A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity

Vol. 40, N 5, 2006, pp. 843–869 www.edpsciences.org/m2an

DOI: 10.1051/m2an:2006036

A RESIDUAL BASED A POSTERIORI ERROR ESTIMATOR FOR AN AUGMENTED MIXED FINITE ELEMENT METHOD

IN LINEAR ELASTICITY

Tom´ as P. Barrios

, Gabriel N. Gatica

, Mar´ ıa Gonz´ alez

and Norbert Heuer

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 and efficienta posteriorierror estimator for the case of pure Dirichlet boundary condi- tions. In addition, several numerical experiments confirming the theoretical properties of the estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution, are also reported.

Mathematics Subject Classification. 65N15, 65N30, 65N50, 74B05.

Received: November 9, 2005.

1. Introduction

A new stabilized mixed finite element method for plane linear elasticity was presented and analyzed recently in [10]. The approach there is based on the introduction of suitable Galerkin least-squares terms arising from the constitutive and equilibrium equations, and from the relation defining the rotation in terms of the displacement.

The 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] and [11]. It is shown in [10] that the continuous and discrete augmented formulations are well-posed, and that the latter becomes locking-free and asymptotically locking-free for Dirichlet and 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-Thomas spaces of lowest order for the stress tensor, piecewise linear elements for the displacement, and 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 and phrases. Mixed finite element, augmented formulation,a posteriorierror estimator, linear elasticity.

∗ This research was partially supported by CONICYT-Chile through the FONDAP Program in Applied Mathematics, by the Direcci´on de Investigaci´on of the Universidad de Concepci´on through the Advanced Research Groups Program, and by the Universidade da Coru˜na through the Research Grants Program.

1 Facultad de Ingenier´ıa, Universidad Cat´olica de la Sant´ısima Concepci´on, Casilla 297, Concepci´on, Chile.tomas@ucsc.cl

2 GI²MA, Departamento de Ingenier´ıa Matem´atica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile.

ggatica@ing-mat.udec.cl

3 Departamento de Matem´aticas, Universidade da Coru˜na, Campus de Elvi˜na s/n, 15071 A Coru˜na, Spain.mgtaboad@udc.es

4BICOM and 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 and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2006036 Article published by EDP Sciences and available at http://www.edpsciences.org/m2anor http://dx.doi.org/10.1051/m2an:2006036

the trace of the displacement as a suitable Lagrange multiplier. This 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] and also throughout the present paper (see, in particular, Sect. 5 below).

According to the above, and strongly motivated by the competitive character of our augmented formulation, we now feel the need of deriving correspondinga posteriorierror estimators. More precisely, the purpose of this work is to develop a residual baseda posteriori error analysis for the augmented mixed finite element scheme from [10] in the case of pure Dirichlet boundary conditions. A posteriorierror analyses of the traditional mixed finite element methods for the elasticity problem can be seen in [5] and the references therein. The rest of this paper is organized as follows. In Section 2 we recall from [10] the continuous and discrete augmented formulations of the corresponding boundary value problem, state the well-posedness of both schemes, and provide the associateda priorierror estimate. The kernel of the present work is given by Sections 3 and 4, where we develop the residual baseda posteriori error analysis. Indeed, in Section 3 we employ a suitable auxiliary problem and apply integration by parts and the local approximation properties of the Cl´ement 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 and the localization technique based on triangle-bubble and 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, and hence our proof of efficiency needs to previously establish more general versions of some technical lemmas concerning inverse estimates and piecewise polynomials. Finally, several numerical results confirming reliability, efficiency, and 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 and 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 spaceU,U²andU^2×2denote, respectively, the space of vectors and square matrices of order 2 with entries inU. In addition,Iis the identity matrix ofR^2×2, and givenτ := (τij),ζ:= (ζij)∈R^2×2, we write as usual

τ^t := (τji), tr(τ) :=

2 i=1

τii, τ^d:=τ−1

2tr(τ)I, and τ :ζ :=

2 i,j=1

τijζij.

