An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity

Comput. Methods Appl. Mech. Engrg. 195 (2006) 3224–3246

www.elsevier.com/locate/cma

An hp-adaptive mixed discontinuous Galerkin FEMfor nearly incompressible linear elasticity

Paul Houston a,*,1, Dominik Schotzau b,2, Thomas P. Wihler c,3

a School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UKb Mathematics Department, University of British Columbia, Vancouver, BC, Canada V6T 1Z2

c Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2K6, Canada

Accepted 17 June 2005

Abstract

We develop the a posteriori error estimation of mixed hp-version discontinuous Galerkin finite element methods fornearly incompressible elasticity problems in two space dimensions. Computable upper and lower bounds on the errormeasured in terms of a natural (mesh-dependent) energy norm are derived. The bounds are explicit in the local meshsizes and approximation orders, and are independent of the locking parameter. A series of numerical experiments arepresented which demonstrate the performance of the proposed error estimator within an automatic hp-adaptive refine-ment procedure.� 2005 Elsevier B.V. All rights reserved.

Keywords: Discontinuous Galerkin methods; A posteriori error estimation; hp-Adaptivity; Linear elasticity; Volume locking

1. Introduction

One of the main challenges in the design of finite element methods for linear elasticity problems is theirrobustness with respect to nearly incompressible materials. Indeed, it is well known that the performance ofstandard low-order elements can significantly deteriorate if the compressibility parameter tends to a certain

0045-7825/$ - see front matter � 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.cma.2005.06.012

* Corresponding author.E-mail addresses: [email protected] (P. Houston), [email protected] (D. Schotzau), [email protected]

(T.P. Wihler).1 Supported by the EPSRC (Grant GR/R76615).2 Supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).3 Supported by the Swiss National Science Foundation, Project PBEZ2-102321.

mailto:[email protected]



P. Houston et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 3224–3246 3225

critical limit; this phenomenon is referred to as volume locking. In order to avoid this effect, several remedieshave been proposed within the literature. For example, certain non-conforming and discontinuous Galer-kin methods are free of volume locking; see, e.g., [4,21,29,30], and the references cited therein. Another classof locking-free methods is obtained in a natural way by the use of mixed finite elements; here, we only men-tion [5] and the references cited therein. Mixed approaches are based on replacing the divergence of the dis-placement by an additional variable (Lagrange multiplier) that is discretized separately.

The finite element approximation of linear elasticity and, more generally, partial differential equations ofelliptic type, poses another major challenge: in polygonal or polyhedral domains arising in applications ofengineering interest, the analytical solution of the underlying partial differential equations often exhibitstrong singularities in the vicinity of corners or edges of the domain. Typically, these singularities can onlybe accurately resolved by employing meshes that are locally refined towards vertices, edges and/or faces ofthe 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 combi-nation can achieve exponential rates of convergence, even in the presence of corner/edge singularities; fordetails, we refer the reader to [2,11,12,27] and the references cited therein.

For the practical realization of hp-discretizations, so-called discontinuous Galerkin (DG, for short) finiteelement methods provide an ideal computational framework. Indeed, in comparison to standard conform-ing finite element methods, DG schemes support the design of more general finite element spaces. Forexample, within a DG approach, non-matching grids containing hanging nodes and non-uniform, evenanisotropic, approximation degree distributions can easily be handled. For recent surveys on the designand analysis of DG methods in a range of applications, we refer the reader to [1,6–8], and the referencescited therein.

In this paper, we propose and study a mixed hp-adaptive DG method for nearly incompressible linearelasticity problems in two dimensional polygonal domains. The method is based on mixed discontinuouselements, whose hp-version stability and a priori convergence properties were previously studied in [25,26]in the context of the Stokes equations; see also [13] for an h-version approach. In the current article, wepresent an hp-version a posteriori error analysis for this DG method; in particular, we derive computableupper and lower energy norm a posteriori error bounds that are explicit in the local mesh sizes andapproximation orders. We emphasize that, since our discretization is based on employing the mixed formof the equations of linear elasticity, we naturally obtain reliability and efficiency constants for the a pos-teriori bounds that are independent of the compressibility parameter. In particular, this implies that theydo not deteriorate in the incompressible limit, which, in the mixed setting, corresponds to the Stokesproblem.

The a posteriori error analysis presented in this article is a continuation of our work in [15,16], whereenergy norm a posteriori error estimation of mixed h-version DG approximations for the Stokes equationsand hp-version DG methods for Poisson�s equation, respectively, was recently developed. Here, the proof ofthe 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 [1,24,25], and employing a norm equivalence result forhp-version DG spaces; see [16, Section 5]. This crucial result has been obtained by establishing an approx-imation property in the spirit of the h-version results in [14,20]. The lower (efficiency) bounds are derivedbased on employing the techniques developed in [23] for conforming hp-version finite element methods. Asin [23], reliability and efficiency of our error bounds cannot be established uniformly with respect to thepolynomial degree, since the proof of efficiency relies on employing inverse estimates which are suboptimalin the spectral order. Finally, we note that, for the Stokes problem, the results of this paper have been pre-viously announced in the conference article [17].

The outline of this article is as follows. In Section 2, we introduce the hp-DG method for the numericalapproximation of linear elasticity problems in mixed form. In Section 3, our a posteriori error bounds are

3226 P. Houston et al. / Comput. Methods Appl. Mech. Engrg. 195 (2006) 3224–3246

presented and discussed; here, both upper and lower energy norm bounds will be considered. The proofs ofthese results will be presented in Section 4. In Section 5, we present a series of numerical experiments toillustrate the performance of the proposed error estimators within an automatic hp-mesh refinementalgorithm. Finally, in Section 6, we summarize the work presented in this paper and draw someconclusions.

Throughout this article, we use the following notation: for an interval D � R or a bounded Lipschitzdomain D � R2, let L2(D) be the Lebesgue space of square integrable functions, endowed with the usualnorm k Æ k0,D. The standard Sobolev space of functions with integer regularity exponent s > 0 is denotedby Hs(D). We write k Æ ks,D and j Æ js,D for the corresponding norms and semi-norms, respectively. As usual,we define H 1

0ðDÞ as the subspace of functions in H1(D) with zero trace on oD. Furthermore, for a functionspace X(D), let X(D)2 and X(D)2·2 be the spaces of all vector and tensor fields whose components belong toX(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 2 R2, and matricesr; s 2 R2�2, we use the standard notation ($v)ij = ojvi, ðr � rÞi ¼

P2j¼1ojrij, and r : s ¼

P2i;j¼1rijsij. Further-

more, let v � w be the matrix whose ijth component is viwj. With this notation, we note that the followingidentity holds v � r � w ¼

P2i;j¼1virijwj ¼ r : ðv� wÞ.

2. Mixed hp-DG method for linear elasticity

In this section, we introduce the equations governing linear elasticity considered in this article and pres-ent a mixed hp-DG method for their discretization.

2.1. Linear elasticity problems

On a given polygonal domain X � R2 with boundary C = oX, we consider the linear elasticity problem:find a vector field (displacement) u ¼ ðu1; u2Þ 2 H 1

0ðXÞ2 such that

� Du� 1

1� 2mrðr � uÞ ¼ f in X; ð1Þ

u ¼ 0 on C. ð2Þ

Here, $Æ is the divergence operator, m 2 0; 12

� �is the Poisson ratio, and f 2 L2(X)2 is an external force (scaled

by 2(1 + m)/E, where E > 0 is Young�s modulus).In order to write Eqs. (1) and (2) in mixed form, we introduce the additional variable (Lagrange

multiplier)

p ¼ � 1

1� 2mr � u.

