Tomás P. Barrios 1, Gabriel N. Gatica 2,María González 3 and Norbert Heuer Introduction

Transcription

1 ESAIM: M2AN Vol. 40, N o 5, 2006, pp DOI: /m2an: ESAIM: Mathematical Modelling Numerical Analysis A RESIDUAL BASED A POSERIORI ERROR ESIMAOR FOR AN AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY omás P. Barrios 1, Gabriel N. Gatica 2,María González 3 Norbert Heuer 4 Abstract. In this paper we develop a residual based a posteriori error analysis for an augmented mixed finite element method applied to the problem of linear elasticity in the plane. More precisely, we derive a reliable efficient a posteriori error estimator for the case of pure Dirichlet boundary conditions. In addition, several numerical experiments confirming the theoretical properties of the estimator, illustrating the capability of the corresponding adaptive algorithm to localize the singularities the large stress regions of the solution, are also reported. Mathematics Subject Classification. 65N15, 65N30, 65N50, 74B05. Received: November 9, Introduction A new stabilized mixed finite element method for plane linear elasticity was presented analyzed recently in [10]. he approach there is based on the introduction of suitable Galerkin least-squares terms arising from the constitutive equilibrium equations, from the relation defining the rotation in terms of the displacement. he resulting augmented method, which is easily generalized to 3D, can be viewed as an extension to the elasticity problem of the non-symmetric procedures utilized in [8] [11]. It is shown in [10] that the continuous discrete augmented formulations are well-posed, that the latter becomes locking-free asymptotically locking-free for Dirichlet mixed boundary conditions, respectively. Moreover, the augmented variational formulation introduced in [10], being strongly coercive in the case of Dirichlet boundary conditions, allows the utilization of arbitrary finite element subspaces for the corresponding discrete scheme, which constitutes one of its main advantages. In particular, Raviart-homas spaces of lowest order for the stress tensor, piecewise linear elements for the displacement, piecewise constants for the rotation can be used. In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, which yields the introduction of Keywords phrases. Mixed finite element, augmented formulation, a posteriori error estimator, linear elasticity. his research was partially supported by CONICY-Chile through the FONDAP Program in Applied Mathematics, by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program, by the Universidade da Coruña through the Research Grants Program. 1 Facultad de Ingeniería, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile. tomas@ucsc.cl 2 GI 2 MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. ggatica@ing-mat.udec.cl 3 Departamento de Matemáticas, Universidade da Coruña, Campus de Elviña s/n, A Coruña, Spain. mgtaboad@udc.es 4 BICOM Department of Mathematical Sciences, Brunel University, Uxbridge, UB8 3PH, UK. norbert.heuer@brunel.ac.uk c EDP Sciences, SMAI 2007 Article published by EDP Sciences available at or

2 844.P. BARRIOS E AL. the trace of the displacement as a suitable Lagrange multiplier. his trace is then approximated by piecewise linear elements on an independent partition of the Neumann boundary whose mesh size needs to satisfy a compatibility condition with the mesh size associated with the triangulation of the domain. Further details on the advantages of the augmented method can be found in [10] also throughout the present paper (see, in particular, Sect. 5 below). According to the above, strongly motivated by the competitive character of our augmented formulation, we now feel the need of deriving corresponding a posteriori error estimators. More precisely, the purpose of this work is to develop a residual based a posteriori error analysis for the augmented mixed finite element scheme from [10] in the case of pure Dirichlet boundary conditions. A posteriori error analyses of the traditional mixed finite element methods for the elasticity problem can be seen in [5] the references therein. he rest of this paper is organized as follows. In Section 2 we recall from [10] the continuous discrete augmented formulations of the corresponding boundary value problem, state the well-posedness of both schemes, provide the associated apriorierror estimate. he kernel of the present work is given by Sections 3 4, where we develop the residual based a posteriori error analysis. Indeed, in Section 3 we employ a suitable auxiliary problem apply integration by parts the local approximation properties of the Clément interpolant to derive a reliable a posteriori error estimator. In other words, the method that we use to prove reliability combines a technique utilized in mixed finite element schemes with the usual procedure applied to primal finite element methods. It is important to remark that just one of these approaches by itself would not be enough in this case. In addition, up to our knowledge, this combined analysis seems to be applied here for the first time. Next, in Section 4 we make use of inverse inequalities the localization technique based on triangle-bubble edge-bubble functions to show that the estimator is efficient. We remark that, because of the new Galerkin least-squares terms employed, most of the residual terms defining the error indicator are new, hence our proof of efficiency needs to previously establish more general versions of some technical lemmas concerning inverse estimates piecewise polynomials. Finally, several numerical results confirming reliability, efficiency, robustness of the estimator with respect to the Poisson ratio, are provided in Section 5. In addition, the capability of the corresponding adaptive algorithm to localize the singularities the large stress regions of the solution is also illustrated here. We end this section with some notations to be used below. Given any Hilbert space U, U 2 U 2 2 denote, respectively, the space of vectors square matrices of order 2 with entries in U. In addition, I is the identity matrix of R 2 2,givenτ := (τ ij ), ζ := (ζ ij ) R 2 2, we write as usual τ t := (τ ji ), tr(τ ):= 2 τ ii, τ d := τ tr(τ ) I, τ : ζ := τ ij ζ ij. i=1 i,j=1 Also, in what follows we utilize the stard terminology for Sobolev spaces norms, employ 0 to denote a generic null vector, use C c, with or without subscripts, bars, tildes or hats, to denote generic constants independent of the discretization parameters, which may take different values at different places. 2. he augmented formulations First we let be a simply connected domain in R 2 with polygonal boundary Γ :=. Our goal is to determine the displacement u stress tensor σ of a linear elastic material occupying the region. In other words, given a volume force f [L 2 ()] 2, we seek a symmetric tensor field σ a vector field u such that σ = C e(u), div(σ) = f in, u = 0 on Γ. (1) Hereafter, e(u) := 1 2 ( u ( u)t ) is the strain tensor of small deformations C is the elasticity tensor

3 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 845 determined by Hooke s law, that is C ζ := λ tr(ζ) I 2µ ζ ζ [L 2 ()] 2 2, (2) where λ, µ > 0 denote the corresponding Lamé constants. It is easy to see from (2) that the inverse tensor C 1 reduces to C 1 ζ := 1 2 µ ζ λ 4 µ (λ µ) tr(ζ) I ζ [L2 ()] 2 2. (3) We now define the spaces H = H(div ;) := {τ [L 2 ()] 2 2 : div(τ ) [L 2 ()] 2 }, H 0 := {τ H : tr(τ ) = 0}, note that H = H 0 R I, that is for any τ H there exist unique τ 0 H 0 d := 1 2 tr(τ ) R such that τ = τ 0 d I. In addition, we define the space of skew-symmetric tensors [L 2 ()] 2 2 skew := { η [L 2 ()] 2 2 : ηη t = 0} introduce the rotation γ := 1 2 ( u ( u)t )) [L 2 ()] 2 2 skew as an auxiliary unknown. hen, given positive parameters κ 1, κ 2,κ 3, independent of λ, we consider from [10] the following augmented variational formulation for (1): Find (σ, u, γ) H 0 := H 0 [H0 1 ()] 2 [L 2 ()] 2 2 skew such that A((σ, u, γ), (τ, v, η)) = F (τ, v, η) (τ, v, η) H 0, (4) where the bilinear form A : H 0 H 0 R the functional F : H 0 R are defined by A((σ, u, γ), (τ, v, η)) := C 1 σ : τ u div(τ ) γ : τ v div(σ) η : σ ( κ 1 e(u) C 1 σ ) : ( e(v)c 1 τ ) κ 2 div(σ) div(τ ) ( κ 3 γ 1 ) ( 2 ( u ( u)t ) : η 1 ) 2 ( v ( v)t ), (5) F (τ, v, η) := f ( v κ 2 div(τ )). (6) he well-posedness of (4) was proved in [10]. More precisely, we have the following result. heorem 2.1. Assume that (κ 1,κ 2,κ 3 ) is independent of λ such that 0 < κ 1 < 2 µ, 0 < κ 2, 0 <κ 3 <κ 1. hen, there exist positive constants M, α, independent of λ, such that A((σ, u, γ), (τ, v, η)) M (σ, u, γ) H0 (τ, v, η) H0 (7) A((τ, v, η), (τ, v, η)) α (τ, v, η) 2 H 0 (8) for all (σ, u, γ), (τ, v, η) H 0. In particular, taking κ 1 = C 1 µ, κ 2 = 1 µ ( 1 κ ) 1, κ 3 = 2 µ C 3 κ 1, (9) with any C 1 ]0, 2[ any C 1 3 ]0, 1[, thisyieldsm α depending only on µ, µ,. herefore, the augmented variational formulation (4) has a unique solution (σ, u, γ) H 0, there exists a positive constant C, independent of λ, such that (σ, u, γ) H0 C F C f [L 2 ()] 2. Proof. See heorems in [10].