Also, in what follows we utilize the standard terminology for Sobolev spaces and norms, employ0to denote a generic null vector, and useCandc, 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. The augmented formulations

First we let Ω be a simply connected domain in R² with polygonal boundary Γ := ∂Ω. Our goal is to determine the displacementuand stress tensorσ of a linear elastic material occupying the region Ω. In other words, given a volume forcef ∈[L²(Ω)]², we seek a symmetric tensor fieldσ and a vector fieldusuch that

σ = Ce(u), div(σ) = −f in Ω, and u = 0 on Γ. (1) Hereafter, e(u) := ¹₂(∇u+ (∇u)^t) is the strain tensor of small deformations and C is the elasticity tensor

determined by Hooke’s law, that is

Cζ := λtr(ζ)I + 2µζ ∀ζ ∈ [L²(Ω)]^2×2, (2) where λ, µ > 0 denote the corresponding Lam´e constants. It is easy to see from (2) that the inverse tensor C⁻¹ reduces to

C⁻¹ζ := 1

2µζ − λ

4µ(λ+µ)tr(ζ)I ∀ζ ∈ [L²(Ω)]^2×2. (3) We now define the spaces H = H(div; Ω) := {τ ∈ [L²(Ω)]^2×2 : div(τ) ∈ [L²(Ω)]²}, H0 := {τ ∈ H :

Ωtr(τ) = 0}, and note that H = H0 ⊕ RI, that is for any τ ∈ H there exist unique τ0 ∈ H0 and d := _2|Ω|¹

Ωtr(τ) ∈ R such that τ = τ₀+dI. In addition, we define the space of skew-symmetric tensors [L²(Ω)]^2×2_skew := {η∈[L²(Ω)]^2×2:η+η^t = 0}and introduce the rotationγ := ¹₂(∇u−(∇u)^t))∈[L²(Ω)]^2×2_skewas an auxiliary unknown. Then, given positive parametersκ1,κ2, andκ3, independent ofλ, we consider from [10]

the following augmented variational formulation for (1): Find (σ,u,γ) ∈ H₀ := H0×[H₀¹(Ω)]²×[L²(Ω)]^2×2_skew such that

A((σ,u,γ),(τ,v,η)) = F(τ,v,η) ∀(τ,v,η)∈H₀, (4) where the bilinear formA:H₀×H₀→Rand the functionalF:H₀→Rare defined by

A((σ,u,γ),(τ,v,η)) :=

ΩC⁻¹σ:τ+

Ω

u·div(τ) +

Ωγ:τ−

Ω

v·div(σ)−

Ωη:σ + κ1

Ω

e(u)− C⁻¹σ :

e(v) +C⁻¹τ + κ2

Ωdiv(σ)·div(τ) + κ3

Ω

γ−1

2(∇u−(∇u)^t)

:

η+1

2(∇v−(∇v)^t)

, (5)

and

F(τ,v,η) :=

Ω

f·(v −κ2div(τ) ). (6) The well-posedness of (4) was proved in [10]. More precisely, we have the following result.

Theorem 2.1. Assume that (κ1, κ2, κ3) is independent of λ and such that 0 < κ1 < 2µ, 0 < κ2, and 0< κ3< κ1. Then, there exist positive constants M, α, independent ofλ, such that

|A((σ,u,γ),(τ,v,η))| ≤ M (σ,u,γ)H0(τ,v,η)H0 (7) and

A((τ,v,η),(τ,v,η)) ≥ α(τ,v,η)²_H0 (8) for all(σ,u,γ), (τ,v,η) ∈ H₀. In particular, taking

κ1 = ˜C1µ , κ2= 1 µ

1− κ1

2µ

, and κ3 = ˜C3κ1, (9)

with any C˜1 ∈ ]0,2[ and any C˜3 ∈ ]0,1[, this yields M and α depending only on µ, _µ¹, and Ω. Therefore, the augmented variational formulation (4) has a unique solution (σ,u,γ) ∈ H₀, and there exists a positive constant C, independent of λ, such that

(σ,u,γ)H0 ≤ CF ≤ Cf[L²(Ω)]².