We note that p 2 L20ðXÞ, where

L20ðXÞ ¼ q 2 L2ðXÞ :

ZX

qdx ¼ 0

� �.

Hence, (1) and (2) is equivalent to finding ðu; pÞ 2 H 10ðXÞ

2 � L20ðXÞ such that

� Duþrp ¼ f in X;

r � uþ ð1� 2mÞp ¼ 0 in X;

u ¼ 0 on C.

ð3Þ


The standard variational formulation of (3) is then given by: find ðu; pÞ 2 H 10ðXÞ

Z
Aðu; vÞ þ Bðv; pÞ ¼X

f � vdx;

� Bðu; qÞ þ Cðp; qÞ ¼ 0

ð4Þ

for all ðv; qÞ 2 H 10ðXÞ

2 � L20ðXÞ, where

Aðu; vÞ ¼Z

Xru : rv dx; Bðv; qÞ ¼ �

ZX

qr � vdx;

and

Cðp; qÞ ¼ ð1� 2mÞZ

Xpqdx.

More compactly, (4) can be written as follows: find ðu; pÞ 2 H 10ðXÞ


aðu; p; v; qÞ ¼Z

Xf � vdx; 8ðv; qÞ 2 H 1

0ðXÞ2 � L2

0ðXÞ; ð5Þ

with

aðu; p; v; qÞ ¼ Aðu; vÞ þ Bðv; pÞ � Bðu; qÞ þ Cðp; qÞ.
We note that, for m 2 0; 1
2

� �, the form a is coercive on H 1

0ðXÞ2 � L2

0ðXÞ and hence the solutionðu; pÞ 2 H 1

0ðXÞ2 � L2

0ðXÞ of (4) exists and is unique.

Remark 1. It can be seen from (1) and (2) that the following stability estimate holds:

kuk21;X þ

1

1� 2mkr � uk2

0;X 6 Ckf k20;X;

with a constant C > 0 that is independent of the Poisson ration m 2 0; 12

� �. This immediately implies that, as

m! 12, the constraint $ Æ u! 0 naturally arises, which corresponds to (nearly) incompressible materials. It is

well-known (see [3]) that this incompressibility constraint may cause a loss of uniformity (with respect to m)in the asymptotic convergence regime of finite element methods based on discretizing the primal variables in(1) and (2). This does not mean that those methods do not converge at all; however, it may happen that theconvergence begins to take place at such high numbers of degrees of freedom that, in certain cases, themethod is not feasible in practice. This lack of robustness of the FEM with respect to incompressible mate-rials is referred to as volume locking. In this paper, we will present a mixed DG method based on (3) that isable to overcome this effect in a natural manner. For considerations on locking-free DG methods based onemploying primal variables, we refer to [13,29,30].

Remark 2. We note that the case when m ¼ 12

in (3) corresponds to the standard Stokes problem for incom-pressible fluid flow. Due to the continuous inf–sup condition

inf06¼q2L2

0ðXÞ

sup06¼v2H1

0ðXÞ2

�R

X qr � vdx

krvk0;Xkqk0;X

P K > 0; ð6Þ

where K is the inf–sup constant, depending only on X, the mixed variational formulation (4) is still well-posed and has a unique solution ðu; pÞ 2 H 1

0ðXÞ2 � L2

0ðXÞ for m ¼ 12; see [5,10] for details. The inf–sup con-

dition (6) also ensures the robustness of the mixed problem (3) with respect to m.


2.2. Meshes and trace operators

In this section, we introduce the notation required for the definition of the hp-DG method.

2.2.1. MeshesThroughout, we assume that the domain X can be subdivided into conforming, shape-regular affine

meshes Th ¼ fKgK2Thconsisting of triangles and/or parallelograms. For each K 2Th, we denote by nK

the unit outward normal vector to the boundary oK, and by hK the elemental diameter. Furthermore,we assign to each element K 2Th an approximation order kK P 1. The local quantities hK and kK arestored in the vectors h ¼ fhKgK2Th

and k ¼ fkKgK2Th, respectively.

We denote by EIðThÞ the set of all interior edges of Th, by EBðThÞ the set of all boundary edges, anddefine EðThÞ ¼ EIðThÞ [ EBðThÞ. We assume that the local mesh sizes and approximation degrees are ofbounded variation, i.e. there is a constant . 6 1 such that

.hK 6 hK 0 6 .�1hK ; .kK 6 kK 0 6 .�1kK ; ð7Þ
whenever K and K 0 share a common edge.
2.2.2. Averages and jumps

Next, we define average and jump operators. To this end, let K+ and K� be two adjacent elements of Th;furthermore, let x be an arbitrary point of the interior edge j ¼ oKþ \ oK� 2 EIðThÞ. Moreover, for sca-lar-, vector-, and matrix-valued functions q, v, and s, respectively, that are smooth inside each element K±,we denote by (q±,v±,s±) the traces of (q,v,s) on j taken from within the interior of K±, respectively. Then,we define the following averages at x 2 j:

q ¼ 1

2ðqþ þ q�Þ; v ¼ 1

2ðvþ þ v�Þ; s ¼ 1

2ðsþ þ s�Þ.

Similarly, the jumps at x 2 j are given by

sqt ¼ qþnKþ þ q�nK� ; svt ¼ vþ � nKþ þ v� � nK� ;

svt ¼ vþ � nKþ þ v� � nK� ; sst ¼ sþnKþ þ s�nK� .

On boundary edges j 2 EBðThÞ, we set q = q, v = v, s = s, as well as sqt = qn, svt = v Æ n,svt = v � n, and sst = sn. Here, n denotes the unit outward normal vector to C.

2.3. Mixed hp-discontinuous Galerkin discretization

We now define an hp-DG method for the approximation of the linear elasticity problem (3) in mixedform and discuss its well-posedness.

Given a mesh Th on X and a degree vector k = {kK}, kK P 1, we approximate (3) by finite element func-tions (uh,ph) 2 Vh · Qh, where

Vh ¼ v 2 L2ðXÞ2 : vjK 2SkK ðKÞ2; K 2Th

n o; ð8Þ

Qh ¼n

q 2 L20ðXÞ : qjK 2SkK�1ðKÞ; K 2Th

o.

Here, for k P 0, SkðKÞ denotes the space PkðKÞ of polynomials of total degree at most k on K, if K is atriangle, and the space QkðKÞ of polynomials of degree at most k in each variable on K, if K is aquadrilateral.


We consider the mixed hp-DG method: find (uh,ph) 2 Vh · Qh such that

Ahðuh; vÞ þ Bhðv; phÞ ¼Z

Xf � vdx;

� Bhðuh; qÞ þ Chðph; qÞ ¼ 0

ð9Þ

for all (v,q) 2 Vh · Qh. The forms Ah, Bh and Ch are given, respectively, by

Ahðu; vÞ ¼Z

Xrhu : rhvdx�

Xj2EðThÞ

Zjrhv : sutþ rhu : svt

� �dsþ

Xj2EðThÞ

Zjcsut : svtds;

Bhðv; qÞ ¼ �Z

Xqrh � vdxþ

Xj2EðThÞ

Zj

q svtds;

Chðp; qÞ ¼ ð1� 2mÞZ

Xpqdx.