4 846.P. BARRIOS E AL. Now, given a finite element subspace H 0,h (σ h, u h, γ h ) H 0,h such that H 0, the Galerkin scheme associated with (4) reads: Find A((σ h, u h, γ h ), (τ h, v h, η h )) = F (τ h, v h, η h ) (τ h, v h, η h ) H 0,h, (10) where κ 1, κ 2,κ 3, being the same parameters employed in the formulation (4), satisfy the assumptions of heorem 2.1. Since A becomes bounded strongly coercive on the whole space H 0, we remark that the well posedness of (10) is guaranteed for any arbitrary choice of the subspace H 0,h. In fact, the following result is also established in [10]. heorem 2.2. Assume that the parameters κ 1, κ 2,κ 3 satisfy the assumptions of heorem 2.1 let H 0,h be any finite element subspace of H 0. hen, the Galerkin scheme (10) has a unique solution (σ h, u h, γ h ) H 0,h, there exist positive constants C, C, independent of h λ, such that (σ h, u h, γ h ) H0 C sup (τ h,v h,η h ) H 0,h (τ h,v h,η h ) 0 F (τ h, v h, η h ) (τ h, v h, η h ) H0 C f [L 2 ()] 2, (σ, u, γ) (σ h, u h, γ h ) H0 C inf (σ, u, γ) (τ h, v h, η (τ h,v h,η h ) H0. (11) h ) H 0,h Proof. It follows from heorem 2.1, Lax-Milgram s Lemma, Céa s estimate. It is important to emphasize here that the main advantage of the augmented approach (10), as compared with the traditional mixed finite element schemes for the linear elasticity problem (see e.g. [3]), is the possibility of choosing any finite element subspace H 0,h of H 0. On the other h, an inmediate consequence of the definition of the continuous discrete augmented formulations is the Galerkin orthogonality A((σ σ h, u u h, γ γ h ), (τ h, v h, η h )) = 0 (τ h, v h, η h ) H 0,h. (12) Next, we recall the specific space H 0,h introduced in [10], which is the simplest finite element subspace of H 0. o this end, we first let { h } h>0 be a regular family of triangulations of the polygonal region by triangles of diameter h with mesh size h := max{ h : }, such that there holds = { : }. In addition, given an integer l 0 a subset S of R 2,wedenotebyP l (S) the space of polynomials in two variables defined in S of total degree at most l, foreach we introduce the local Raviart-homas space of order zero (cf. [3, 12]), {( ) ( ) ( )} 1 0 x 1 R 0 ( ):=span,, [P 1 ( )] 2, 0 1 x 2 where ( x 1 x 2 ) is a generic vector of R 2. hen, defining H σ h := { τ h H(div ;): τ h [R 0 ( ) t ] 2 }, (13) X h := { v h C( ) : v h P 1 ( ) }, (14)

5 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 847 H u h := X h X h, (15) we take H 0,h := H σ 0,h H u 0,h H γ h, (16) where H σ { 0,h := τ h H σ } h : tr(τ h )=0, (17) H0,h u := { v h Hh u : v h = 0 on Γ }, (18) H γ h := { η h [L 2 ()] 2 2 skew : η h [P 0 ( )] 2 2 }. (19) he following theorem provides the rate of convergence of (10) when the specific finite element subspace (16) is utilized. heorem 2.3. Let (σ, u, γ) H 0 (σ h, u h, γ h ) H 0,h := H σ 0,h Hu 0,h Hγ h betheuniquesolutionsofthe continuous discrete augmented mixed formulations (4) (10), respectively. Assume that σ [H r ()] 2 2, div(σ) [H r ()] 2, u [H r1 ()] 2,γ [H r ()] 2 2, for some r (0, 1]. hen there exists C > 0, independent of h λ, such that (σ, u, γ) (σ h, u h, γ h ) H0 Ch r { σ [H r ()] 2 2 div(σ) [H r ()] 2 u [H r1 ()] 2 γ [H r ()] 2 2 }. Proof. It is a consequence of Céa s estimate, the approximation properties of the subspaces defining H 0,h, suitable interpolation theorems in the corresponding function spaces. See Section 4.1 in [10] for more details. 3. A residual based A POSERIORI error estimator In this section we derive a residual based a posteriori error estimator for (10). First we introduce several notations. Given,weletE()bethesetofitsedges,letE h be the set of all edges of the triangulation h.henwewritee h = E h () E h (Γ), where E h () := {e E h : e } E h (Γ) := {e E h : e Γ}. In what follows, h e sts for the length of edge e E h. Further, given τ [L 2 ()] 2 2 (such that τ C( ) on each ), an edge e E( ) E h (), the unit tangential vector t along e, weletj[τ t ]bethe corresponding jump across e, thatis,j[τt ]:=(τ τ ) e t,where is the other triangle of h having e as an edge. Abusing notation, when e E h (Γ), we also write J[τ t ]:=τ e t. We recall here that t := ( ν 2,ν 1 ) t where ν := (ν 1,ν 2 ) t is the unit outward normal to. Analogously, we define the normal jumps J[τν ]. In addition, given scalar, vector, tensor valued fields v, ϕ := (ϕ 1,ϕ 2 ), τ := (τ ij ), respectively, we let curl(v) := ( v x 2 v x 1 ), curl(ϕ) := ( curl(ϕ 1 ) t curl(ϕ 2 ) t ), curl(τ ):= ( τ12 x 1 τ 22 x 1 τ11 x 2 τ21 x 2 ).