Proof. See Theorems 3.1 and 3.2 in [10].

Now, given a finite element subspace H_0,h ⊆ H₀, the Galerkin scheme associated with (4) reads: Find (σh,u_h,γ_h) ∈ H_0,hsuch that

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 κ1, κ2, and κ3, being the same parameters employed in the formulation (4), satisfy the assumptions of Theorem 2.1. SinceAbecomes bounded and strongly coercive on the whole spaceH₀, 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].

Theorem 2.2. Assume that the parametersκ1,κ2, andκ3satisfy the assumptions of Theorem2.1and letH_0,h be any finite element subspace ofH₀. Then, the Galerkin scheme(10)has a unique solution(σh,u_h,γ_h)∈H_0,h, and there exist positive constants C,C˜, independent ofhandλ, such that

(σh,u_h,γ_h)H0 ≤ C sup

(τh,vh,η_h^)∈H0,h

(τh,vh,η_h)=0

|F(τh,v_h,η_h)| (τh,v_h,η_h)H0

≤ Cf_[L²_(Ω)]²,

and

(σ,u,γ)−(σ_h,u_h,γ_h)H0 ≤ C˜ inf

(τh,vh,ηh)∈H0,h (σ,u,γ)−(τ_h,v_h,η_h)H0. (11) Proof. It follows from Theorem 2.1, Lax-Milgram’s Lemma, and C´ea’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 (seee.g.[3]), is the possibility of choosing any finite element subspaceH_0,h ofH₀.

On the other hand, an inmediate consequence of the definition of the continuous and 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,hintroduced in [10], which is the simplest finite element subspace of H₀. To this end, we first let{Th}h>0be a regular family of triangulations of the polygonal region ¯Ω by trianglesT of diameterhT with mesh sizeh:= max{hT : T ∈ Th}, and such that there holds ¯Ω = ∪ {T : T ∈ Th}.

In addition, given an integer≥0 and a subsetS ofR², we denote byP(S) the space of polynomials in two variables defined in S of total degree at most , and for each T ∈ Th we introduce the local Raviart-Thomas space of order zero (cf. [3, 12]),

RT0(T) := span

1 0

, 0

1

, x1

x2 ⊆ [P1(T)]², where (x1

x2) is a generic vector ofR². Then, defining Hσ

h :=

τ_h∈H(div; Ω) : τ_h|_T ∈ [RT₀(T)^t]² ∀T ∈ T_h

, (13)

Xh :=

vh ∈ C( ¯Ω) : vh|_T ∈ P₁(T) ∀T ∈ T_h

, (14)

and

H_h^u := Xh×Xh, (15)

we take

H_0,h := Hσ

0,h×H_0,h^u ×Hγ

h , (16)

where

Hσ

0,h : =

τh ∈ Hσ

h :

Ωtr(τh) = 0

, (17)

H_0,h^u : = {v_h ∈ H_h^u: v_h = 0 on Γ} , (18) and

Hγ

h :=

η_h ∈ [L²(Ω)]^2×2_skew: η_h|_T ∈ [P₀(T)]^2×2 ∀T ∈ T_h

. (19)

The following theorem provides the rate of convergence of (10) when the specific finite element subspace (16) is utilized.

Theorem 2.3. Let(σ,u,γ)∈H₀ and(σh,u_h,γ_h)∈H_0,h := Hσ

0,h×H_0,h^u ×Hγ

h be the unique solutions of the continuous and discrete augmented mixed formulations(4)and(10), respectively. Assume thatσ∈[H^r(Ω)]^2×2, div(σ) ∈ [H^r(Ω)]², u ∈ [H^r+1(Ω)]², and γ ∈ [H^r(Ω)]^2×2, for some r ∈ (0,1]. Then there exists C > 0, independent of hand λ, such that

(σ,u,γ)−(σh,u_h,γ_h)H0 ≤ C h^r

σ_[H^r_(Ω)]^2×2+div(σ)_[H^r_(Ω)]²+u_[H^r+1_(Ω)]²+γ_[H^r_(Ω)]^2×2 .