ð10Þ
Here, $h and $hÆ denote the discrete gradient and divergence operators, respectively, defined elementwise.The function c 2 L1ðEðThÞÞ is the so-called discontinuity stabilization function that is chosen as follows.Define the functions h 2 L1ðEðThÞÞ and k 2 L1ðEðThÞÞ by
hðxÞ :¼minfhK ; hK 0 g; x 2 j ¼ oK \ oK 0 2 EIðThÞ;hK ; x 2 j ¼ oK \ C 2 EBðThÞ;

�kðxÞ :¼

maxfkK ; kK 0 g; x 2 j ¼ oK \ oK 0 2 EIðThÞ;kK ; x 2 j ¼ oK \ C 2 EBðThÞ.

�
Then we set
c ¼ ch�1k

2; ð11Þ
with a parameter c > 0 that is independent of h, k, and m.
As in (5), the discrete formulation (9) is equivalent to finding (uh,ph) 2 Vh · Qh such that

ahðuh; ph; v; qÞ ¼Z

Xf � vdx ð12Þ

for all (v,q) 2 vh · Qh, where

ahðu; p; v; qÞ ¼ Ahðu; vÞ þ Bhðv; pÞ � Bhðu; qÞ þ Chðp; qÞ. ð13Þ
Henceforth, we assume that c is chosen sufficiently large; under this condition, the problem (9), cf., also,(12), is uniquely solvable.
Proposition 2.1. Let m 2 0; 12

� �. Then, there is a constant cmin > 0 such that, for all c P cmin, the mixed hp-DG

method (9), cf., also, (12), possesses a unique solution (uh,ph) 2 Vh · Qh.

Proof. This readily follows from the coercivity of the forms Ah (for c sufficiently large) and Ch. h

Remark 3. In the limiting case m ¼ 12, the discrete problem (9) is also uniquely solvable. This follows from

the discrete inf–sup conditions in [25,13] that show that the form Bh is stable uniformly in h. While on quad-rilateral meshes the discrete inf–sup constant has been shown to decay at the most like 1/max{kK} in theapproximation orders kK, a similar hp-version result on triangular meshes still needs to be established.


Remark 4. The form Ah corresponds to the so-called symmetric interior penalty discretization of theLaplace operator; for a detailed review of a wide class of DG methods for diffusion problems and theStokes system, we refer to the articles [1,25], respectively.

Remark 5. In the case of inhomogeneous Dirichlet boundary conditions, u = g on C, with a datum g sat-isfying the compatibility condition �Cg Æ nds = 0, the functional on the right-hand side of the first equationin (9) must be replaced by

F hðvÞ ¼Z

Xf � vdx�

Xj2EBðThÞ

Zjðg � nÞ : rhvdsþ

Xj2EBðThÞ

Zjcg � vds.

Additionally, the right-hand side of the second equation in (9) is set equal to

GhðqÞ ¼ �X

j2EBðThÞ

Zj

qg � nds.

3. 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 theerror of the hp-DG method (9), measured in terms of the energy norm jjj Æ jjjDG given by

jjjðv; qÞjjj2DG ¼ krhvk20;X þ

Xj2EðThÞ

Zjcjsvtj2 dsþ ð2� 2mÞkqk2

0;X.

The proofs of the corresponding a posteriori error bounds, Theorems 3.1 and 3.2, will be given in Section 4.

3.1. Weighted error indicators

In order to study the dependence on the polynomial degree in the a posteriori error analysis for conform-ing hp-finite element methods, the use of weighted local error indicators gK was recently proposed in [23].Following that approach, we derive a family of weighted estimators ga;K, K 2Th, a 2 [0, 1], for the mixedhp-DG method proposed in this paper. We note that, as for conforming hp-methods, simultaneous reliabil-ity and efficiency—uniformly in the polynomial degrees—cannot be achieved for any fixed a 2 [0, 1].

On a reference element bK , we define the weight function UKðxÞ ¼ distðx; obK Þ. For an arbitrary element

K 2Th, we set UK ¼ cKUK � F �1K , where F K : bK ! K is the elemental transformation and cK is a scaling

factor chosen such that �KUK dx = meas(K). Similarly, on the reference interval bI ¼ ð�1; 1Þ, we definethe weight function UIðxÞ ¼ 1� x2. For an interior edge j, the weight Uj is then defined byUj ¼ cjUI � F �1

j , where Fj is the affine transformation that maps (�1,1) onto j and cj is chosen such that�jUjds = length(j).

As in [23], for each element K 2Th and a 2 [0,1], we introduce the weighted local error indicator ga;K,which is given by

g2a;K ¼ k2a

K g2a;RK ;1

þ g2a;EK

� �þ g2

RK ;2þ g2

JK. ð14Þ

Here,

g2a;RK ;1

¼ k�2K h2

K ðPh f þ Duh �rphÞUa=2K

�� 2

0;K; ð15Þ

g2RK ;2¼ kr � uh þ ð1� 2mÞphk

20;K ; ð16Þ


are residual terms corresponding to the first and second equations in (3), respectively, and Phf denotes theelementwise L2-projection of f onto the space SkK�1ðKÞ, K 2T. Furthermore,

g2a;EK¼ 1

2

Xj2oKnC

k�1=2

h1=2ðspht� srhuhtÞUa=2

j

�� 2

0;j; ð17Þ

is a weighted edge residual. Finally, the term

g2JK¼Xj2oK

c1=2suht

�� 2

0;jð18Þ

measures the discontinuities (jumps) of the approximate displacement uh over element edges.

3.2. Reliability and efficiency

The aim of this section is to present the a posteriori error estimates for the mixed hp-DG method (9). Thefirst result is concerned with establishing the reliability of the error indicators ga;K.

Theorem 3.1. Let a 2 [0,1]. Furthermore, let ðu; pÞ 2 H 10ðXÞ

2 � L20ðXÞ be the solution of the linear elasticity

problem (3) and (uh,ph) 2 Vh · Qh its mixed hp-DG approximation obtained by (9). Then, the following a

posteriori error bound holds

jjjðu� uh; p � phÞjjjDG 6 CEST

XK2Th

g2a;K

!1=2

þ COSC

XK2Th

OSCðf ;KÞ !1=2

;

where the elemental error indicators ga;K are given by (14) and

OSCðf ;KÞ ¼ h2Kk�2

K kf �Ph f k20;K

is a data oscillation term. The constants CEST,COSC > 0 are independent of m, h, k, c, and a.

Remark 6. We stress that the constants CEST and COSC in Theorem 3.1 do not depend on c, provided that cis chosen sufficiently large. In addition, we emphasize that these constants are completely independent ofthe Poisson ratio m; in particular, they do not deteriorate as m! 1

2, thereby clearly indicating the robustness

of the hp-DG method with respect to volume locking. In fact, our a posteriori analysis also applies to thecase when m ¼ 1

2, which corresponds to the Stokes system for incompressible fluid flow (see also [17]); cf.

Remark 3.