6 848.P. BARRIOS E AL. hen, for (σ, u, γ) H 0 (σ h, u h, γ h ) H 0,h being the solutions of the continuous discrete formulations (4) (10), respectively, we define an error indicator θ as follows: θ 2 := f div (σ h ) 2 [L 2 ( )] σ 2 h σ t h 2 [L 2 ( )] γ 2 2 h 1 2 ( u h ( u h ) t ) 2 [L 2 ( )] 2 2 } h 2 { curl(c 1 σ h γ h ) 2[L2( )]2 curl(c 1 (e(u h ) C 1 σ h )) 2[L2( )]2 { } h e J[(C 1 σ h u h γ h )t ] 2 [L 2 (e)] 2 J[(C 1 (e(u h ) C 1 σ h ))t ] 2 [L 2 (e)] 2 e E( ) h 2 div (e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t )) 2 [L 2 ( )] 2 h 2 div (γ h 1 2 ( u h ( u h ) t )) 2 [L 2 ( )] 2 e E( ) E h () e E( ) E h () h e J[(e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t ))ν ] 2 [L 2 (e)] 2 h e J[(γ h 1 2 ( u h ( u h ) t ))ν ] 2 [L 2 (e)] 2. (20) he residual character of each term on the right-h side of (20) is quite clear. In addition, we observe that some of these terms are known from residual estimators for the non-augmented mixed finite element method in linear elasticity (see e.g. [5]), but most of them are new since, as we show below, they arise from the new Galerkin least-squares terms introduced in the augmented formulation. We also mention that, as usual, the expression θ := { } θ 2 1/2 is employed as the global residual error estimator. he following theorem is the main result of this paper. heorem 3.1. Let (σ, u, γ) H 0 (σ h, u h, γ h ) H 0,h be the unique solutions of (4) (10), respectively. hen there exist C eff,c rel > 0, independent of h λ, such that C eff θ (σ σ h, u u h, γ γ h ) H0 C rel θ. (21) he so-called efficiency (lower bound in (21)) is proved below in Section 4 the reliability estimate (upper bound in (21)) is derived throughout the rest of the present section. he method that we use to prove reliability combines a procedure employed in mixed finite element schemes (see e.g. [4, 5]), where an auxiliary problem needs to be defined, with the integration by parts Clément interpolant technique usually applied to primal finite element methods (see [13]). We emphasize that just one of these approaches by itself would not suffice. We begin with the following preliminary estimate. Lemma 3.1. here exists C>0, independent of h λ, such that C (σ σ h, u u h, γ γ h ) H0 sup 0 (τ,v,η) H 0 div (τ )=0 A((σ σ h, u u h, γ γ h ), (τ, v, η)) (τ, v, η) H0 f div (σ h ) [L2 ()] 2. (22) Proof. Let us define σ = e(z), where z [H0 1()]2 is the unique solution of the boundary value problem: div (e(z)) = f div (σ h ) in, z = 0 on Γ. It follows that σ H 0, the corresponding continuous dependence result establishes the existence of c>0 such that σ H(div ;) c f div (σ h ) [L 2 ()] 2. (23)

7 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 849 In addition, it is easy to see that div (σ σ h σ )=0 in. hen, using the coercivity of A (cf. (8)), we find that α (σ σ h σ, u u h, γ γ h ) 2 H 0 A((σ σ h σ, u u h, γ γ h ), (σ σ h σ, u u h, γ γ h )) = A((σ σ h, u u h, γ γ h ), (σ σ h σ, u u h, γ γ h )) A((σ, 0, 0), (σ σ h σ, u u h, γ γ h )), which, employing the boundedness of A (cf. (7)), yields α (σ σ h σ, u u h, γ γ h ) H0 A((σ σ h, u u h, γ γ sup h ), (τ, v, η)) M σ H(div ;). (24) 0 (τ,v,η) H 0 (τ, v, η) H0 div (τ )=0 Hence, (22) follows straightforwardly from the triangle inequality, (23), (24). It remains to bound the first term on the right-h side of (22). o this end, we will make use of the well known Clément interpolation operator I h : H 1 () X h (cf. [7]), with X h given by (14), which satisfies the stard local approximation properties stated below in Lemma 3.2. It is important to remark that I h is defined in [7] so that I h (v) X h H 1 0 () for all v H1 0 (). Lemma 3.2. here exist constants c 1,c 2 > 0, independent of h, such that for all v H 1 () there holds v I h (v) L2 ( ) c 1 h v H1 ( ( )) h, v I h (v) L 2 (e) c 2 h 1/2 e v H 1 ( (e)) e E h, where ( ):= { h : }, (e) := { h : e }. Proof. See [7]. We now let (τ, v, η) H 0,(τ, v, η) 0, be such that div (τ )=0 in. Since is connected, there exists a stream function ϕ := (ϕ 1,ϕ 2 ) [H 1 ()] 2 such that ϕ 1 = ϕ 2 =0τ = curl(ϕ). hen, denoting ϕ h := (ϕ 1,h,ϕ 2,h ), with ϕ i,h := I h (ϕ i ), i {1, 2}, theclément interpolant of ϕ i, we define τ h := curl(ϕ h ). Note that there holds the decomposition τ h = τ h,0 d h I,whereτ h,0 H σ 0,h d h = tr(τ h) 2 R. Fromthe orthogonality relation (12) it follows that A((σ σ h, u u h, γ γ h ), (τ, v, η)) = A((σ σ h, u u h, γ γ h ), (τ τ h,0, v v h, η)), (25) where v h := (I h (v 1 ),I h (v 2 )) H0,h u is the vector Clément interpolant of v := (v 1,v 2 ) [H0 1()]2. Since tr(σ σ h)=0u u h = 0 on Γ, we deduce, using the orthogonality between symmetric skewsymmetric tensors, that A((σ σ h, u u h, γ γ h ), (d h I, 0, 0)) = 0. Hence, (25) (4) give A((σ σ h, u u h, γ γ h ), (τ, v, η)) = A((σ σ h, u u h, γ γ h ), (τ τ h, v v h, η)) = F (τ τ h, v v h, η) A((σ h, u h, γ h ), (τ τ h, v v h, η)).

8 850.P. BARRIOS E AL. According to the definitions of the forms A F (cf. (5), (6)), noting that div (τ τ h )=div curl(ϕ ϕ h )= 0, using again the above mentioned orthogonality, we find, after some algebraic manipulations, that A((σ σ h, u u h, γ γ h ), (τ, v, η)) = (f div (σ h )) (v v h ) { ( 1 2 (σ h σ t h ) κ 3 γ h 1 )} 2 ( u h ( u h ) t ) : η { } (C 1 σ h u h γ h )κ 1 C 1 (e(u h ) C 1 σ h ) :(τ τ h ) ( {κ 1 e(u h ) 1 ) ( 2 (C 1 σ h (C 1 σ h ) t ) κ 3 γ h 1 )} 2 ( u h ( u h ) t ) : (v v h ). (26) he rest of the proof of reliability consists in deriving suitable upper bounds for each of the terms appearing on the right-h side of (26). We begin by noticing that direct applications of the Cauchy-Schwarz inequality give 1 2 (σ h σ t h):η σ h σ t h [L2 ()] η 2 2 [L 2 ()] 2 2, (27) (γ h 1 2 ( u h ( u h ) t )) : η γh 1 2 ( u h ( u h ) t ) [L2 η ()] 2 2 [L 2 ()] 2 2. (28) he decomposition = h the integration by parts formula on each element are employed next to hle the terms from the third fourth rows of (26). We first replace (τ τ h )bycurl(ϕ ϕ h ) use that curl( u h )=0 in each triangle,toobtain (C 1 σ h u h γ h ):(τ τ h )= = (C 1 σ h u h γ h ):curl(ϕ ϕ h ) curl(c 1 σ h γ h ) (ϕ ϕ h ) e E h J[(C 1 σ h u h γ h )t ], ϕ ϕ h [L 2 (e)] 2, (29) C 1 (e(u h ) C 1 σ h ):(τ τ h )= C 1 (e(u h ) C 1 σ h ):curl(ϕ ϕ h ) = h curl(c 1 (e(u h ) C 1 σ h )) (ϕ ϕ h ) J[(C 1 (e(u h ) C 1 σ h ))t ], ϕ ϕ h [L2 (e)] 2. (30) e E h