Proof. It is a consequence of C´ea’s estimate, the approximation properties of the subspaces definingH_0,h, and suitable interpolation theorems in the corresponding function spaces. See Section 4.1 in [10] for more details.

3. A residual based

error estimator

In this section we derive a residual based a posteriorierror estimator for (10). First we introduce several notations. GivenT ∈ Th, we letE(T) be the set of its edges, and letEhbe the set of all edges of the triangulation Th. Then we writeEh=Eh(Ω)∪Eh(Γ), whereEh(Ω) :={e∈Eh: e⊆Ω}andEh(Γ) :={e∈Eh: e⊆Γ}.

In what follows,hestands for the length of edgee∈Eh. Further, givenτ ∈[L²(Ω)]^2×2(such thatτ|T ∈C(T) on each T ∈ Th), an edgee∈E(T)∩Eh(Ω), and the unit tangential vectort_T alonge, we let J[τt_T] be the corresponding jump acrosse, that is,J[τt_T] := (τ|T−τ|T)|et_T, whereTis the other triangle ofThhavingeas an edge. Abusing notation, whene∈Eh(Γ), we also writeJ[τt_T] :=τ|et_T. We recall here thatt_T := (−ν2, ν1)^t where νT := (ν1, ν2)^tis the unit outward normal to∂T. Analogously, we define the normal jumpsJ[τ νT]. In addition, given scalar, vector, and tensor valued fieldsv,ϕ:= (ϕ1, ϕ2), andτ := (τij), respectively, we let

curl(v) := −_∂x^∂v₂

∂x∂v1

, curl(ϕ) := curl(ϕ1)^t curl(ϕ2)^t

, and curl(τ) :=

∂τ12

∂x1 − ^∂τ_∂x¹¹₂

∂τ22

∂x1 − ^∂τ_∂x²¹₂

.

Then, for (σ,u,γ)∈H₀and (σh,u_h,γ_h)∈H_0,hbeing the solutions of the continuous and discrete formulations (4) and (10), respectively, we define an error indicatorθT as follows:

θ_T² := f+div(σ_h)²_[L2(T)]² +σ_h−σ^t_h²_[L2(T)]^2×2 +γ_h−1

2(∇u_h−(∇u_h)^t)²_[L2(T)]^2×2

+ h²_T

curl(C⁻¹σ_h+γ_h)²_[L2(T)]² +curl(C⁻¹(e(u_h)− C⁻¹σ_h))²_[L2(T)]²

+

e∈E(T)

he

J[(C⁻¹σh− ∇u_h+γ_h)t_T]²_[L2(e)]² +J[(C⁻¹(e(u_h)− C⁻¹σh))t_T]²_[L2(e)]²

+ h²_T div(e(u_h)−1

2(C⁻¹σ_h+ (C⁻¹σ_h)^t))²_[L2(T)]²

+ h²_T div(γ_h−1

2(∇u_h−(∇u_h)^t))²_[L2(T)]²

+

e∈E(T)∩Eh(Ω)

he J[(e(u_h)−1

2(C⁻¹σ_h+ (C⁻¹σ_h)^t))ν_T]²_[L2(e)]²

+

e∈E(T)∩Eh(Ω)

he J[(γ_h−1

2(∇u_h−(∇u_h)^t))ν_T]²_[L2(e)]². (20) The residual character of each term on the right-hand 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 expressionθ :=

T∈Thθ_T²_1/2

is employed as the global residual error estimator.

The following theorem is the main result of this paper.

Theorem 3.1. Let(σ,u,γ)∈H₀and(σ_h,u_h,γ_h)∈H_0,hbe the unique solutions of(4)and(10), respectively.