Remark 7. In order to incorporate inhomogeneous boundary conditions u = g on C, the error indicatorsga;K are simply adjusted by modifying the jump indicators kc1=2suhtk2

0;oK on oK \ C; cf. [16].

Next, we discuss the efficiency of the error indicators ga;K.

Theorem 3.2. Let ðu; pÞ 2 H 10ðXÞ

2 � L20ðXÞ be the analytical solution of (3) and (uh,ph) 2 Vh · Qh its mixed

hp-DG approximation obtained by (9). Writing ga;K to denote the weighted error indicators defined in (14), we

have the following bounds:

(a) Let a 2 [0,1]. For any e > 0, there is a constant Ce, independent of m, h, k, c, a, and K 2Th, such that

g2a;RK ;1

6 Ce k2ð1�aÞK krðu� uhÞk2

0;K þ kp � phk20;K

� �þ kmaxð1þ2e�2a;0Þ

K k�2K h2

Kkf �Phf k20;K

� �.


(b) There is a constant C, independent of m, h, k, c, and K 2Th, such that

g2RK ;26 C krðu� uhÞk2

0;K þ kp � phk20;K

� �.

(c) Let a 2 [0,1]. For any e > 0, there is a constant Ce, independent of m, h, k, c, a, and K 2Th, such that

g2a;EK6 Cek

maxð1þ2e�2a;0ÞK kK krhðu� uhÞk2

0;dKþ kp � phk

20;dK

� �þ k2e

K k�2K h2

Kkf �Phf k20;dK

� �;

where dK ¼SfK 0 2Th : K 0 and K share a common edgeg.

(d) The jump term satisfies the following equality

g2JK¼Xj2oK

kc1=2su� uhtk20;j.

Remark 8. From Theorem 3.1, we see that the best upper (reliable) a posteriori error bounds are obtainedfor a = 0, while the best efficiency bounds arise when a = 1. As for conforming hp-FEM, cf. [23], for exam-ple, simultaneous reliability and efficiency cannot be achieved for any fixed a 2 [0, 1]. This is perhaps notsurprising, since in the special case when singularities present in the underlying analytical solution ariseon inter-element boundaries, the energy norm of the error may decay to zero at a superconvergent rateas the polynomial degree is increased; see, for example, [27], where hp-approximation results in weightedSobolev norms are developed.

4. Proofs

4.1. Proof of Theorem 3.1

The proof of Theorem 3.1 will be outlined in the following sections; it is based on employing the recenthp-decomposition result for discontinuous spaces from [16], together with a non-consistent reformulationof the hp-DG method (9) using lifting operators (see, e.g., [1]). A similar approach has also been developedfor the a posteriori error analysis of the h-version of the interior penalty DG method for the Stokes problemin the article [15].

4.1.1. Decomposition of hp-DG spaces

We split the DG space Vh from (8) into a conforming part

Vch :¼ Vh \ H 1

0ðXÞ2;

and a purely non-conforming part V?h . Here, V?h is defined as the orthogonal complement of Vch in Vh with

respect to the norm

kvk21;h ¼ krhvk2

0;X þX

j2EðThÞ

Zjh�1k

2jsvtj2 ds.

With this notation, the following norm equivalence result holds; see [16], for details.

Proposition 4.1. The expression

v 7!X

j2EðThÞ

Zjk

2h�1jsvtj2 ds

!1=2


is a norm on V?h . This norm is equivalent to the norm k Æ k1,h and there is a constant CP > 0, independent of hand k, such that !

kvk1;h 6 CP

Xj2EðThÞ

Zjk

2h�1jsvtj2 ds

1=2

6 CPkvk1;h

for all v 2 V?h .

4.1.2. Lifting operators and extended forms

In this section, we define a suitable extension of the forms Ah and Bh from (10) to the continuous levelusing the lifting operators introduced in [1]; see also [24,25]. To this end, we define the space

VðhÞ ¼ H 10ðXÞ

2 þ Vh. ð19Þ
Furthermore, by using the auxiliary space
Rh ¼ s 2 L2ðXÞ2�2 : sjK 2 SkK ðKÞ2�2;K 2Th

n o;

we introduce the lifting operator L : VðhÞ ! Rh by
ZXLðvÞ : sdx ¼
Xj2EðThÞ

Zj

svt : s ds; 8s 2 Rh.

In addition, we define M : VðhÞ ! Qh by
ZXMðvÞqdx ¼
Xj2EðThÞ

Zj

svt q ds; 8q 2 Qh.

The above lifting operators are stable; see [24,25] for details. More precisely, for any v 2 V(h), the followingbounds hold:

kLðvÞk20;X 6 CL

Xj2EðThÞ

Zjk

2h�1jsvtj2 ds;

kMðvÞk20;X 6 CL

Xj2EðThÞ

Zjk

2h�1jsvtj2 ds;

ð20Þ

where the constant CL > 0 is independent of h and k.We are now in a position to introduce the following extended forms:
eAhðu; vÞ ¼
ZXrhu : rhvdx�

ZXðLðuÞ : rhvþLðvÞ : rhuÞdxþ

Xj2EðThÞ

Zjcsut : svtds;

eBhðv; qÞ ¼ �Z

Xqrh � vdxþ

ZXMðvÞqdx;

eChðp; qÞ ¼ ð1� 2mÞZ

Xpqdx.

Moreover, we define

~ahðu; p; v; qÞ ¼ eAhðu; vÞ þ eBhðv; pÞ � eBhðu; qÞ þ eChðp; qÞ.
We emphasize that, in contrast to the form ah, defined in (13), the form eah is well-defined on(V(h) · L2(X))2. Furthermore, we observe that
~ah � ah on ðVh � QhÞ2; ~ah � a on ðH 1

0ðXÞ2 � L2ðXÞÞ2. ð21Þ


4.1.3. Stability results

In this section, we present some basic stability properties of the form ~ah; firstly, we have the followingcontinuity result.

Lemma 4.1. For any ðu; pÞ 2 V?h � L2ðXÞ and ðv; qÞ 2 H10ðXÞ

2 � L2ðXÞ, we have

j~ahðu; p; v; qÞj 6 CCjjjðu; pÞjjjDGjjjðv; qÞjjjDG;

where CC = 2max(5,3CLc�1)1/2, and CL is the constant from (20).

Proof. We first note that for v 2 H 10ðXÞ

2, we have LðvÞ � svt � 0 and MðvÞ � 0, from the stability of thelifting operators (20). Employing these results, together with the Cauchy–Schwarz inequality, and theinequality

kr � vk0;X 6

ffiffiffi2pkrvk0;X; 8v 2 H 1ðXÞ2;

we deduce that

j~ahðu;p;v;qÞj6 jeAhðu;vÞjþ jeBhðv;pÞjþ jeBhðu;qÞjþ jeChðp;qÞj

6 krhuk0;Xkrvk0;XþkLðuÞk0;Xkrvk0;Xþkpk0;Xkr� vk0;Xþ1ffiffiffi2p kqk0;X

ffiffiffi2pkrh �uk0;X

þffiffiffi2pkMðuÞk0;X

1ffiffiffi2p kqk0;Xþð1�2mÞkpk0;Xkqk0;X

6 5krhuk20;Xþ3CLc

�1X

j2EðThÞkc1=2sut

2

0;jþð2�2mÞkpk2

0;X

!1=2

4krvk20;Xþð2�2mÞkqk2

0;X

� �1=2

6 2maxð5;3CLc�1Þ1=2jjjðu;pÞjjjDGjjjðv;qÞjjjDG;

which completes the proof. h

Next, we prove a stability result for the form ~ah restricted to H 10ðXÞ

2 � L20ðXÞ. This result is a direct con-

sequence of the definition of the extended forms eAh, eBh and eCh and the inf–sup condition in (6).