9 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 851 On the other h, using that v v h = 0 on Γ, we easily get (e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t )) : (v v h ) = div (e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t )) (v v h ) (γ h 1 2 ( u h ( u h ) t )) : (v v h ) = e E h () J[(e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t ))ν ], v v h [L 2 (e)] 2, (31) div (γ h 1 2 ( u h ( u h ) t )) (v v h ) e E h () J[(γ h 1 2 ( u h ( u h ) t ))ν ], v v h [L 2 (e)] 2. (32) In what follows, we apply again the Cauchy-Schwarz inequality, Lemma 3.2, the fact that the numbers of triangles in ( ) (e) are bounded, independently of h, to derive the estimates for the expression (f div σ h ) (v v h ) in (26) the right-h sides of (29), (30), (31), (32), with constants C independent of h λ. Indeed, we easily have (f div σ h ) (v v h ) f div σ h [L2 ( )] 2 v v h [L2 ( )] 2 c 1 C f div σ h [L2 ( )] 2 h v [H1 ( ( )] 2 { h 2 f div σ h 2 [L 2 ( )] 2 In addition, for the terms containing the stream function ϕ (cf. (29), (30)), we get c 1 C } 1/2 v [H1()]2. (33) curl(c 1 σ h γ h ) (ϕ ϕ h ) curl(c 1 σ h γ h ) [L2 ( )] 2 ϕ ϕ h [L 2 ( )] 2 curl(c 1 σ h γ h ) [L2 ( )] 2 h ϕ [H1 ( ( )] 2 { h 2 curl(c 1 σ h γ h ) 2 [L 2 ( )] 2 } 1/2 ϕ [H1()]2, (34)

10 852.P. BARRIOS E AL. curl(c 1 (e(u h ) C 1 σ h )) (ϕ ϕ h ) h curl(c 1 (e(u h ) C 1 σ h )) [L 2 ( )] 2 ϕ ϕ h [L 2 ( )] 2 c 1 curl(c 1 (e(u h ) C 1 σ h )) [L 2 ( )] 2 h ϕ [H 1 ( ( )] 2 { } 1/2 C h 2 curl(c 1 (e(u h ) C 1 σ h )) 2 [L 2 ( )] ϕ, (35) 2 [H1()]2 J[(C 1 σ h u h γ h )t ], ϕ ϕ h [L2 (e)] 2 e E h J[(C 1 σ h u h γ h )t ] [L2 (e)] 2 ϕ ϕ h [L2 (e)] 2 e E h c 2 J[(C 1 σ h u h γ h )t ] [L 2 (e)] 2 h1/2 e ϕ [H 1 ( (e))] 2 e E h { } 1/2 C h e J[(C 1 σ h u h γ h )t ] 2 [L 2 (e)] ϕ, (36) 2 [H1()]2 e E h J[(C 1 (e(u h ) C 1 σ h ))t ], ϕ ϕ h [L2 (e)] 2 e E h J[(C 1 (e(u h ) C 1 σ h ))t ] [L2 (e)] 2 ϕ ϕ h [L2 (e)] 2 e E h c 2 J[(C 1 (e(u h ) C 1 σ h ))t ] [L 2 (e)] 2 h1/2 e ϕ [H 1 ( (e))] 2 e E h { } 1/2 C h e J[(C 1 (e(u h ) C 1 σ h ))t ] 2 [L 2 (e)] ϕ. (37) 2 [H1()]2 e E h We observe here, thanks to the equivalence between ϕ [H 1 ()] 2 ϕ [H 1 ()] 2,that ϕ [H 1 ()] 2 C ϕ [H 1 ()] 2 = C curl(ϕ) [L 2 ()] 2 = C τ H(div ;), (38) which allows to replace ϕ [H 1 ()] 2 by τ H(div ;) in the above estimates (34) (37).

11 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 853 Similarly, for the terms on the right-h side of (31) (32), we find that div (e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t )) (v v h ) div (e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t )) [L2 ( )] 2 v v h [L2 ( )] 2 c 1 div (e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t )) [L2 ( )] 2 h v [H1 ( ( )] 2 { } 1/2 C h 2 div (e(u h) 1 2 (C 1 σ h (C 1 σ h ) t )) 2 [L 2 ( )] v, (39) 2 [H1()]2 div (γ h 1 2 ( u h ( u h ) t )) (v v h ) div (γ h 1 2 ( u h ( u h ) t )) [L2 ( )] 2 v v h [L2 ( )] 2 c 1 div (γ h 1 2 ( u h ( u h ) t )) [L 2 ( )] 2 h v [H 1 ( ( )] 2 { } 1/2 C h 2 div (γ h 1 2 ( u h ( u h ) t )) 2 [L 2 ( )] v, (40) 2 [H1()]2 e E h () J[(e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t ))ν ], v v h [L2 (e)] 2 J[(e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t ))ν ] [L 2 (e)] 2 v v h [L 2 (e)] 2 e E h () c 2 C e E h () e E h () J[(e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t ))ν ] [L 2 (e)] 2 h1/2 e v [H 1 ( (e))] 2 h e J[(e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t ))ν ] 2 [L 2 (e)] 2 1/2 v [H1 ()] 2, (41)

12 854.P. BARRIOS E AL. e E h () J[(γ h 1 2 ( u h ( u h ) t ))ν ], v v h [L 2 (e)] 2 J[(γ h 1 2 ( u h ( u h ) t ))ν ] [L2 (e)] 2 v v h [L2 (e)] 2 e E h () c 2 C e E h () e E h () J[(γ h 1 2 ( u h ( u h ) t ))ν ] [L 2 (e)] 2 h1/2 e v [H 1 ( (e))] 2 h e J[(γ h 1 2 ( u h ( u h ) t ))ν ] 2 [L 2 (e)] 2 1/2 v [H1 ()] 2. (42) herefore, placing (34) (37) (resp. (39) (42)) back into (29) (30) (resp. (31) (32)), employing the estimates (27), (28), (33), using the identities e E h () e = 1 2 e E( ) E h () e e E e h = e E h () e, e E( ) E h (Γ) e we conclude from (26) that A((σ σ h, u u h, γ γ sup h ), (τ, v, η)) C θ. 0 (τ,v,η) H 0 (τ, v, η) H0 div (τ )=0 his inequality Lemma 3.1 complete the proof of reliability of θ. We end this section by remarking that when the finite element subspace H 0,h is given by (16), that is when σ h [R 0 ( ) t ] 2, u h [P 1 ( )] 2 γ h [P 0 ( )] 2 2, then the expression (20) for θ 2 simplifies to θ 2 := f div (σ h ) 2 [L 2 ( )] σ 2 h σ t h 2 [L 2 ( )] γ 2 2 h 1 2 ( u h ( u h ) t ) 2 [L 2 ( )] 2 2 } h 2 { curl(c 1 σ h ) 2[L2( )]2 curl(c 1 (C 1 σ h )) 2[L2( )]2 { } h e J[(C 1 σ h u h γ h )t ] 2 [L 2 (e)] 2 J[(C 1 (e(u h ) C 1 σ h ))t ] 2 [L 2 (e)] 2 e E( ) h 2 div (1 2 (C 1 σ h (C 1 σ h ) t )) 2 [L 2 ( )] 2 h e J[(e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t ))ν ] 2 [L 2 (e)] 2 e E( ) E h () e E( ) E h () h e J[(γ h 1 2 ( u h ( u h ) t ))ν ] 2 [L 2 (e)] 2. (43)

