete the proof of Theorem.1; to this end, we first note that the following inverse estimates hold (cf. [, Lemma. and Theorem.5]) η 0;RK,1 Ck α K η α;r K,1, η 0;EK Ck α K η α;e K, K T h, with a constant C that is independent of the local mesh sizes and the polynomial degrees. Hence, for the proof of Theorem.1, it is sufficient to consider the case α = 0 only; thereby, for notational simplicity, we drop the subscript α in this section. By (e u, e p ) = (u u h, p p h ) we denote the error of the hp-dg approximation. Furthermore, we decompose u h into u h = u c h + u h, in accordance with the decomposition from Section.1.1; we then set e c u = u u c h. Noting that [e u ] = [u h ], gives (e u, e p ) DG (e c u, e p ) DG + max(1, γ 1 / ) u h 1,h. () Furthermore, from Lemma., we obtain (v, q) H0 1() L 0 () such that (e c u, e p ) DG ã h (e c u, e p ; v, q), (v, q) DG C S. () Then, due to (1), (5) and (1), we have, for v h V h arbitrary, (e c u, e p) DG ã h (e c u, e p; v, q) = ã h (u, p; v, q) ã h (u h, p h ; v, q) + ã h (u h, 0; v, q) = f (v v h ) dx ã h (u h, p h ; v v h, q) + ã h (u h, 0; v, q). (5) 16

17 By the continuity of the form ã h, cf. Lemma.1, and since (v, q) DG C S, the last term in the above inequality can be estimated as follows: ã h (u h, 0; v, q) C CC S (u h, 0) DG C C C S max(1, γ 1 / ) u h 1,h. (6) Thus, combining (), (5) and (6), leads to (e u, e p ) DG T 1 + T, (7) where T 1 = f (v v h ) dx ã h (u h, p h ; v v h, q), T = (C C C S + 1) max(1, γ 1 / ) u h 1,h. It remains to bound the terms T 1 and T. Estimation of T 1 : In order to estimate the term T 1, we let v h in (5) be the (conforming) hp-scott-zhang interpolant of v in () constructed in []. It satisfies v h V c h, as well as the approximation property K T h ( ) k K h K v v h 0,K + (v v h ) 0,K + k 1 / h 1/ (v v h ) 0, K C I v 0,, (8) with an interpolation constant C I that is independent of h and k, and depends solely on the shape regularity of the mesh and the constant ϱ in (7). Therefore, setting ξ v = v v h, we have T 1 = f ξ v dx Ãh(u h, ξ v ) B h (ξ v, p h ) + B h (u h, q) C h (p h, q). (9) Integration by parts and the definition of the lifting operator L leads to Ãh(u h, ξ v ) = K T h + ( K E(T h ) = K T h + u h ξ v dx K ) h u h : (ξ v n K )ds L(u h ) : ξ v dx + L(ξ v ) : h u h dx c [u h ] : [ξ v ] ds K u h ξ v dx L(u h ) : ξ v dx E I (T h ) E(T h ) [ h u h ] {ξ v } ds c [u h ] : [ξ v ] ds. 17

18 Similarly, we obtain B h (ξ v, p h ) + B h (u h, q) = K T K h K T K h p h ξ v dx + q u h dx + E I (T h ) M(u h )q dx. [p h ] {ξ v } ds Substituting the above expressions into (9) and noting that, since ξ v V c h, [ξ v ] = 0 and {ξ v } = ξ v, we get T 1 = ( ) Πh f + u h p h ξv dx K T K h + + E I (T h ) ( [ph ] [ h u h ] ) ξ v ds + M(u h ) q dx (1 ν) p h q dx + K T K h q u h dx L(u h ) : ξ v dx (f Π h f) ξ v dx. Employing the stability bounds from (0), yields T 1 h K kk 1 Π hf + u h p h 0,K h 1 K k K ξ v 0,K K T h + u h + (1 ν)p h 0,K q 0,K K T h + h 1 / k 1/ ([p h ] [ h u h ]) 0, h 1/ k 1 / ξ v 0, E I (T h ) + C 1 / L γ 1/ + C 1 / L γ 1/ E(T h ) E(T h ) c 1 / [u h ] 0, c 1 / [u h ] 0, 1/ 1/ ξ v 0, q 0, + h K kk 1 f Π hf 0,K h 1 K k K ξ v 0,K. K T h Applying the Cauchy-Schwarz inequality and using that, for ν [0, 1 /], we 18