Lemma 4.2. There exists a positive stability constant CS such that, for any ðu; pÞ 2 H10ðXÞ

2 � L20ðXÞ, there is

ðv; qÞ 2 H10ðXÞ

2 � L20ðXÞ with

~ahðu; p; v; qÞP jjjðu; pÞjjjDG; jjjðv; qÞjjjDG 6 CS. ð22Þ
The constant CS is independent of m, c, k, and h, but depends on the inf–sup constant K in (6).
Proof. We proceed as in [28, Lemma 2.1] (and the references therein). Let p 2 L20ðXÞ. Then, by (6), there

exists w 2 H 10ðXÞ

2 such thatZ
�
Xpr � wdx P Kkpk2

0;X; krwk0;X 6 kpk0;X;

where K > 0 is the inf–sup constant from (6). Hence, since LðuÞ �LðwÞ � 0 and MðuÞ �MðwÞ � 0, weobtain Z Z Z

~ahðu; p; w; pÞ ¼Xru : rwdx�

Xpr � wdxþ

Xpr � udxþ ð1� 2mÞkpk2

0;X

P �ffiffiffiffiffiffi2

K

rkruk0;X

ffiffiffiffiffiffiK

2

rkrwk0;X þKkpk2

0;X �ffiffiffiffiffiffiK

2

rkpk0;X

ffiffiffiffiffiffi2

K

rkr � uk0;X þ ð1� 2mÞkpk2

0;X

P �K�1kruk20;X �K�1kr � uk2

0;X þK

2þ 1� 2m

�kpk2

0;X.


Furthermore, the fact that kr � uk20;X 6 2kruk2

0;X, leads to

~ahðu; p; w; pÞP �3K�1kruk20;X þ

K

2þ 1� 2m

�kpk2

0;X

P �3K�1kruk20;X þmin

K

2; 1

�ð2� 2mÞkpk2

0;X.

Additionally, we have that

~ahðu; p; u; pÞP kruk20;X.

Thus, a pair (v,q) satisfying (22) may be found by choosing an appropriate linear combination of (u,p) and(w,p) (independent of m). h

4.1.4. A posteriori error estimation

We now proceed to complete the proof of Theorem 3.1; to this end, we first note that the following in-verse estimates hold (cf. [23, Lemma 2.4 and Theorem 2.5])

g0;RK ;16 Cka

Kga;RK ;1; g0;EK

6 CkaKga;EK

; K 2Th;

with a constant C that is independent of the local mesh sizes and the polynomial degrees. Hence, for theproof of Theorem 3.1, it is sufficient to consider the case a = 0 only; thereby, for notational simplicity,we drop the subscript �a� in this section.

By (eu,ep) = (u � uh,p � ph) we denote the error of the hp-DG approximation. Furthermore, we decom-pose uh into uh ¼ uc

h þ u?h , in accordance with the decomposition from Section 4.1.1; we then set ecu ¼ u� uc

h.Noting that seut ¼ su?h t, gives

jjjðeu; epÞjjjDG 6 jjjðecu; epÞjjjDG þmaxð1; c1=2Þku?h jjj1;h. ð23Þ

Furthermore, from Lemma 4.2, we obtain ðv; qÞ 2 H 10ðXÞ


jjjðecu; epÞjjjDG 6 ~ahðec

u; ep; v; qÞ; jjjðv; qÞjjjDG 6 CS . ð24Þ
Then, due to (21), (5) and (12), we have, for vh 2 Vh arbitrary,
jjjðecu; epÞjjjDG 6 ~ahðec

u; ep; v; qÞ ¼ ~ahðu; p; v; qÞ � ~ahðuh; ph; v; qÞ þ ~ahðu?h ; 0; v; qÞ

¼Z

Xf � ðv� vhÞdx� ~ahðuh; ph; v� vh; qÞ þ ~ahðu?h ; 0; v; qÞ. ð25Þ

By the continuity of the form ~ah, cf. Lemma 4.1, and since jjj(v,q)jjjDG 6 CS, the last term in the aboveinequality can be estimated as follows:

~ahðu?h ; 0; v; qÞ 6 CCCS jjjðu?h ; 0ÞjjjDG 6 CCCS maxð1; c1=2Þku?h k1;h. ð26Þ

Thus, combining (23), (25) and (26), leads to

jjjðeu; epÞjjjDG 6 T 1 þ T 2; ð27Þ
where
T 1 ¼Z

Xf � ðv� vhÞdx� ~ahðuh; ph; v� vh; qÞ;

T 2 ¼ ðCCCS þ 1Þmaxð1; c1=2Þku?h k1;h.

It remains to bound the terms T1 and T2.


Estimation of T1: In order to estimate the term T1, we let vh in (25) be the (conforming) hp-Scott–Zhanginterpolant of v in (24) constructed in [22]. It satisfies vh 2 Vc

h, as well as the approximation property

XK2Th

k2Kh�2

K kv� vhk20;K þ krðv� vhÞk2

0;K þ kk1=2h�1=2ðv� vhÞk2

0;oK

� �6 C2

I krvk20;X; ð28Þ

with an interpolation constant CI that is independent of h and k, and depends solely on the shape–regularityof the mesh and the constant . in (7).

Therefore, setting nv = v � vh, we have

T 1 ¼Z

Xf � nv dx� eAhðuh; nvÞ � eBhðnv; phÞ þ eBhðuh; qÞ � eChðph; qÞ. ð29Þ

Integration by parts and the definition of the lifting operator L leads to

�eAhðuh; nvÞ ¼X

K2Th

ZK

Duh � nv dx�Z

oKrhuh : ðnv � nKÞds

�

þZ

XLðuhÞ : rnv dxþ

ZXLðnvÞ : rhuh dx

�X

j2EðThÞ

Zjcsuht : snvtds

¼X

K2Th

ZK

Duh � nv dx�X

j2EIðThÞ

Zj

srhuht � nv ds

þZ

XLðuhÞ : rnv dx�

Xj2EðThÞ

Zjcsuht : snvtds.

Similarly, we obtain

�eBhðnv; phÞ þ eBhðuh; qÞ ¼ �X

K2Th

ZKrph � nv dxþ

Xj2EIðThÞ

Zj

spht � nv ds�X

K2Th

ZK

qr � uh dx

þZ

XMðuhÞqdx.

Substituting the above expressions into (29) and noting that, since nv 2 Vch, snvt = 0 and nv = nv,

we get

T 1 ¼X

K2Th

ZKðPhf þ Duh �rphÞ � nv dx�

XK2Th

ZK

qr � uh dx

þX

j2EIðThÞ

Zjðspht� srhuhtÞ � nv dsþ

ZXLðuhÞ : rnv dx

þZ

XMðuhÞqdx� ð1� 2mÞ

ZX

phqdxþZ

Xðf �Phf Þ � nv dx.