13 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY Efficiency of the A POSERIORI error estimator In this section we proceed as in [4] [5] (see also [9]) apply inverse inequalities (see [6]) the localization technique introduced in [14], which is based on triangle-bubble edge-bubble functions, to prove the efficiency of our a posteriori error estimator θ (lower bound of the estimate (21)) Preliminaries We begin with some notations preliminary results. Given e E( ), we let ψ ψ e be the usual triangle-bubble edge-bubble functions, respectively (see (1.5) (1.6) in [14]). In particular, ψ satisfies ψ P 3 ( ), supp(ψ ), ψ =0on, 0 ψ 1in. Similarly, ψ e P 2 ( ), supp(ψ e ) w e := { h : e E( )}, ψ e =0on \e, 0 ψ e 1inw e. We also recall from [13] that, given k N {0}, there exists an extension operator L : C(e) C( ) that satisfies L(p) P k ( ) L(p) e = p p P k (e). Additional properties of ψ, ψ e,l are collected in the following lemma. Lemma 4.1. For any triangle there exist positive constants c 1, c 2, c 3 c 4, depending only on k the shape of, such that for all q P k ( ) p P k (e), there hold ψ q 2 L 2 ( ) q 2 L 2 ( ) c 1 ψ 1/2 q 2 L 2 ( ), (44) ψ e p 2 L 2 (e) p 2 L 2 (e) c 2 ψe 1/2 p 2 L 2 (e), (45) c 4 h e p 2 L 2 (e) ψ1/2 e L(p) 2 L 2 ( ) c 3 h e p 2 L 2 (e). (46) Proof. See Lemma 1.3 in [13]. he following inverse estimate will also be used. Lemma 4.2. Let l, m N {0} such that l m. hen, for any triangle,thereexistsc>0, depending only on k, l, m the shape of, such that q H m ( ) ch l m q Hl ( ) q P k ( ). (47) Proof. See heorem in [6]. Our goal is to estimate the 11 terms defining the error indicator θ 2 (cf. (20)). Using f = div σ, the symmetry of σ, γ = 1 2 ( u ( u)t ), we first observe that there hold f div (σ h ) 2 [L 2 ( )] 2 = div (σ σ h) 2 [L 2 ( )] 2, (48) σ h σ t h 2 [L 2 ( )] 4 σ σ h [L 2 ( )] 2 2, (49) γ h 1 { } 2 ( u h ( u h ) t ) 2 [L 2 ( )] 2 γ γ 2 2 h 2 [L 2 ( )] u u h [H 1 ( )]. (50) 2 he upper bounds of the remaining 8 residual terms, which depend on the mesh parameters h h e, will be derived in Section 4.2 below. o this end we prove four lemmas establishing, in a sufficiently general way, some results concerning inverse inequalities piecewise polynomials. hey will be used to estimate the terms involving curl div operators, the normal tangential jumps. he result required for the curl operator is given first. Lemma 4.3. Let ρ h [L 2 ()] 2 2 be a piecewise polynomial of degree k 0 on each. In addition, let ρ [L 2 ()] 2 2 be such that curl(ρ) =0 on each.hen,thereexistsc>0, independent of h, such that for any curl(ρ h ) [L2 ( )] 2 ch 1 ρ ρ h [L 2 ( )] 2 2. (51)

14 856.P. BARRIOS E AL. Proof. We proceed as in the proof of Lemma 6.3 in [5]. Applying (44), integrating by parts, observing that ψ =0on, using the Cauchy-Schwarz inequality, we obtain c 1 1 curl(ρ h ) 2 [L 2 ( )] 2 ψ1/2 curl(ρ h ) 2 [L 2 ( )] = ψ 2 curl(ρ h ) curl(ρ h ρ) = (ρ ρ h ):curl(ψ curl(ρ h )) ρ ρ h [L 2 ( )] 2 2 curl(ψ curl(ρ h )) [L 2 ( )] 2 2. (52) Next, the inverse inequality (47) the fact that 0 ψ 1give curl(ψ curl(ρ h )) [L 2 ( )] 2 2 ch 1 ψ curl(ρ h ) [L2 ( )] 2 ch 1 curl(ρ h ) [L 2 ( )] 2, which, together with (52), yields (51). he tangential jumps across the edges of the triangulation will be hled by employing the following estimate. Lemma 4.4. Let ρ h [L 2 ()] 2 2 be a piecewise polynomial of degree k 0 on each.hen,thereexists c>0, independent of h, such that for any e E h J[ρ h t ] [L 2 (e)] 2 ch 1/2 e ρ h [L2 (w e)] 2 2. (53) Proof. Givenanedgee E h,wedenotebyw h := J[ρ h t ] the corresponding tangential jump of ρ h. hen, employing (45) integrating by parts on each triangle of w e,weobtain c 1 2 w h 2 [L 2 (e)] 2 ψ1/2 e w h 2 [L 2 (e)] = 2 ψ1/2 e L(w h ) 2 [L 2 (e)] 2 = ψ e L(w h ) J[ρ h t ] = curl(ρ h ) ψ e L(w h ) ρ h : curl(ψ e L(w h )), (54) e w e w e which, using the Cauchy-Schwarz inequality, yields c 1 2 w h 2 [L 2 (e)] 2 curl(ρ h) [L 2 (w e)] 2 ψ e L(w h ) [L 2 (w e)] 2 ρ h [L2 (w e)] 2 2 curl(ψ e L(w h )) [L2 (w e)] 2 2. (55) Now, applying Lemma 4.3 with ρ = 0 using that h 1 h 1 e, we find that curl(ρ h ) [L2 (w e)] 2 Ch 1 e ρ h [L2 (w e)] 2 2. (56) On the other h, employing (46) the fact that 0 ψ e 1, we deduce that ψ e L(w h ) [L2 (w e)] 2 Ch1/2 e w h [L2 (e)] 2, (57) whereas the inverse estimate (47) (46) yield curl(ψ e L(w h )) [L 2 (w e)] 2 2 Ch 1/2 e w h [L2 (e)] 2. (58) Finally, (53) follows easily from (55) (58), which completes the proof. he estimate required for the terms involving the div operator is provided next.

15 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 857 Lemma 4.5. Let ρ h [L 2 ()] 2 2 be a piecewise polynomial of degree k 0 on each.hen,thereexists c>0, independent of h, such that for any div (ρ h ) [L 2 ( )] 2 ch 1 ρ h [L 2 ( )] 2 2. (59) Proof. Applying (44), integrating by parts, then employing the Cauchy-Schwarz inequality, we find that c 1 1 div (ρ h) 2 [L 2 ( )] 2 ψ1/2 div (ρ h ) 2 [L 2 ( )] = ψ 2 div (ρ h ) div (ρ h ) = ρ h : (ψ div (ρ h )) ρ h [L2 ( )] 2 2 (ψ div (ρ h )) [L2 ( )] 2 2. (60) Next, the inverse estimate (47) the fact that 0 ψ 1in imply that (ψ div (ρ h )) [L 2 ( )] 2 2 ch 1 ψ div (ρ h ) [L2 ( )] 2 ch 1 div (ρ h) [L2 ( )] 2, which, together with (60), yields (59). Finally, the estimate required for the normal jumps across the edges of the triangulation is established as follows. Lemma 4.6. Let ρ h [L 2 ()] 2 2 be a piecewise polynomial of degree k 0 on each.hen,thereexists c>0, independent of h, such that for any e E h J[ρ h ν ] [L 2 (e)] 2 ch 1/2 e ρ h [L2 (w e)] 2 2. (61) Proof. We proceed similarly as in the proof of Lemma 4.4. Given an edge e E h, we now denote by w h := J[ρ h ν ] the corresponding normal jump of ρ h. hen, employing (45) integrating by parts on each triangle of w e,weobtain = e c 1 2 w h 2 [L 2 (e)] 2 ψ1/2 e w h 2 [L 2 (e)] 2 = ψ1/2 e L(w h ) 2 [L 2 (e)] 2 ψ e L(w h ) J[ρ h ν ] = which, using the Cauchy-Schwarz inequality, yields w e w e div (ρ h ) ψ e L(w h ) ρ h : (ψ e L(w h )), (62) c 1 2 w h 2 [L 2 (e)] 2 div (ρ h) [L 2 (w e)] 2 ψ e L(w h ) [L 2 (w e)] 2 ρ h [L2 (w e)] 2 2 (ψ e L(w h )) [L2 (w e)] 2 2. (63) Now, applying Lemma 4.5 using that h 1 h 1 e, we deduce that div (ρ h ) [L 2 (w e)] 2 Ch 1 e ρ h [L2 (w e)] 2 2. (64) On the other h, employing (46) the fact that 0 ψ e 1, we deduce that ψ e L(w h ) [L 2 (w e)] 2 Ch1/2 e w h [L 2 (e)] 2, (65) whereas the inverse estimate (47) (46) yield (ψ e L(w h )) [L 2 (w e)] 2 2 Ch 1/2 e w h [L2 (e)] 2. (66) Finally, (61) follows easily from (63) (66), which completes the proof.