19 have 1 ν, implies T 1 K T h + ( h K kk Π hf + u h p h 0,K + u ) h + (1 ν)p h 0,K E I (T h ) h 1 / k 1/ ([p h ] [ h u h ]) 0, + C L γ 1 + 1/ h KkK f Π h f 0,K K T h ( h K k K ξ v 0,K + ξ v 0,K + ) q 0,K K T h + E I (T h ) h 1/ k 1 / ξ v 0, 1/ E(T h ) max(1, C 1 / L γ 1/ ) 1/ ηk + OSC(f, K) K T h ( h K k K ξ v 0,K + ξ v 0,K + ) h 1/ k 1 / ξ v 0, K K T h + ( ν) q 0, 1/. c 1 / [u h ] 0, Recalling the approximation property (8) and the second bound in (), gives T 1 C S max(1, C 1 / L γ 1/ ) max(1, C I ) K T h η K + OSC(f, K) 1/. (0) Estimation of T : We use the norm equivalence result from Proposition.1 and the fact that [u h ] = [u h ]; thereby, we obtain T C P (C C C S + 1) max(1, γ 1 / ) C P (C C C S + 1) max(1, γ 1/ ) C P (C C C S + 1) max(1, γ 1/ ) E(T h ) E(T h ) 1/ ηk. K T h h 1 k [u h ] c [u h ] 1/ 1/ (1) Combining the estimates from (7), (0), and (1) completes the proof of Theorem.1. 19

20 . Proof of Theorem. We present the proofs of the four assertions in Theorem. separately. The proofs of (a) and (c) are analogous to the corresponding bounds derived in [, Lemma. and Lemma.5]; see also [16]. For the sake of completeness, we present the main steps. Assertion (a): We first consider the case α > 1 /. To this end, we set v K = (Π h f + u h p h )Φ α K. Then, using that u + p = f in L (K), we obtain by elementary manipulations v K Φ α/ K 0,K = (Π h f + u h p h ) v K dx K = ( (u u h ) + (p p h )) v K dx+ (Π h f f) v K dx K K = (u u h ) : v K dx (p p h )( v K ) dx K K + (Π h f f) v K dx K C ( (u u h ) 0,K + p p h 0,K ) vk 0,K + (f Π h f)φ α / K 0,K v K Φ α/ K 0,K. From the proof of [, Lemma.], we have Since v K Φ α/ K ( η α;rk,1 C v K 0,K Ck (1 α) K kkh K v K Φ α/ K 0,K. 0,K = k K h 1 K η α;rk,1, we readily obtain that k 1 α K ( ) (u uh ) 0,K + p p h 0,K +k 1 K h K f Π h f 0,K ). () This shows the assertion for α > 1 /. For α [0, 1 /], we first use that η α;rk,1 C ε k β α K η β;r K,1, for β = 1 / + ε with ε > 0, and apply the bound in () to η β;rk,1. Assertion (b): In order to bound η RK,, we simply use that u+(1 ν)p = 0 in L (K). This yields η R K, = u h + (1 ν)p h 0,K = (u u h ) + (1 ν)(p p h ) 0,K C ( (u u h ) 0,K + p p h 0,K). Assertion (c): Again, we first consider the case α > 1 / and let be an edge shared by two elements K 1 and K. Set δ := (K 1 K ) ; Lemma.6 of [] ensures the existence of a function w H 1 0(δ ) with w = ([p h ] [ h u h ])Φ α, 0

21 w δ = 0 and ( w 0,δ Ch 1 ( α) K σk K + σ 1) ([p h ] [ u h ])Φ α / 0,, w 0,δ Ch K σ ([p h ] [ u h ])Φ α / 0,, () for any σ (0, 1]. Using that [ u] [p] = 0 on interior edges and that u + p = f on each element K, it can be readily seen that ([p h ] [ u h ])Φ α / 0, = ([p h ] [ u h ]) w ds = ([p h p] [ (u h u)]) w ds ( ) ( ) = (ph p)n K1 w ds + (ph p)n K w ds K 1 K ( ) ( ) (uh u) n K1 w ds (uh u) n K w ds K 1 K = h (u h u) : w dx + (p h p)( w ) dx δ δ (f + u h p h ) w dx δ C ( h (u u h ) 0,δ + p p h 0,δ ) w 0,δ + ( Π h f + u h p h 0,δ + f Π h f 0,δ ) w 0,δ. Here, n K1 and n K denote the unit outward normal vectors on the boundaries K 1 and K, respectively. By summing up this estimate over all edges of a given element K, invoking the bounds for w 0,δ and w 0,δ from (), and using assertion (a), we obtain η α;e K ( ) ( h C kk 1 (σk ( α) ) K + σ 1 ) + kkσ (u u h ) 0,δ K + p p h 0,δ K + Cσk (1+ε) K kk h K f Π hf 0,δ K. Setting σ = kk proves the assertion for α > 1 /. For α [0, 1 /], we have that η α;ek Ck β α K η β;ek and use the above argument for η β;ek with β = 1 / + ε to obtain the assertion. Assertion (d): This is a simple consequence of the fact that the jump of u vanishes over interior edges and that u = 0 on Γ. Hence, for E(T h ), we have that kh 1/ [u h ] 0, = kh 1/ [u u h ] 0,, which completes the proof of Theorem.. 1

22 5 Numerical Experiments In this section we present a series of numerical examples to illustrate the practical performance of the proposed a posteriori error estimator derived in Theorem.1, with α = 0, within an automatic hp adaptive refinement procedure which is based on employing 1 irregular quadrilateral elements. The hp adaptive meshes are constructed by first marking the elements for refinement/derefinement according to the size of the local error indicators η 0;K ; this is done by employing the fixed fraction strategy, with refinement and derefinement fractions set to 5% and 10%, respectively. Once an element K T h has been flagged for refinement or derefinement, a decision must be made whether the local mesh size h K or the local degree k K of the approximating polynomial should be adjusted accordingly. The choice to perform either h refinement/derefinement or p refinement/derefinement is based on estimating the local smoothness of the (unknown) analytical solution. To this end, we employ the hp adaptive strategy developed in [19], where the local regularity of the analytical solution is estimated from truncated local Legendre expansions of the computed numerical solution; see, also, [9,18]. Here, the emphasis will be to demonstrate the robustness of the proposed a posteriori error indicator with respect to the Poisson ratio ν and the parameter γ arising in the definition of the discontinuity stabilization function c, cf. (11). Indeed, we shall show that the proposed a posteriori error indicator converges to zero at the same asymptotic rate as the energy norm of the actual error on a sequence of non-uniform hp adaptively refined meshes for a range of γ and ν. For simplicity, we set the constant C EST arising in Theorem.1 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 bound and the energy norm of the actual error. In general, to ensure the reliability of the error estimator, C EST must be determined numerically for the underlying problem at hand. In all of our numerical experiments, data oscillation will be neglected within our computations. Finally, we note that for both of the numerical examples considered in this section, closed form analytical solutions are not readily available; thereby, suitably accurate approximations have been computed on highly refined hp meshes. 5.1 Example 1 Here, we let be the unit square (0, 1) ; further, we select f = 0 and enforce the inhomogeneous boundary condition u = (g, 0) on Γ, where

23 Sfrag replacements Sfrag replacements Sfrag replacements Error Bounds 10 True Errors γ=10 γ= PSfrag replacements (Degrees of Freedom) 1 / Error Bounds True Errors γ=10 γ= (Degrees of Freedom) 1 / Effectivity Index (a) ν = Effectivity Index PSfrag replacements Error Bounds True Errors γ=10 γ= (Degrees of Freedom) 1 / 5 1 γ=10 γ= (b) ν = 0.5 Effectivity Index PSfrag replacements Mesh Number 1 γ=10 γ= (c) ν = 0. Mesh Number 1 γ=10 γ= Mesh Number Fig. 1. Example 1. Left: Comparison of the actual and estimated energy norm of the error with respect to the (third root of the) number of degrees of freedom; Right: Effectivity indices. sin (πx) for (x, y) (0, 1) {1}, g(x, y) = 0 otherwise. This represents a slight modification of the example considered in [0].

24 (a) (b) Fig.. Example 1. h and hp meshes after 9 hp-adaptive refinements, with 09 elements and 180 degrees of freedom, for ν = and γ = 10. In Figure 1 we present a comparison of the actual and estimated energy norm of the error versus the third root of the number of degrees of freedom in the finite element space V h Q h on a linear log scale, for the sequence of meshes generated by our hp adaptive algorithm. Here, numerical experiments are presented for ν = 0.999, 0.5, 0. for different values of the parameter γ arising in the definition of the discontinuity stabilization function c, cf. (11). We remark that the third root of the number of degrees of freedom is chosen on the basis of the a priori error analysis carried out in the article [6]. For each value of ν, we observe that the error bound over-estimates the true error by a (reasonably) consistent factor for both γ = 10, 100; indeed, from Figure 1, we see that all of the computed effectivity indices lie in the range between 5. Here, and in the following example, the mesh number is used to distinguish each of the hp finite element meshes produced by our adaptive mesh refinement algorithm; here, the initial hp mesh has mesh number 1. Moreover, we notice that the proposed mixed hp-dg method and our a posteriori error estimator are robust with respect to both the Poisson ratio, as this approaches the critical value of 1 /, and the discontinuity stabilization function, as γ is increased. Finally, from Figure 1 we observe that the convergence lines using hp-refinement are (roughly) straight on a linear-log scale, which indicates that exponential convergence is attained in all cases. In Figures,, and we show the mesh generated using the proposed a posteriori error indicator after 9 hp adaptive refinement steps for ν = 0.999, 0.5, 0., respectively, with γ = 10. For clarity, in each case we show the h mesh alone, as well as the corresponding distribution of the polynomial degree on this mesh. For ν = 0.999, we see that the mesh has been h refined along the two sides, as well as the bottom of the domain, where (mild) boundary

25 (a) (b) Fig.. Example 1. h and hp meshes after 9 hp-adaptive refinements, with 598 elements and 0699 degrees of freedom, for ν = 0.5 and γ = 10. (a) (b) Fig.. Example 1. h and hp meshes after 9 hp-adaptive refinements, with 51 elements and 178 degrees of freedom, for ν = 0. and γ = 10. layers are located. Additionally, a horizontal line of h refined elements occurs towards the top of the domain, which coincides with the location of the center of the recirculation of the displacement vector. Once the h mesh has adequately captured the structure of the solution, the hp adaptive algorithm increased the degree of the approximating polynomial within the rest of the computational domain. Although, for ν = 0.5, 0., we see a fairly similar structure in the hp mesh distribution to that for ν = 0.999, there are a couple of noticeable differences. Firstly, the imposition of the second equation in () leads to the development of a singularity in p at the corners (0, 1) and 5

26 6 Sfrag replacements γ=10 γ=100 True Errors Error Bounds (Degrees of Freedom) 1 / Effectivity Index PSfrag replacements (a) ν = γ=10 γ= Mesh Number Sfrag replacements γ=10 γ=100 True Errors Error Bounds (Degrees of Freedom) 1 / Effectivity Index PSfrag replacements 5 (b) ν = γ=10 γ= Mesh Number Fig. 5. Example. Left: Comparison of the actual and estimated energy norm of the error with respect to the (third root of the) number of degrees of freedom; Right: Effectivity indices. (1, 1) of. Indeed, as the boundary conditions lead to u = 0 at (0, 1) and (1, 1), the divergence of the displacement u is expected to be small in the regions adjacent to these corners. However, since the term 1 ν in the second equation of () is small as well, p is not necessarily close to zero in the vicinity of these vertices. Consequently, this leads to additional h refinement in the region containing these singularities of p. Secondly, as ν decreases, the center of the recirculation of the displacement vector moves downwards, leading to the corresponding horizontal strip of h refined elements being slightly lower in each case. 5. Example In this section, we consider the example of a singular solution in a non-convex domain. To this end, we let be the L shaped domain ( 1, 1) \ ([0, 1) ( 1, 0]), set f = 1, and impose homogeneous Dirichlet boundary conditions 6

27 5 5 Fig. 6. Example. hp-mesh after 6 hp-adaptive refinements, with 6 elements and 1780 degrees of freedom, for ν = and γ = 10. for u on the whole of Γ. We emphasize that the analytical solution (u, p) is analytic in, but u exhibits singularities at the corners of. In Figure 5, we show the history of the actual and estimated energy norm of the error on each of the meshes generated by our hp adaptive algorithm for ν = 0.999, 0.5 and γ = 10, 100. As in the previous example, we observe that the a posteriori bound over estimates the true error by a consistent factor between 6, though here we do see, that for this example, the effectivity indices do initially grow very slightly as the mesh is refined in all but one of the cases considered; though, asymptotically they seem to be tending towards a constant value. Additionally, we clearly observe that both our mixed hp- DG method and the corresponding a posteriori error estimator are free from volume locking. For both values of the Poisson ratio, we observe exponential convergence of the energy norm of the error using hp refinement: for each value of ν, on a linear log scale, the convergence lines are on average straight for γ = 10, 100. Finally, in Figure 6 we show the mesh generated using the local error indicators η 0;K after 6 hp adaptive refinement steps for ν = and γ = 10. Here, we see that the h mesh has been largely refined in the vicinity of the corners of the domain; in particular, stronger refinement has occurred in the vicinity of the reentrant corner located at the origin, as well as in the region adjacent to this singular point. From the zoom of the hp mesh in the vicinity of the origin, we see that this h refinement has occurred in the diagonal direction x = y, while in the other diagonal direction, x = y, p refinement has been 7

28 performed, cf. [16]. Away from the corners, we see that the polynomial degrees have been increased, since the underlying analytical solution is smooth in this region. An analogous hp refined mesh is also generated for the case when ν = 0.5; for brevity, we omit the details. 6 Conclusions In this article, we have derived both upper and lower residual based energy norm a posteriori error bounds for mixed hp DG approximations to the equations governing linear elasticity in two spatial dimensions. The analysis is based on employing a non-consistent reformulation of the hp DG method using lifting operators, together with a decomposition result for the underlying discontinuous space. We emphasize that our error analysis is robust with respect to the Poisson ratio; indeed, numerical experiments presented in this article clearly demonstrate that both the proposed hp DG method and the corresponding a posteriori estimator is free from volume locking. The extension of the current analysis to three dimensional problems follows analogously for the h version of the DG method, based on employing the decomposition result for DG spaces derived in the articles [1,0]; the extension of this result to hp DG methods in D is the subject of future research. Additional future work will be devoted to the extension of our analysis to irregular meshes and, in particular, anisotropic hp adaptive discontinuous Galerkin methods. References [1] D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal., 9: , 001. [] I. Babuška and M. Suri. The p and h p versions of the finite element method, basic principles and properties. SIAM Review, 6:578 6, 199. [] I. Babuška and M. Suri. On locking and robustness in the finite element method. SIAM J. Numer. Anal., 9(5):161 19, 199. [] S.C. Brenner and L. Sung. Linear finite element methods for planar linear elasticity. Math. Comp., 59:1 8, 199. [5] F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. In Springer Series in Computational Mathematics, volume 15. Springer Verlag, New York,

29 [6] B. Cockburn. Discontinuous Galerkin methods for convection-dominated problems. In T. Barth and H. Deconink, editors, High-Order Methods for Computational Physics, volume 9, pages 69. Springer Verlag, [7] B. Cockburn, G.E. Karniadakis, and C.-W. Shu, editors. Discontinuous Galerkin Methods. Theory, Computation and Applications, volume 11 of Lect. Notes Comput. Sci. Eng. Springer Verlag, 000. [8] B. Cockburn and C.-W. Shu. Runge Kutta discontinuous Galerkin methods for convection dominated problems. J. Sci. Comput., 16:17 61, 001. [9] T. Eibner and J.M. Melenk. An adaptive strategy for hp-fem based on testing for analyticity. Technical Report 1/00, University of Reading, 00. [10] V. Girault and P.A. Raviart. Finite element methods for Navier Stokes equations. Springer Verlag, New York, [11] B. Guo and I. Babuška. The hp version of the finite element method. Part I: The basic approximation method. Comp. Mech., 1:1 1, [1] B. Guo and I. Babuška. The hp version of the finite element method. Part II: General results and applications. Comp. Mech., 1:0 0, [1] P. Hansbo and M.G. Larson. Discontinuous finite element methods for incompressible and nearly incompressible elasticity by use of Nitsche s method. Comput. Methods Appl. Mech. Engrg., 191: , 00. [1] P. Houston, I. Perugia, and D. Schötzau. Mixed discontinuous Galerkin approximation of the Maxwell operator. SIAM J. Numer. Anal., (1): 59, 00. [15] P. Houston, D. Schötzau, and T. Wihler. Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem. J. Sci. Comput., (1):57 80, 005. [16] P. Houston, D. Schötzau, and T. Wihler. Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems. Technical Report 00-, University of Leicester, 00. Submitted for publication. [17] P. Houston, D. Schötzau, and T. Wihler. hp-adaptive discontinuous Galerkin finite element methods for the Stokes problem. In P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, and D. Knörzer, editors, Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, Volume II, 00. [18] P. Houston, B. Senior, and E. Süli. Sobolev regularity estimation for hp adaptive finite element methods. In F. Brezzi, A. Buffa, S. Corsaro, and A. Murli, editors, Numerical Mathematics and Advanced Applications ENUMATH 001, pages Springer, 00. [19] P. Houston and E. Süli. A note on the design of hp adaptive finite element methods for elliptic partial differential equations. Comput. Methods Appl. Mech. Engrg., 19(-5):9,

30 [0] O.A. Karakashian and F. Pascal. A posteriori error estimation for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal., 1:7 99, 00. [1] R. Kouhia and R. Stenberg. A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg., 1:195 1, [] J.M. Melenk. hp Interpolation of non-smooth functions. Technical Report NI0050, Isaac Newton Institute for the Mathematical Sciences, 00. [] J.M. Melenk and B.I. Wohlmuth. On residual-based a posteriori error estimation in hp-fem. Adv. Comp. Math., 15:11 1, 001. [] I. Perugia and D. Schötzau. An hp-analysis of the local discontinuous Galerkin method for diffusion problems. J. Sci. Comput., 17: , 00. [5] D. Schötzau, C. Schwab, and A. Toselli. Mixed hp-dgfem for incompressible flows. SIAM J. Numer. Anal., 0:171 19, 00. [6] D. Schötzau and T. P. Wihler. Exponential convergence of mixed hp-dgfem for Stokes flow in polygons. Numer. Math., 96():9 61, 00. [7] C. Schwab. p- and hp-fem Theory and Application to Solid and Fluid Mechanics. Oxford University Press, Oxford, [8] M. Vogelius. An analysis of the p-version of the finite element method for nearly incompressible materials. Numer. Math., 1:9 5, 198. [9] T. P. Wihler. Locking-free adaptive discontinuous Galerkin FEM for elasticity problems. To appear in Math. Comp., 00. [0] T. P. Wihler. Locking-free DGFEM for elasticity problems in polygons. IMA J. Numer. Anal., :5 75, 00. 0