Employing the stability bounds from (20), yields

jT 1j 6X

K2Th

hKk�1K kPhf þ Duh �rphk0;Kh�1

K kKknvk0;K

þX

K2Th

kr � uh þ ð1� 2mÞphk0;Kkqk0;K

þX

j2EIðThÞkh1=2

k�1=2ðspht� srhuhtÞk0;jkh�1=2

k1=2nvk0;j

þ C1=2L c�1=2

Xj2EðThÞ

kc1=2suhtk20;j

!1=2

krnvk0;X

þ C1=2L c�1=2

Xj2EðThÞ

kc1=2suhtk20;j

!1=2

kqk0;X

þX

K2Th

hKk�1K kf �Phf k0;Kh�1

K kKknvk0;K .

Applying the Cauchy–Schwarz inequality and using that, for m 2 0; 12

� , we have 1 6 2 � 2m, implies

jT 1j 6X

K2Th

h2Kk�2

K kPhf þ Duh �rphk20;K þ kr � uh þ ð1� 2mÞphk

20;K

� �

þX

j2EIðThÞkh1=2

k�1=2ðspht� srhuhtÞk2

0;j þ 2CLc�1

Xj2EðThÞ

kc1=2suhtk20;j þ

XK2Th

h2Kk�2

K kf �Phf k20;K

!1=2

�X

K2Th

2h�2K k2

Kknvk20;K þ krnvk

20;K þ 2kqk2

0;K

� �þ

Xj2EIðThÞ

kh�1=2k

1=2nvk20;j

!1=2

6

ffiffiffi2p

max 1;ffiffiffi2p

C1=2L c�1=2

� � XK2Th

g2K þOSCðf ;KÞ

!1=2

�X

K2Th

h�2K k2

Kknvk20;K þ krnvk

20;K þ kh�1=2

k1=2nvk

20;oK

� �þ ð2� 2mÞkqk2

0;X

!1=2

.

Recalling the approximation property (28) and the second bound in (24), gives

jT 1j 6ffiffiffi2p

CS maxð1;ffiffiffi2p

C1=2L c�1=2Þmaxð1;CIÞ

XK2Th

g2K þOSCðf ;KÞ

!1=2

. ð30Þ

Estimation of T2: We use the norm equivalence result from Proposition 4.1 and the fact that su?h t ¼ suht;thereby, we obtain


T 2 6 CP ðCCCS þ 1Þmaxð1; c1=2ÞX

j2EðThÞ

Zjh�1k

2 jsuhtj2 !1=2

6 CP ðCCCS þ 1Þmaxð1; c�1=2ÞX

j2EðThÞ

Zjc jsuhtj2

!1=2

6 CP ðCCCS þ 1Þmaxð1; c�1=2ÞX

K2Th

g2K

!1=2

. ð31Þ

Combining the estimates from (27), (30), and (31) completes the proof of Theorem 3.1.

4.2. Proof of Theorem 3.2

We present the proofs of the four assertions in Theorem 3.2 separately. The proofs of (a) and (c) areanalogous to the corresponding bounds derived in [23, Lemmas 3.4 and 3.5]; see also [16]. For the sakeof completeness, we present the main steps.

Assertion (a): We first consider the case a > 12. To this end, we set vK ¼ ðPhf þ Duh �rphÞUa

K . Then,using that �Du + $p = f in L2(K)2, we obtain by elementary manipulations

kvKU�a=2K k2

0;K ¼Z

KðPhf þ Duh �rphÞ � vK dx

¼Z

Kð�Dðu� uhÞ þ rðp � phÞÞ � vK dxþ

ZKðPhf � f Þ � vK dx

¼Z

Krðu� uhÞ : rvK dx�

ZKðp � phÞðr � vKÞdxþ

ZKðPhf � f Þ � vK dx

6 Cðkrðu� uhÞk0;K þ kp � phk0;KÞkrvKk0;K þ kðf �Phf ÞUa=2K k0;KkvKU�a=2

K k0;K .

From the proof of [23, Lemma 3.4], we have

krvKk20;K 6 Ck2ð1�aÞ

K k2Kh�2

K kvKU�a=2K k2

0;K .

Since kvKU�a=2K k0;K ¼ kKh�1

K ga;RK ;1, we readily obtain that

ga;RK ;1 6 C k1�aK ðkrðu� uhÞk0;K þ kp � phk0;KÞ þ k�1

K hKkf �Phf k0;K

� �. ð32Þ

This shows the assertion for a > 12. For a 2 0; 1

2

� , we first use that ga;RK ;1

6 Cekb�aK gb;RK ;1

, for b ¼ 12þ e with

e > 0, and apply the bound in (32) to gb;RK ;1.Assertion (b): In order to bound gRK,2, we simply use that $ Æ u + (1 � 2m)p = 0 in L2(K). This yields

g2RK ;2¼ kr � uh þ ð1� 2mÞphk

20;K ¼ kr � ðu� uhÞ þ ð1� 2mÞðp � phÞk

20;K

6 C krðu� uhÞk20;K þ kp � phk

20;K

� �.

Assertion (c): Again, we first consider the case a > 12

and let j be an edge shared by two elements K1 andK2. Set dj :¼ ðK1 [ K2Þ�; Lemma 2.6 of [23] ensures the existence of a function wj 2 H 1

0ðdjÞ2 withwjjj ¼ ðspht� srhuhtÞUa

j, wjjodj= 0 and


krwjk20;dj6 Ch�1

K rk2ð2�aÞK þ r�1

� �kðspht� sruhtÞUa=2

j k20;j;

kwjk20;dj6 ChKrkðspht� sruhtÞUa=2

j k20;j;

ð33Þ

for any r 2 (0, 1]. Using that s$ut � spt = 0 on interior edges and that �Du + $p = f on each element K, itcan be readily seen that

kðspht� sruhtÞUa=2j k

20;j ¼

Zjðspht� ½ruhtÞ � wj ds ¼

Zjðsph � pt� srðuh � uÞtÞ � wj ds

¼Z

oK1

ððph � pÞnK1Þ � wj dsþ

ZoK2

ððph � pÞnK2Þ � wj ds

�Z

oK1

ðrðuh � uÞ � nK1Þ � wj ds�

ZoK2

ðrðuh � uÞ � nK2Þ � wj ds

¼ �Z

dj

rhðuh � uÞ : rwj dxþZ

dj

ðph � pÞðr � wjÞdx

�Z

dj

ðf þ Duh �rphÞ � wj dx

6 C krhðu� uhÞk0;djþ kp � phk0;dj

� �krwjk0;dj

þ ðkPhf þ Duh �rphk0;djþ kf �Phf k0;dj

Þkwjk0;dj.

Here, nK1and nK2

denote the unit outward normal vectors on the boundaries oK1 and oK2, respectively. Bysumming up this estimate over all edges of a given element K, invoking the bounds for krwjk0;dj

andkwjk0;dj

from (33), and using assertion (a), we obtain

g2a;EK6 Cðk�1

K ðrk2ð2�aÞK þ r�1Þ þ k3

KrÞðkrhðu� uhÞk20;dKþ kp � phk

20;dKÞ þ Crk2ð1þeÞ

K k�2K h2

Kkf �Phf k20;dK

.