16 858.P. BARRIOS E AL. We note that the generality of Lemmas allows to apply them not only in the present context, but also in the a posteriori error analysis of other primal mixed finite element methods he main efficiency estimates As already announced, we now complete the proof of efficiency of θ by conveniently applying Lemmas to the corresponding terms defining θ 2. Lemma 4.7. here exist C 1,C 2 > 0, independent of h λ, such that for any } h 2 curl(c 1 σ h γ h ) 2 [L 2 ( )] C 2 1 { σ σ h 2[L2( γ γ h 2[L2( )]2 2 )]2 2 (67) h 2 curl(c 1 (e(u h ) C 1 σ h )) 2 [L 2 ( )] 2 C 2 { u u h 2[H1( )]2 σ σ h 2[L2( )]2 2 }. (68) Proof. Applying Lemma 4.3 with ρ h := C 1 σ h γ h ρ := u = C 1 σ γ, then using the triangle inequality the continuity of C 1,weobtain curl(c 1 σ h γ h ) [L 2 ( )] 2 ch 1 (C 1 σ γ) (C 1 σ h γ h ) [L2 ( )] 2 2 = ch 1 C 1 (σ σ h )(γ γ h ) [L 2 ( )] 2 2 } Ch 1 { σ σ h [L2 ( )] 2 2 γ γ h [L2 ( )] 2 2, which yields (67). Similarly, (68) follows from Lemma 4.3 with ρ h := C 1 (e(u h ) C 1 σ h )ρ:= C 1 (e(u) C 1 σ)=0. Lemma 4.8. here exist C 3,C 4 > 0, independent of h λ, such that for any e E h h e J[(C 1 σ h u h γ h )t ] 2 [L 2 (e)] 2 C 3 { σ σ h 2 [L 2 (w e)] 2 2 u u h 2 [H 1 (w e)] 2 γ γ h 2 [L 2 (w e)] 2 2 } (69) h e J[(C 1 (e(u h ) C 1 σ h ))t ] 2 [L 2 (e)] 2 C 4 { u u h 2[H1(we)]2 σ σ h 2[L2(we)]2 2 }. (70) Proof. Applying Lemma 4.4 with ρ h := C 1 σ h u h γ h, introducing 0 = C 1 σ u γ in the resulting estimate, then using the triangle inequality the continuity of C 1,weget J[(C 1 σ h u h γ h )t ] [L 2 (e)] 2 ch 1/2 e C 1 σ h u h γ h [L2 (w e)] 2 2 = ch 1/2 e C 1 (σ h σ)( u u h )(γ h γ) [L 2 (w e)] 2 2 } Ch 1/2 e { σ σ h [L 2(we)] 2 2 u u h [H 1(we)] 2 γ γ h [L 2(we)] 2 2, which implies (69). Analogously, the estimate (70) is obtained from Lemma 4.4 defining ρ h := C 1 (e(u h ) C 1 σ h ) then introducing 0 = C 1 (e(u) C 1 σ). Lemma 4.9. here exist C 5,C 6 > 0, independent of h λ, such that for any h 2 div (e(u h) 1 } 2 (C 1 σ h (C 1 σ h ) t )) 2 [L 2 ( )] C 2 5 { u u h 2[H1( σ σ h 2[L2( )]2 )]2 2 (71)

17 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 859 h 2 div (γ h 1 2 ( u h ( u h ) t )) 2 [L 2 ( )] 2 C 6 { γ γ h 2[L2( )]2 2 u u h 2[H1( )]2 }. (72) Proof. We apply Lemma 4.5 with ρ h := e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t ), introduce the expression 0 = e(u) 1 2 (C 1 σ (C 1 σ) t ) in the resulting estimate, then use the triangle inequality the continuity of the operators e C 1,toobtain div (e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t )) [L2 ( )] 2 ch 1 e(u h) 1 2 (C 1 σ h (C 1 σ h ) t ) [L2 ( )] 2 2 = ch 1 Ch 1 (e(u h) e(u)) 1 2 (C 1 (σ σ h )(C 1 }(σ σ h )) t ) [L2 ( )] { u 2 2 u h [H 1( )] 2 σ σ h [L2(, )]2 2 which gives (71). Similarly, applying Lemma 4.5 with ρ h := γ h 1 2 ( u h ( u h ) t ) introducing 0 = γ 1 2 ( u ( u)t ), we obtain (72). Lemma here exist C 7,C 8 > 0, independent of h λ, such that for any e E h h e J[(e(u h ) 1 } 2 (C 1 σ h (C 1 σ h ) t ))ν ] 2 [L 2 (e)] C 2 7 { u u h 2[H1(we)]2 σ σ h 2[L2(we)]2 2 (73) h e J[(γ h 1 2 ( u h ( u h ) t ))ν ] 2 [L 2 (e)] 2 C 8 { γ γ h 2[L2(we)]2 2 u u h 2[H1(we)]2 }. (74) Proof. We apply Lemma 4.6 with ρ h := e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t ), introduce the expression 0 := e(u) 1 2 (C 1 σ (C 1 σ) t ), then employ again the triangle inequality the continuity of the operators e C 1, to find that J[(e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t ))ν ] [L 2 (e)] 2 ch 1/2 e e(u h ) 1 2 (C 1 σ h (C 1 σ h ) t ) [L 2 (w e)] 2 2 = ch 1/2 e (e(u h ) e(u)) 1 2 (C 1 (σ σ h )(C 1 (σ} σ h )) t ) [L 2 (w e)] { u 2 2 u h [H 1(we)] 2 σ σ h [L 2(we)] 2 2, Ch 1/2 e which yields (73). Analogously, the estimate (74) follows also from Lemma 4.6 defining ρ h := γ h 1 2 ( u h ( u h ) t ) then introducing 0 = γ 1 2 ( u ( u)t ). Finally, the efficiency of θ (lower bound of (21)) follows straightforwardly from the estimates (48) (50), (67), (68) (cf. Lem. 4.7), (69), (70) (cf. Lem. 4.8), (71), (72) (cf. Lem. 4.9), (73), (74) (cf. Lem. 4.10), after summing over all using that the number of triangles in each domain w e is bounded by two. 5. Numerical results In this section we provide several numerical results illustrating the performance of the augmented mixed finite element scheme (10) of the a posteriori error estimator θ analyzed in this paper, using the specific finite element subspace H σ h Hu 0,h Hγ h, defined at the end of Section 2 (see (13) (19)). We recall that in this case the local indicator θ 2 reduces to (43).

18 860.P. BARRIOS E AL. Now, before presenting the examples, we would like to remark in advance that, as compared with more traditional mixed methods, besides the fact, already emphasized, of being able to choose any finite element subspace, our augmented approach presents other important advantages, as well. Indeed, let us first observe that in the case of uniform refinements each interior edge (resp. interior node) belongs to 2 (resp. 6) triangles, which yields corresponding correction factors of when counting the global number of degrees of freedom, say N, in terms of the number of triangles, say M. hen, it is not difficult to see that the number of unknowns N of (10) behaves asymptotically as 5M, whereas this behaviour is given by 7.5M when the well-known PEERS from [1] is used in the Galerkin scheme of the non-augmented formulation. In other words, the discrete system using PEERS introduces about 50% more degrees of freedom than our approach at each mesh, therefore the augmented method becomes a much cheaper alternative. Furthermore, it is important to note that the polynomial degrees involved in the definition of H σ h Hu 0,h Hγ h, being 1, 1 0, yield simpler computations than for the PEERS subspace, whose polynomial degrees are 2, 0, 1, respectively. Similarly, as compared with BDM (see e.g. [2, 3]), the augmented scheme also becomes more economical. In fact, as detailed for instance in [2], just the unknowns associated with the displacements rotations of BDM are given by 6M. o this amount, one still needs to add the degrees of freedom for the stresses which are given locally by a 15-dimensional space. Only after a static condensation process, BDM reduces to 6 unknowns per each edge, which yields N behaving as 9M. Naturally, the competitive character of our augmented method has strongly motivated the need of deriving corresponding a posteriori error estimators in this paper. o this respect, because of the introduction of the Galerkin-least squares terms needed to define the augmented formulation, we must recognize that θ is certainly more expensive than, for instance, the error indicator introduced in [2]. However, it is also clear that the reliability efficiency of θ become more advantageous features than the sole reliability of the estimator in [2]. Finally, in connection with the residual-based a posteriori error estimator developed in [5] for PEERS BDM, which is also reliable efficient, we point out that the advantage of θ, though a bit more expensive, is still the freedom to choose the finite element subspaces defining the augmented scheme. On the other h, in order to implement the integral mean zero condition for functions of the space H { σ 0,h = τ h H σ h : tr(τ h)=0 } we introduce, as described in [10], a Lagrange multiplier (ϕ h R below). hat is, instead of (10), we consider the equivalent problem: Find (σ h, u h, γ h,ϕ h ) H σ h Hu 0,h Hγ h R such that A((σ h, u h, γ h ), (τ h, v h, η h )) ϕ h ψ h tr(τ h ) = F (τ h, v h, η h ), tr(σ h ) = 0, (75) for all (τ h, v h, η h,ψ h ) H σ h Hu 0,h Hγ h the equivalence between (10) (75). R. In fact, we recall from [10] the following theorem establishing heorem 5.1. a) Let (σ h, u h, γ h ) H 0,h be the solution of (10). hen(σ h, u h, γ h, 0) is a solution of (75). b) Let (σ h, u h, γ h,ϕ h ) H σ h Hu 0,h Hγ h R be a solution of (75). henϕ h =0 (σ h, u h, γ h ) is the solution of (10).

19 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 861 Proof. See heorem 4.3 in [10]. In what follows, as indicated before, N sts for the total number of degrees of freedom (unknowns) of (75). Also, the individual total errors are denoted by e(σ) := σ σ h H(div ;), e(u) := u u h [H 1 ()] 2, e(γ) := γ γ h [L 2 ()] 2 2, e(σ, u, γ) := { [e(σ)] 2 [e(u)] 2 [e(γ)] 2 } 1/2, respectively, whereas the effectivity index with respect to θ is defined by e(σ, u, γ)/θ. On the other h, we recall that given the Young modulus E the Poisson ratio ν of a linear elastic material, the corresponding Lamé constants are defined by µ := E Eν 2(1ν) λ := (1ν)(1 2ν). hen, in order to emphasize the robustness of the a posteriori error estimator θ with respect to the Poisson ratio, in the examples below we fix E = 1 consider ν =0.4900, ν =0.4999, or both, which yield the following values of µ λ : ν µ λ In addition, since the augmented method was already shown in [10] to be( robust with ) respect to the parameters 1 κ 1, κ 2,κ 3, we simply consider for all the examples (κ 1,κ 2,κ 3 ) = µ, 2 µ, µ 2, which corresponds to the feasible choice described in heorem 2.1 with C 1 =1 C 3 = 1 2. We now specify the data of the five examples to be presented here. We take as either the square ]0, 1[ 2 or the L-shaped domain ] 0.5, 0.5[ 2 \ [0, 0.5] 2, choose the datum f so that ν the exact solution u(x 1,x 2 ):=(u 1 (x 1,x 2 ),u 2 (x 1,x 2 )) t are given in the table below. Actually, according to (1) (2) we have σ = λ div (u) I 2µ e(u), hence simple computations ( show that f := div(σ) = (λµ) (div u) µ u. We also recall that the rotation γ is defined as ) 1 2 u ( u) t. We observe that the solution of Example 3 is singular at the boundary point (0,0). In fact, the behaviour of u in a neighborhood of the origin implies that div (σ) [H 1/3 ()] 2 only, which, according to heorem 2.3, yields 1/3 as the expected rate of convergence for the uniform refinement. On the other h, the solutions of Examples 1, 4, 5 show large stress regions in a neighborhood of the boundary point (1, 1), in a neighborhood of the interior point (1/2, 1/2), around the line x 1 = 0, respectively. Example ν u 1(x 1,x 2)=u 2(x 1,x 2) 1 ]0, 1[ x 1 (x 1 1) x 2 (x 2 1) (x 1 1) 2 (x 2 1) ]0, 1[ x 1 (x 1 1) x 2 (x 2 1) (x 2 1 x 2 2) 1/ ] 0.5, 0.5[ 2 \ [0, 0.5] x 1 x 2 (x ) (x ) (x 2 1 x 2 2) 1/3 4 ]0, 1[ sin(πx 1)sin(πx 2) 1000 (x 1 1/2) (x 2 1/2) ] 0.5, 0.5[ 2 \ [0, 0.5] x 1 x 2 (x ) (x ) (x ) 1/3 he numerical results given below were obtained using a Compaq Alpha ES40 Parallel Computer a Fortran code. he linear system arising from the augmented mixed scheme (75) is implemented as explained in Section 4.3 of [10], the individual errors are computed on each triangle using a Gaussian quadrature rule.

20 862.P. BARRIOS E AL. We first utilize Examples 1 2 to illustrate the good behaviour of the a posteriori error estimator θ in a sequence of uniform meshes generated by equally spaced partitions on the sides of the square ]0, 1[ 2. In ables 5.1 through 5.4 we present the individual total errors, the a posteriori error estimators, the effectivity indices for these examples, with ν = ν =0.4999, for this sequence of uniform meshes. We remark that in both cases, independently of how large the errors could become, there are practically no differences between the effectivity indices obtained with the two values of ν, which numerically shows the robustness of θ with respect to the Poisson ratio ( hence with respect to the Lamé constant λ). Moreover, this index always remains in a neighborhood of 0.89 in Example 1 (resp in Ex. 2), which confirms the reliability efficiency of θ. In fact, as established by our main heorem 3.1, the effectivity index must lie between the constants C eff C rel (see (21)). Next, we consider Examples 3, 4, 5, to illustrate the performance of the following adaptive algorithm based on θ for the computation of the solutions of (75) (see [14]): 1. Start with a coarse mesh h. 2. Solve the Galerkin scheme (75) for the current mesh h. 3. Compute θ for each triangle. 4. Consider stopping criterion decide to finish or go to next step. 5. Use blue-green procedure to refine each element h whose local indicator θ satisfies θ 1 2 max{θ : }. 6. Define resulting mesh as the new h gotostep2. At this point we introduce the experimental rate of convergence, which, given two consecutive triangulations with degrees of freedom N N corresponding total errors e e, is defined by r(e) := 2 log(e/e ) log(n/n ) In ables 5.5 through 5.10 we provide the individual total errors, the experimental rates of convergence, the a posteriori error estimators, the effectivity indices for the uniform adaptive refinements as applied to Examples 3 5. In this case, uniform refinement means that, given a uniform initial triangulation, each subsequent mesh is obtained from the previous one by dividing each triangle into the four ones arising when connecting the midpoints of its sides. We observe from these tables that the errors of the adaptive procedure decrease much faster than those obtained by the uniform one, which is confirmed by the experimental rates of convergence provided there. his fact can also be seen in Figures 5.1 through 5.3 where we display the total error e(σ, u, γ) vs. the degrees of freedom N for both refinements. As shown by the values of r(e), particularly in Example 3 (where r(e) approaches 1/3 for the uniform refinement), the adaptive method is able to recover, at least approximately, the quasi-optimal rate of convergence O(h) for the total error. Furthermore, the effectivity indices remain again bounded from above below, which confirms the reliability efficiency of θ for the adaptive algorithm. On the other h, some intermediate meshes obtained with the adaptive refinement are displayed in Figures 5.4 through 5.6. Note that the method is able to recognize the singularities the large stress regions of the solutions. In particular, this fact is observed in Example 3 (see Fig. 5.4) where the adapted meshes are highly refined around the singular point (0, 0). Similarly, the adapted meshes obtained in Examples 4 5 (see Figs ) concentrate the refinements around the interior point (1/2, 1/2) the segment x 1 = 0, respectively, where the largest stresses occur. Summarizing, the numerical results presented in this section underline the reliability efficiency of θ strongly demonstrate that the associated adaptive algorithm is much more suitable than a uniform discretization procedure when solving problems with non-smooth solutions.

21 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 863 able 5.1. Mesh sizes, individual total errors, a posteriori error estimators, effectivity indices for a sequence of uniform meshes (Ex. 1, ν =0.4900). N h e(σ) e(u) e(γ) e(σ, u, γ) θ e(σ, u, γ)/θ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E able 5.2. Mesh sizes, individual total errors, a posteriori error estimators, effectivity indices for a sequence of uniform meshes (Ex. 1, ν =0.4999). N h e(σ) e(u) e(γ) e(σ, u, γ) θ e(σ, u, γ)/θ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

22 864.P. BARRIOS E AL. able 5.3. Mesh sizes, individual total errors, a posteriori error estimators, effectivity indices for a sequence of uniform meshes (Ex. 2, ν =0.4900). N h e(σ) e(u) e(γ) e(σ, u, γ) θ e(σ, u, γ)/θ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E able 5.4. Mesh sizes, individual total errors, a posteriori error estimators, effectivity indices for a sequence of uniform meshes (Ex. 2, ν =0.4999). N h e(σ) e(u) e(γ) e(σ, u, γ) θ e(σ, u, γ)/θ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

23 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 865 able 5.5. Individual total errors, experimental rates of convergence, a posteriori error estimators, effectivity indices for the uniform refinement (Ex. 3). N e(σ) e(u) e(γ) e(σ, u, γ) r(e) θ e(σ, u, γ)/θ E E E E E E E E E E E E E E E E E E E E able 5.6. Individual total errors, experimental rates of convergence, a posteriori error estimators, effectivity indices for the adaptive refinement (Ex. 3). N e(σ) e(u) e(γ) e(σ, u, γ) r(e) θ e(σ, u, γ)/θ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E uniform refinement adaptive refinement N Figure 5.1. otal errors e(σ, u, γ) vs. degrees of freedom N for the uniform adaptive refinements (Ex. 3).

24 866.P. BARRIOS E AL. able 5.7. Individual total errors, experimental rates of convergence, a posteriori error estimators, effectivity indices for the uniform refinement (Ex. 4). N e(σ) e(u) e(γ) e(σ, u, γ) r(e) θ e(σ, u, γ)/θ E E E E E E E E E E E E E E E E E E E E able 5.8. Individual total errors, experimental rates of convergence, a posteriori error estimators, effectivity indices for the adaptive refinement (Ex. 4). N e(σ) e(u) e(γ) e(σ, u, γ) r(e) θ e(σ, u, γ)/θ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E uniform refinement adaptive refinement N Figure 5.2. otal errors e(σ, u, γ) vs. degrees of freedom N for the uniform adaptive refinements (Ex. 4).

25 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 867 able 5.9. Individual total errors, experimental rates of convergence, a posteriori error estimators, effectivity indices for the uniform refinement (Ex. 5). N e(σ) e(u) e(γ) e(σ, u, γ) r(e) θ e(σ, u, γ)/θ E E E E E E E E E E E E E E E E E E E E able Individual total errors, experimental rates of convergence, a posteriori error estimators, effectivity indices for the adaptive refinement (Ex. 5). N e(σ) e(u) e(γ) e(σ, u, γ) r(e) θ e(σ, u, γ)/θ E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E uniform refinement adaptive refinement N Figure 5.3. otal errors e(σ, u, γ) vs. degrees of freedom N for the uniform adaptive refinements (Ex. 5).

26 868.P. BARRIOS E AL. Figure 5.4. Adapted intermediate meshes with 1783, 3663, 8203, degrees of freedom (Ex. 3). Figure 5.5. Adapted intermediate meshes with 883, 2583, 4988, 9748 degrees of freedom (Ex. 4).

27 AUGMENED MIXED FINIE ELEMEN MEHOD IN LINEAR ELASICIY 869 Figure 5.6. Adapted intermediate meshes with 2383, 4038, 7938, degrees of freedom (Ex. 5). References [1] D.N. Arnold, F. Brezzi J. Douglas, PEERS: A new mixed finite element method for plane elasticity. Japan J. Appl. Math. 1 (1984) [2] D. Braess, O. Klaas, R. Niekamp, E. Stein F. Wobschal, Error indicators for mixed finite elements in 2-dimensional linear elasticity. Comput. Method. Appl. M. 127 (1995) [3] F. Brezzi M. Fortin, Mixed Hybrid Finite Element Methods. Springer-Verlag (1991). [4] C. Carstensen, A posteriori error estimate for the mixed finite element method. Math. Comput. 66 (1997) [5] C. Carstensen G. Dolzmann, A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81 (1998) [6] P.G. Ciarlet, he Finite Element Method for Elliptic Problems. North-Holl, Amsterdam, New York, Oxford (1978). [7] P. Clément, Approximation by finite element functions using local regularisation. RAIRO Anal. Numér. 9 (1975) [8] J. Douglas J. Wan, An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52 (1989) [9] G.N. Gatica, A note on the efficiency of residual-based a posteriori error estimators for some mixed finite element methods. Electronic rans. Numer. Anal. 17 (2004) [10] G.N. Gatica, Analysis of a new augmented mixed finite element method for linear elasticity allowing R 0 P 1 P 0 approximations. ESAIM: M2AN 40 (2006) [11] A. Masud.J.R. Hughes, A stabilized mixed finite element method for Darcy flow. Comput. Method. Appl. M. 191 (2002) [12] J.E. Roberts J.-M. homas, Mixed Hybrid Methods, in Hbook of Numerical Analysis II, Finite Element Methods (Part 1) P.G. Ciarlet J.L. Lions Eds., North-Holl, Amsterdam (1991). [13] R. Verfürth, A posteriori error estimation adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50 (1994) [14] R. Verfürth, A Review of A Posteriori Error Estimation Adaptive Mesh-Refinement echniques. Wiley-eubner (Chichester) (1996).