Then there existCeff, Crel>0, independent of handλ, such that

Ceffθ ≤ (σ−σh,u−u_h,γ−γ_h)H0 ≤ Crelθ. (21) The so-called efficiency (lower bound in (21)) is proved below in Section 4 and the reliability estimate (upper bound in (21)) is derived throughout the rest of the present section. The 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 and Cl´ement 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. There existsC >0, independent ofhandλ, such that C (σ−σh,u−u_h,γ−γ_h)H0

≤ sup

0=(τ,v,η)∈H0

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_(Ω)]². (22)

Proof. Let us define σ^∗ = e(z), where z ∈ [H₀¹(Ω)]² is the unique solution of the boundary value problem:

−div(e(z)) =f+div(σh) in Ω, z = 0 on Γ. It follows thatσ^∗ ∈ H0, and the corresponding continuous dependence result establishes the existence ofc >0 such that

σ^∗_H(div;Ω) ≤ cf+div(σh)_[L²_(Ω)]². (23)

In addition, it is easy to see thatdiv(σ−σh−σ^∗) = 0in Ω. Then, using the coercivity ofA(cf.(8)), we find that

α(σ−σh−σ^∗,u−u_h,γ−γ_h)²_H0

≤A((σ−σ_h−σ^∗,u−u_h,γ−γ_h),(σ−σ_h−σ^∗,u−u_h,γ−γ_h))

=A((σ−σh,u−u_h,γ−γ_h),(σ−σh−σ^∗,u−u_h,γ−γ_h))

−A((σ^∗,0,0),(σ−σh−σ^∗,u−u_h,γ−γ_h)), which, employing the boundedness ofA(cf.(7)), yields

α(σ−σ_h−σ^∗,u−u_h,γ−γ_h)H0

≤ sup

0=(τ,v,η)∈H0

div(τ)=0

A((σ−σ_h,u−u_h,γ−γ_h),(τ,v,η)) (τ,v,η)H0

+ Mσ^∗_H(div;Ω). (24)

Hence, (22) follows straightforwardly from the triangle inequality, (23), and (24).

It remains to bound the first term on the right-hand side of (22). To this end, we will make use of the well known Cl´ement interpolation operator Ih : H¹(Ω)→ Xh (cf. [7]), withXh given by (14), which satisfies the standard local approximation properties stated below in Lemma 3.2. It is important to remark thatIh is defined in [7] so that Ih(v)∈Xh∩H₀¹(Ω) for allv∈H₀¹(Ω).

Lemma 3.2. There exist constants c1, c2 > 0, independent ofh, such that for allv∈H¹(Ω) there holds v−Ih(v)_L²_(T) ≤ c1hTv_H¹_(∆(T)) ∀T ∈ Th,

and

v−Ih(v)_L²_(e) ≤ c2h^1/2_e v_H¹_(∆(e)) ∀e ∈ Eh, where∆(T) := ∪{T∈ T_h: T∩T =∅}, and ∆(e) := ∪{T∈ T_h: T∩e=∅}.

Proof. See [7].

We now let (τ,v,η)∈H₀, (τ,v,η)=0, be such thatdiv(τ) = 0in Ω. Since Ω is connected, there exists a stream functionϕ := (ϕ1, ϕ2)∈[H¹(Ω)]² such that

Ωϕ1 =

Ωϕ2 = 0 andτ =curl(ϕ). Then, denoting ϕ_h := (ϕ1,h, ϕ2,h), with ϕi,h :=Ih(ϕi), i∈ {1,2}, the Cl´ement interpolant ofϕi, we defineτh :=curl(ϕ_h).

Note that there holds the decompositionτ_h=τ_h,0+dhI, whereτ_h,0∈Hσ

0,handdh=

Ωtr(τh)

2|Ω| ∈R. From the orthogonality relation (12) it follows that

A((σ−σh,u−u_h,γ−γ_h),(τ,v,η)) =A((σ−σh,u−u_h,γ−γ_h),(τ−τh,0,v−v_h,η)), (25) where v_h := (Ih(v1), Ih(v2)) ∈ H_0,h^u is the vector Cl´ement interpolant of v := (v1, v2) ∈ [H₀¹(Ω)]². Since

Ωtr(σ−σ_h) = 0 and u−u_h = 0 on Γ, we deduce, using the orthogonality between symmetric and skew- symmetric tensors, that

A((σ−σh,u−u_h,γ−γ_h),(dhI,0,0)) = 0. Hence, (25) and (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)).

According to the definitions of the formsAandF(cf. (5), (6)), noting thatdiv(τ−τh) = div curl(ϕ−ϕ_h) = 0, and using again the above mentioned orthogonality, we find, after some algebraic manipulations, that

A((σ−σh,u−u_h,γ−γ_h),(τ,v,η)) =

Ω(f +div(σh))·(v−v_h) +

Ω

1

2(σ_h−σ^t_h) −κ3

γ_h−1

2(∇u_h−(∇u_h)^t)

:η

−

Ω

(C⁻¹σh− ∇u_h+γ_h) + κ1C⁻¹(e(u_h)− C⁻¹σh)

: (τ−τh)

−

Ω

κ1

e(u_h)−1

2(C⁻¹σh+ (C⁻¹σh)^t)

+ κ3

γ_h−1

2(∇u_h−(∇u_h)^t)

:∇(v−v_h).

(26)

The rest of the proof of reliability consists in deriving suitable upper bounds for each of the terms appearing on the right-hand side of (26). We begin by noticing that direct applications of the Cauchy-Schwarz inequality give

Ω

1

2(σ_h−σ^t_h) :η

≤ σ_h−σ^t_h[L2(Ω)]^2×2 η_[L²_(Ω)]^2×2, (27) and

Ω(γ_h−1

2(∇u_h−(∇u_h)^t)) :η

≤ γ_h−1

2(∇u_h−(∇u_h)^t)

[L²(Ω)]^2×2 η_[L²_(Ω)]^2×2. (28) The decomposition Ω =∪T∈ThT and the integration by parts formula on each element are employed next to handle the terms from the third and fourth rows of (26). We first replace (τ−τh) bycurl(ϕ−ϕ_h) and use that curl(∇u_h) = 0in each triangleT ∈ Th, to obtain

Ω(C⁻¹σ_h− ∇u_h+γ_h) : (τ−τ_h) =

T∈Th

T(C⁻¹σ_h− ∇u_h+γ_h) :curl(ϕ−ϕ_h)

=

T∈Th

T

curl(C⁻¹σh+γ_h)·(ϕ−ϕ_h)

−

e∈Eh

J[(C⁻¹σ_h− ∇u_h+γ_h)t_T],ϕ−ϕ_h[L²_(e)]², (29)

and

ΩC⁻¹(e(u_h)− C⁻¹σh) : (τ−τh) =

T∈Th

T C⁻¹(e(u_h)− C⁻¹σh) :curl(ϕ−ϕ_h)

=

T∈Th

T curl(C⁻¹(e(u_h)− C⁻¹σ_h))·(ϕ−ϕ_h)

−

e∈Eh

J[(C⁻¹(e(u_h)− C⁻¹σh))t_T],ϕ−ϕ_h[L²_(e)]². (30)

On the other hand, using thatv−v_h=0on Γ, we easily get

Ω(e(u_h)−1

2(C⁻¹σh+ (C⁻¹σh)^t)) :∇(v−v_h)

= −

T∈Th

T

div(e(u_h)−1

2(C⁻¹σ_h+ (C⁻¹σ_h)^t))·(v−v_h)

+

e∈Eh(Ω)

J[(e(u_h)−1

2(C⁻¹σ_h+ (C⁻¹σ_h)^t))ν_T],v−v_h[L²_(e)]², (31) and

Ω(γ_h−1

2(∇u_h−(∇u_h)^t)) :∇(v−v_h)

= −

T∈Th

T

div(γ_h−1

2(∇u_h−(∇u_h)^t))·(v−v_h)

+

e∈Eh(Ω)

J[(γ_h−1

2(∇u_h−(∇u_h)^t))νT],v−v_h[L²_(e)]². (32) In what follows, we apply again the Cauchy-Schwarz inequality, Lemma 3.2, and the fact that the numbers of triangles in ∆(T) and ∆(e) are bounded, independently of h, to derive the estimates for the expression

Ω(f +div σh)·(v−v_h) in (26) and the right-hand sides of (29), (30), (31), and (32), with constants C independent ofhand λ. Indeed, we easily have

Ω

(f +divσh)·(v−v_h) ≤

T∈Th

f+div σh_[L²_(T)]²v−v_h[L²_(T)]²

≤c1

T∈Th

f+divσh_[L²_(T)]² hTv_[H¹_(∆(T)]²

≤C

T∈Th

h²_Tf+div σ_h²_[L2(T)]²

_1/2

v_[H¹_(Ω)]². (33)

In addition, for the terms containing the stream functionϕ(cf.(29), (30)), we get

T∈Th

Tcurl(C⁻¹σh+γ_h)·(ϕ−ϕ_h) ≤

T∈Th

curl(C⁻¹σh+γ_h)_[L²_(T)]²ϕ−ϕ_h[L²_(T)]²

≤c1

T∈Th

curl(C⁻¹σh+γ_h)_[L²_(T)]² hTϕ_[H¹_(∆(T)]²

≤C

T∈Th

h²_Tcurl(C⁻¹σh+γ_h)²_[L2(T)]²

_1/2

ϕ_[H¹_(Ω)]², (34)

T∈Th

Tcurl(C⁻¹(e(u_h)− C⁻¹σh))·(ϕ−ϕ_h)

≤

T∈Th

curl(C⁻¹(e(u_h)− C⁻¹σ_h))_[L²_(T)]²ϕ−ϕ_h[L²_(T)]²

≤ c1

T∈Th

curl(C⁻¹(e(u_h)− C⁻¹σh))_[L²_(T)]² hTϕ_[H¹_(∆(T)]²

≤ C

T∈Th

h²_Tcurl(C⁻¹(e(u_h)− C⁻¹σ_h))²_[L2(T)]²

_1/2

ϕ_[H¹_(Ω)]², (35)

e∈Eh

J[(C⁻¹σ_h− ∇u_h+γ_h)t_T],ϕ−ϕ_h[L²_(e)]²

≤

e∈Eh

J[(C⁻¹σh− ∇u_h+γ_h)t_T]_[L²_(e)]² ϕ−ϕ_h[L²_(e)]²

≤ c2

e∈Eh

J[(C⁻¹σh− ∇u_h+γ_h)t_T]_[L²_(e)]² h^1/2_e ϕ_[H¹_(∆(e))]²

≤ C

e∈Eh

heJ[(C⁻¹σh− ∇u_h+γ_h)t_T]²_[L2(e)]²

_1/2

ϕ_[H¹_(Ω)]², (36)

and

e∈Eh

J[(C⁻¹(e(u_h)− C⁻¹σh))t_T],ϕ−ϕ_h[L²_(e)]²

≤

e∈Eh

J[(C⁻¹(e(u_h)− C⁻¹σ_h))t_T]_[L²_(e)]² ϕ−ϕ_h[L²_(e)]²

≤ c2

e∈Eh

J[(C⁻¹(e(u_h)− C⁻¹σh))t_T]_[L²_(e)]² h^1/2_e ϕ_[H¹_(∆(e))]²

≤ C

e∈Eh

heJ[(C⁻¹(e(u_h)− C⁻¹σ_h))t_T]²_[L2(e)]²

_1/2

ϕ_[H¹_(Ω)]². (37)

We observe here, thanks to the equivalence betweenϕ_[H¹_(Ω)]² and|ϕ|_[H¹_(Ω)]², that

ϕ_[H¹_(Ω)]² ≤ C|ϕ|_[H¹_(Ω)]² = Ccurl(ϕ)_[L²_(Ω)]² = Cτ_H(div;Ω), (38)

which allows to replaceϕ_[H¹_(Ω)]² byτ_H(div;Ω)in the above estimates (34)–(37).