Setting r ¼ k�2K proves the assertion for a > 1

2. For a 2 0; 1

2

� , we have that ga;EK

6 Ckb�aK gb;EK

and use theabove argument for gb;EK

with b ¼ 12þ e to obtain the assertion.

Assertion (d): This is a simple consequence of the fact that the jump of u vanishes over interior edges andthat u = 0 on C. Hence, for j 2 EðThÞ, we have that

kkh�1=2suhtk20;j ¼ kkh�1=2su� uhtk2

0;j;

which completes the proof of Theorem 3.2.

5. Numerical experiments

In this section we present a series of numerical examples to illustrate the practical performance of theproposed a posteriori error estimator derived in Theorem 3.1, with a = 0, within an automatic hp-adaptiverefinement procedure which is based on employing 1-irregular quadrilateral elements. The hp-adaptivemeshes are constructed by first marking the elements for refinement/derefinement according to the sizeof the local error indicators g0;K; this is done by employing the fixed fraction strategy, with refinementand derefinement fractions set to 25% and 10%, respectively. Once an element K 2Th has been flaggedfor refinement or derefinement, a decision must be made whether the local mesh size hK or the local degree


kK of the approximating polynomial should be adjusted accordingly. The choice to perform either h-refine-ment/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], wherethe local regularity of the analytical solution is estimated from truncated local Legendre expansions ofthe computed numerical solution; see, also, [9,18].

Here, the emphasis will be to demonstrate the robustness of the proposed a posteriori error indicatorwith respect to the Poisson ratio m and the parameter c arising in the definition of the discontinuity stabil-ization function c, cf. (11). Indeed, we shall show that the proposed a posteriori error indicator convergesto 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 c and m. For simplicity, we set the constant CEST arising in Theorem3.1 equal to one and ensure that the corresponding effectivity indices are roughly constant on all of themeshes employed; here, the effectivity index is defined as the ratio of the a posteriori error bound andthe energy norm of the actual error. In general, to ensure the reliability of the error estimator, CEST mustbe determined numerically for the underlying problem at hand. In all of our numerical experiments, dataoscillation will be neglected within our computations. Finally, we note that for both of the numerical exam-ples considered in this section, closed form analytical solutions are not readily available; thereby, suitablyaccurate approximations have been computed on highly refined hp-meshes.

5.1. Example 1

Here, we let X be the unit square (0, 1)2; further, we select f = 0 and enforce the inhomogeneous bound-ary condition u ¼ ðg; 0Þ> on C, where

gðx; yÞ ¼ sin2ðpxÞ for ðx; yÞ 2 ð0; 1Þ � f1g;0 otherwise.

(

This represents a slight modification of the example considered in [30].In Fig. 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 Vh · Qh on a linear–log scale, for thesequence of meshes generated by our hp-adaptive algorithm. Here, numerical experiments are presented form = 0.4999, 0.45, 0.4 for different values of the parameter c arising in the definition of the discontinuity sta-bilization function c, cf. (11). We remark that the third root of the number of degrees of freedom is chosenon the basis of the a priori error analysis carried out in the article [26]. For each value of m, we observe thatthe error bound over-estimates the true error by a (reasonably) consistent factor for both c = 10, 100; in-deed, from Fig. 1, we see that all of the computed effectivity indices lie in the range between 3 and 5. Here,and in the following example, the mesh number is used to distinguish each of the hp-finite element meshesproduced by our adaptive mesh refinement algorithm; here, the initial hp-mesh has mesh number 1. More-over, we notice that the proposed mixed hp-DG method and our a posteriori error estimator are robust withrespect to both the Poisson ratio, as this approaches the critical value of 1/2, and the discontinuity stabil-ization function, as c is increased. Finally, from Fig. 1 we observe that the convergence lines using hp-refine-ment are (roughly) straight on a linear-log scale, which indicates that exponential convergence is attained inall cases.

In Figs. 2–4 we show the mesh generated using the proposed a posteriori error indicator after 9 hp-adap-tive refinement steps for m = 0.4999, 0.45, 0.4, respectively, with c = 10. For clarity, in each case we showthe h-mesh alone, as well as the corresponding distribution of the polynomial degree on this mesh. Form = 0.4999, we see that the mesh has been h-refined along the two sides, as well as the bottom of the domain,where (mild) boundary layers are located. Additionally, a horizontal line of h-refined elements occurs to-

5 10 15 20 25 3010-4

10-3

10-2

10-1

100

γ=10γ=100

Error Bounds

True Errors

(Degrees of Freedom)1/3

0 2 4 6 8 10 120

1

2

3

4

5

6

γ=10γ=100

Eff

ectiv

ity I

ndex

Mesh Number

5 10 15 20 25 30 3510-4

10-3

10-2

10-1

100

γ=10γ=100

Error Bounds

True Errors


0 2 4 6 8 10 120

1

2

3

4

5

6

γ=10γ=100

Eff

ectiv

ity I

ndex

Mesh Number

5 10 15 20 25 3010-4

10-3

10-2

10-1

100

γ=10γ=100

Error Bounds

True Errors


0 2 4 6 8 10 120

1

2

3

4

5

6

γ=10γ=100

Eff

ectiv

ity I

ndex

Mesh Number

(a)

(b)

(c)

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. (a) m = 0.4999, (b) m = 0.45, and (c) m = 0.4.


wards the top of the domain, which coincides with the location of the center of the recirculation of the dis-placement vector. Once the h-mesh has adequately captured the structure of the solution, the hp-adaptivealgorithm increased the degree of the approximating polynomial within the rest of the computational do-main. Although, for m = 0.45, 0.4, we see a fairly similar structure in the hp-mesh distribution to that for

4444443

3

3

33

4

4

4

44

4

222

2

4

4

4

4

4

4 4

4

3

3 2

3

4

4

3

3

33

3333

33

3 333

33333333

2

3

3

3

3

3

3 3

333

3 3

3

3

2

33

3 3

3 3 3 3

3

33

3

3

3

33

3

3

3

3 3

3

3

3

3

3 3

3

33

3 3

3 3

33

44

4 4

2 222

4

44

4

3

3 3

3

4

44

4

3 333

333

3

3

33

3 3 3

33

33

3 333

3

3

223

33 333

333

33 333

2 222

333

3

3

33

3

22 2

2

44

4 4

2 222

22 2

2

22222222

222 2

222

2

2222

2222

2 222

222

2

2222

2222

222

22 222

222 2

222

2

222 2

2 222

22 2

2

2 222

22 2

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2

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2222

222 2

222

2

22 2

2

2 222

2222

222 2

222

2

2 222

222

2

22222222

2222

2222

2222

4

33

4

33

4 4

2 222

222

2

44 4

432232222

222222222222

444 43322

2222

222222222222

(a) (b)

Fig. 2. Example 1. h- and hp-meshes after 9 hp-adaptive refinements, with 409 elements and 13,820 degrees of freedom, for m = 0.4999and c = 10.

4

44

44

44

43

3

3 333 4

4

3

4

3 3

43

33

4

4 4

4

4 3

44

4

44

3

444444

4 4

3 3

33

4

44

4

3 3 33

3

33

33 3

33

3

33

3

33

3

3 3

33

3 3

3

3333

3

3

4 3

3

3333

3

33

3

3

3

3

4

44

4

3 333

3

33

3

3

3 3

33

3333

33 3

3

333 332 2

3

2 222

33

3

33

3

332 2

3 3

222

2

222 2

333

2 22

22 2

2 333

222

222 2

333322223223

44 4

4 3333222222222222 44

4 43333

2222

222

2

3 3

33

222

22 222

222 2

222

2

2 222

222

2222

2

2 222

222

2

222 2

22 2

2

222 23

3 3

3 22 2

2

222

2

2 222

222

2 2 222

222

2

22 2

2

2 222

22 2

2

2 222

33

3 3

222 2

444 4

3333

22 2

2

2 222

2 222

22 2

2

3333

222 2

222

2

222

22 222

232222222222222

3 3

33 222 2

222 2

222

22 222

44

4 4332

2

4 4

44

3 333

2222 2222

4

44

4

333

3

2222 2222

33 3

3

4

4 4

4

3 223

22 2

2

4

4 4

4

222222222222

2222

333

3

2222 2222

3 333

2222 2222

333

3

2222 2222

4 4

44

3 333

2222 222222222222

22222222

22222222

44

4 4

44

4 422222222222222222222

4444

4 444

444

4

(a) (b)

Fig. 3. Example 1. h- and hp-meshes after 9 hp-adaptive refinements, with 598 elements and 20,699 degrees of freedom, for m = 0.45and c = 10.


m = 0.4999, there are a couple of noticeable differences. Firstly, the imposition of the second equation in (3)leads to the development of a singularity in p at the corners (0,1) and (1,1) of X. Indeed, as the boundaryconditions lead to $ Æ u = 0 at (0, 1) and (1, 1), the divergence of the displacement u is expected to be small inthe regions adjacent to these corners. However, since the term 1 � 2m in the second equation of (3) is smallas well, p is not necessarily close to zero in the vicinity of these vertices. Consequently, this leads to addi-tional h-refinement in the region containing these singularities of p. Secondly, as m decreases, the center ofthe recirculation of the displacement vector moves downwards, leading to the corresponding horizontalstrip of h-refined elements being slightly lower in each case.

33

4

44

44

44

4

4 3

44

4

4 4

4

43

4

4 4

4

4

44

34 4

44

3 333

44

4 4

343

3 3

3

3

44

3

33

33

3

3 333

3

3

3

3 3

4

44

4

443

3

3

3

3 33

3 3

3 3

3

3 3

3 33

3 3

4 334

33

3 3

3

3 3

333

3 3

3

33

3

4 4

33

3 3

33

33 3

3 3333

3333

3 3

33 3 3

33

3 333

333

33 333

333

3

3

33

3

22 2

23 3

33

333 3

33 3

3333 3

33 3

3333

322

2 2

3 3

33

3

33

3

333223

33332222 333333223 44

4 433332222

232

222 2

22 2

2

2 222

222

2

22 2

23 3

33

222 2

22 2

2

2 222

222

2

222 2

22 2

2

222

2

222 2

2 222

222 2

22 2

2

2 222

222

2

22 2

2

2 222

222

2

222 2

222

2

222 2

3 3

33 22 2

2

2 222

2 222 3

33

3

222 2

22 2

2

2 222

222

2

22 2

2 222 2

222

22 222

332222322222222

3 333

333

3

222

22 222

222

2

222 2

22 2

22 2

22222

2

222 2

22 2

2

2 222

222

222

2 222 2

2

3333 33334

44

4

333

333

3 3

2222

44

4 4

44

4 422222222222222222222

(a) (b)

Fig. 4. Example 1. h- and hp-meshes after 9 hp-adaptive refinements, with 514 elements and 17,824 degrees of freedom, for m = 0.4 andc = 10.

5 10 15 20 25 30 35 40

10-5

10-4

10-3

γ=10γ=100

Error Bounds

True Errors


0 2 4 6 8 10 120

1

2

3

4

5

6

γ=10γ=100

Eff

ectiv

ity I

ndex

Mesh Number

(a)

5 10 15 20 25 30 35

10-3

10-2

10-1

γ=10γ=100

Error Bounds

True Errors

(Degrees of Freedom)1/30 2 4 6 8 10 12

0

1

2

3

4

5

6

γ=10γ=100

Eff

ectiv

ity I

ndex

Mesh Number(b)

Fig. 5. Example 2. 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. (a) m = 0.4999 and (b) m = 0.45.



5.2. Example 2

In this section, we consider the example of a singular solution in a non-convex domain. To this end, welet X be the L-shaped domain (�1,1)2n([0,1) · (�1,0]), set f = 1, and impose homogeneous Dirichletboundary conditions for u on the whole of C. We emphasize that the analytical solution (u,p) is analyticin X, but $u exhibits singularities at the corners of X.

In Fig. 5, we show the history of the actual and estimated energy norm of the error on each of the meshesgenerated by our hp-adaptive algorithm for m = 0.4999, 0.45 and c = 10, 100. As in the previous example,we observe that the a posteriori bound over-estimates the true error by a consistent factor between 3 and 6,though here we do see, that for this example, the effectivity indices do initially grow very slightly as the meshis refined in all but one of the cases considered; though, asymptotically they seem to be tending towards aconstant value. Additionally, we clearly observe that both our mixed hp-DG method and the correspondinga posteriori error estimator are free from volume locking. For both values of the Poisson ratio, we observeexponential convergence of the energy norm of the error using hp-refinement: for each value of m, on a lin-ear–log scale, the convergence lines are on average straight for c = 10, 100.

Finally, in Fig. 6 we show the mesh generated using the local error indicators g0;K after 6 hp-adaptiverefinement steps for m = 0.4999 and c = 10. Here, we see that the h-mesh has been largely refined in thevicinity of the corners of the domain; in particular, stronger refinement has occurred in the vicinity ofthe reentrant corner located at the origin, as well as in the region adjacent to this singular point. Fromthe zoom of the hp-mesh in the vicinity of the origin, we see that this h-refinement has occurred in the diag-onal direction x = y, while in the other diagonal direction, x = �y, p-refinement has been performed, cf.[16]. Away from the corners, we see that the polynomial degrees have been increased, since the underlyinganalytical solution is smooth in this region. An analogous hp-refined mesh is also generated for the casewhen m = 0.45; for brevity, we omit the details.

Fig. 6. Example 2. hp-mesh after 6 hp-adaptive refinements, with 462 elements and 13,780 degrees of freedom, for m = 0.4999 andc = 10.


6. Conclusions

In this article, we have derived both upper and lower residual-based energy norm a posteriori errorbounds for mixed hp-DG approximations to the equations governing linear elasticity in two spatial dimen-sions. The analysis is based on employing a non-consistent reformulation of the hp-DG method using liftingoperators, together with a decomposition result for the underlying discontinuous space. We emphasize thatour error analysis is robust with respect to the Poisson ratio; indeed, numerical experiments presented inthis article clearly demonstrate that both the proposed hp-DG method and the corresponding a posterioriestimator are free from volume locking. The extension of the current analysis to three-dimensional prob-lems follows analogously for the h-version of the DG method, based on employing the decomposition resultfor DG spaces derived in the articles [14,20]; the extension of this result to hp-DG methods in 3D is thesubject of future research. Additional future work will be devoted to the extension of our analysis to irreg-ular meshes and, in particular, anisotropic hp-adaptive discontinuous Galerkin methods.

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An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity