A posteriori error estimates for mixed FEM in elasticity

Numer. Math. (1998) 81: 187–209

Numerische Mathematik

c Springer-Verlag 1998

A posteriori error estimates for mixed FEM in elasticity Carsten Carstensen1 , Georg Dolzmann2 1 2

Mathematisches Seminar, Christian-Albrechts-Universit¨at zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany; e-mail: [email protected] Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany; e-mail: [email protected]

Received July 17, 1997

Summary. A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from inverse estimates. The constants in both estimates are independent of the Lam´e constant λ, and so locking phenomena for λ → ∞ are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement. Mathematics Subject Classification (1991): 65N30, 65N15, 73C35

1. Introduction The fundamental problem in linear elasticity is usually modelled as follows [Ci2, Va]: Let Ω ⊂ Rd be the reference configuration of the elastic body under consideration with boundary ∂Ω = ΓD ∪ ΓN , ΓD not empty and connected, ΓD ∩ΓN = ∅. Given a volume force f : Ω → Rd , a displacement uD : ΓD → Rd and a traction g : ΓN → Rd , find a displacement u : Ω → d×d : τ = τ T } satisfying Rd and a stress tensor σ : Ω → Md×d sym := {τ ∈ M (1.1) (1.2)

−div σ = f, σ = CE(u) in Ω, u = uD on ΓD , σn = g on ΓN ,

Correspondence to: C. Carstensen

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where the fourth order elasticity tensor C is bounded, positive definite, and satisfies the symmetry conditions Cijkl = Cjikl = Cijlk = Cklij . We write E(v) = 12 (∇v + (∇v)T ) for the infinitesimal strain tensor. In the following we restrict ourselves to the model of plane strain, i.e. (1.3)

CE(u) = λtr(E(u))Id + 2µE(u),

where λ and µ are the Lam´e constants, tr(A) = A11 + . . . + Add is the trace of the matrix A and Id is the d × d identity matrix. (Using ideas from [AF] it is easy to see that our estimates hold also for more general tensors C.) It is a consequence of Korn’s inequality and the Lax-Milgram lemma that problem (1.1)–(1.2) has a unique solution (σ, u) ∈ L2 (Ω; Md×d sym ) × 1,2 d W (Ω; R ) which satisfies the a priori estimate kuk1,2;Ω + kσk2;Ω ≤ c1 kf k2;Ω . In addition, the error estimate for the displacement requires the following regularity assumption kuk2,2;Ω + kσk1,2;Ω ≤ c2 (kf k2;Ω + kuD kH 3/2 (ΓD ) + kgkH 1/2 (ΓN )). (1.4) A realistic hypothesis for (1.4) to hold is 0 < dist(ΓD ; ΓN ), i.e., the boundary condition does not change at some boundary point. Furthermore, the constant c2 is supposed to be independent of λ (see Theorem 2.1 in [ADG] and Lemma A.1 in [Vo] for the cases ΓN = ∅ and ΓD = ∅, respectively; the general statement does not seem to be available in the literature). Mixed methods are a powerful tool for the numerical solution of the system (1.1)–(1.2). They provide at the same time an approximation of the displacement and the stress tensor. A priori estimates have been established for a wide choice of different methods which satisfy the Babuˇska-Brezzi condition. A subtle choice of the discrete spaces avoids the common phenomenon of locking (i.e., the estimates are independent of the parameter λ in (1.3)). A difficulty in the design of stable numerical schemes is linked to the symmetry of the stress tensor σ and therefore Fraeijs de Veubeke [FdV] and following his ideas Brezzi-Douglas-Marini [BDM], Arnold-BrezziDouglas [ABD] and Stenberg [St] weakened the symmetry condition and reformulated the elasticity problem: Find u : Ω → Rd , σ : Ω → Md×d d×d : η + η T = 0}, such that and γ : Ω → Md×d skew := {η ∈ M (1.5) (1.6)

σ = C(∇u − γ), σ = σ T , −div σ = f in Ω, u = uD on ΓD , σn = g on ΓN .

In the following we will assume uD = 0. In the corresponding variational formulation one seeks (σ, u, γ) ∈ Σg × U × W such that (1.7)

a(σ, τ ) + b(τ ; u, γ) = 0 and

b(σ; v, η) = −(f, v),

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for all (τ, v, η) ∈ Σ0 × U × W. Here, the linear and bilinear forms and the function spaces Σt , U, W are defined by Z C−1 σ : τ dx, a(σ, τ ) = Ω Z b(σ; u, γ) = (hdiv σ, ui + σ : γ)dx, Ω Z (f, v) = hf, vidx, Ω

Σt = {σ ∈ L2 (Ω; Md×d ) : div σ ∈ L2 (Ω; Rd ), σn = t on ΓN }, d×d ), U × W = L2 (Ω; Rd ) × L2 (Ω; Mskew for t = 0 and t = g. In this approach, the symmetry of the stress tensor σ is relaxed and only imposed by means of the Lagrange multiplier γ. Let Σt,h , Uh , Wh be finite dimensional spaces approximating Σt , U, and W. Then the corresponding discrete solution (σh , uh , γh ) ∈ Σg,h × Uh × Wh is characterised by (1.8) a(σh , τh ) + b(τh ; uh , γh ) = 0 and

b(σh ; vh , ηh ) = −(f, vh ),

for all (τh , vh , ηh ) ∈ Σ0,h × Uh × Wh . In this formulation, σh satisfies only the weak symmetry condition Z (1.9) σh : γh dx = 0 ∀γh ∈ Wh , Ω

which does not imply σh = σhT if σh − σhT 6∈ Wh . In two dimensions existence, uniqueness, and a priori estimates for several choices of discrete spaces have been proven in [St] which include the low order PEERS (plane elasticity element with reduced symmetry) constructed by ArnoldBrezzi-Douglas [ABD] and a modification of the Brezzi-DouglasMarini element BDMk by Stenberg (which we will refer to as BDMSk element). A posteriori estimates in the natural norms, on the other hand, do not seem to be available in the literature (see, however, [BKNSW] for estimates in mesh dependent norms and [RS] for results concerning stabilised dual–mixed formulations). In this paper, we propose an a posteriori error estimator for the errors ε = σ − σh and e = u − uh for the PEERS and the BDMSk method (see Sect. 2 for details). Our analysis relies on a decomposition of symmetric tensors in the spirit of a generalised Helmholtz decomposition. Helmholtz decomposition was first used in [Ca,A] to prove efficiency and reliability of error estimators for mixed finite elements. The estimator accounts for the

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residues on the triangles T and the jumps across the element boundaries E. More precisely, we define (see Sect. 2 for the notation used below) ηT2 = h2T kdiv εk22;T + h2T kcurl(C−1 σh + γh )k22;T +h2T inf kC−1 σh + γh − ∇vh k22;T + kSkw(σh )k22;T , vh ∈Uh  hE kJ((C−1 σh + γh )t)k22;E if E ⊂ Ω ∪ ΓD , 2 = ηE hE k(σ − σh )nk22;E if E ⊂ ΓN , X X 2 (1.10) η 2 = ηT2 + ηE . T ∈Th

E∈Eh

The main result of this paper states reliability and efficiency of the estimator η. All constants in the estimates are under the regularity assumption (1.4) independent of h and λ. In particular, the common locking phenomena are avoided. Theorem 1.1. Let Th be a shape-regular triangulation of Ω ⊂ R2 and let (σh , uh , γh ) be the solution of (1.8) for the PEERS or the BDMSk element. Assume that the regularity assumption (1.4) holds. Then there exists a constant c3 , which depends only on Ω, µ, and the polynomial degree of the elements, such that ku − uh k2;Ω + kγ − γh k2;Ω + kC−1/2 (σ − σh )k2;Ω ≤ c3 η. Theorem 1.2. Assume in addition that curl(C−1 σh +γh )|T is a polynomial for all T ∈ Th and (σ − σh )n|E for all E ⊂ ΓN . Then there exists a constant c4 , which depends only on Ω, µ, and the polynomial degree of the elements, such that  η ≤ c4 ku − uh k2;Ω + kC−1 (σ − σh ) + γ − γh k2;Ω  + kσ − σh k2;Ω + khT div εk2;Ω . Remarks. 1. It follows from (1.7) that C−1 σ+γ = ∇u. Therefore the terms in the estimator are natural residuals: curl(C−1 σh +γh ) and J((C−1 σh +γh )t) are zero if C−1 σh + γh is a gradient. The distance of this term to gradients is also measured by the expression inf kC−1 σh + γh − ∇vh k. 2. The term inf vh ∈Uh kC−1 σh + γh − ∇vh k can be replaced by its upper bound kC−1 σh + γh − ∇uh k which still satisfies the efficiency estimate of Theorem 1.2. 3. Since −div ε = f +div σh is a known quantity, we can replace kC−1/2 (σ −σh )k2;Ω by the (weighted) norm kσ − σh kH(div;Ω) = kC−1/2 (σ − σh )k2;Ω + kdiv(σ − σh )k2;Ω

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on the left hand side in Theorem 1.1, but we lose the factor hT in the estimator above in front of the term kdiv εk2;T . 4. The regularity assumption (1.4) is not needed for the estimate of kC−1/2 (σ −σh )k2;Ω in Theorem 1.1, but in the duality argument in the estimation of ku−uh k2;Ω . Hence, if we suppress ku−uh kL2 (Ω) then Theorem 1.1 remains true even if dist(ΓD ; ΓN ) = 0. 5. According to the triangle inequality and the preceding remarks, the error ku − uh k2;Ω + kγ − γh k2;Ω + kC−1/2 (σ − σh )k2;Ω and the error indicator η are equivalent in the sense that their quotient is bounded from below and above independently of the material parameter λ and the mesh–size h. In particular, the estimates are robust with respect to λ → ∞ for (nearly) incompressible materials. 6. The estimator justifies an adaptive finite element scheme which refines a given grid only in regions where the error is relatively large. A standard algorithm for efficient mesh-design is as follows: For each mesh ThL with a Galerkin solution (phL , uhL ) and local error estimators η(T ) =: ηT + P E⊆∂T ηE , we refine T ∈ ThL (e.g., by halving its largest side) if (for example) max η(T 0 )/2 ≤ η(T ). 0 T ∈ThL

Then, further refinements to avoid hanging nodes lead to a new mesh ThL+1 from which we start again. 7. The estimates are stated for the elements of practical importance only. The arguments used in the proofs rely only on the following properties (with L00 the piecewise constant functions on Ω and L11 the continuous piecewise affine ones) L00 ⊂ Uh ,

L00 ∩ H(div; Ω)2 ⊆ Σ0,h ,

and

L11 ⊆ Wh .

To obtain estimates for the displacements, we further require a commutation property for some (Fortin–) interpolation operator πh (of (2.1)—(2.4) below). We refer to [Ca] for a discussion in the general framework (for Laplace’s equation). 2. Preliminaries We assume that Ω is a bounded domain in R2 with polygonal boundary. Let Th be a regular triangulation of Ω in the sense of [Ci1], which satisfies the minimum angle condition, i.e., there exists a constant c5 > 0 such that c5−1 h2T ≤ |T | ≤ c5 h2T . Here, |T | is the area and hT is the diameter of T ∈ Th . The set of all element sides in Th is denoted by Eh and hE is the

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length of the edge E ∈ Eh . We assume in addition that ΓN is a finite union of connected components Γi , i = 0, . . . , M , and that ΓD and ΓN have positive distance. Thus we have Eh = EΩ ∪ ED ∪ EN where EΩ is the set of all interior element sides and ED and EN is the collection of all edges contained in ΓD and ΓN , respectively. We write Eh0 = EΩ ∪ EN . It is useful to define a function hT on Ω by hT |T = hT and a function hE on the union of all element sides by hE |E = hE . We write u ∈ W m,p (Th ) and v ∈ W m,p (Eh ) if u|T ∈ W m,p (T ) for all T ∈ Th and v|E ∈ W m,p (E) for all E ∈ Eh . For each E ∈ Eh we fix a normal nE to E such that nE coincides with the exterior normal to ∂Ω if E ⊂ ∂Ω. This allows us to define a mapping J : W 1,2 (Th ) → L2 (Eh ) by J(v)|E = (v|T + )|E − (v|T − )|E if E = T¯+ ∩ T¯− and nE is the exterior normal to T + on E and J(v)|E = (v|T )|E if E = T¯ ∩ ∂Ω. Finally we define for Φ ∈ W 1,2 (Ω), u = (u1 , u2 ) ∈ W 1,2 (Ω; R2 ), and σ ∈ W 1,2 (Ω; M2×2 ) Curl Φ = (Φ,2 , −Φ,1 ), 

u1,2 −u1,1 u2,2 −u2,1



, curl u = u2,1 − u1,2 , Curl u =     σ12,1 − σ11,2 σ11,1 + σ12,2 curl σ = , div σ = . σ22,1 − σ21,2 σ21,1 + σ22,2 We use the standard notation for the Lebesgue spaces Lp (Ω) with norm k · kp;Ω and the Sobolev spaces W m,p (Ω) with norm k · km,p;Ω and seminorm | · |m,p;Ω . The closure of Cc∞ (Ω), the space of infinitely often differentiable functions with compact support, with respect to k·km,p;Ω is denoted by W0m,p (Ω). The definition of the finite element spaces involves the bubble function bT = λ1 λ2 λ3 on a triangle T ∈ Th , where λi are the barycentric coordinates of T . The PEERS is based on the following function spaces Uh = {vh ∈ U : vh|T ∈ P0 (T ; R2 ) ∀ T ∈ Th }, 2×2 Wh = {γh ∈ W ∩ C 0 (Ω; M2×2 skew ) : γh|T ∈ P1 (T ; Mskew ) ∀ T ∈ Th },

Σh = {σh ∈ L2 (Ω; M2×2 ) : div σh ∈ U, σh|T ∈ RT0 (T ) ⊕ B0 (T ) ∀T ∈ Th }, Σt,h = {σh ∈ Σh : σh n = t˜ on ΓN },

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where t˜ is the orthogonal projection of t in L2 (E) onto P0 (E; R2 ) for all edges E ⊂ ΓN . Here, RT0 is the Raviart-Thomas space of lowest degree, and RT0 (T ) = {σ ∈ L2 (T ; M2×2 ) : σ = τ + a ⊗ x, τ ∈ M2×2 , a ∈ R2 }, B0 (T ) = {σ ∈ L2 (T ; M2×2 ) : σ = a ⊗ Curl bT , a ∈ R2 }, BDMk (Ω) = {σh ∈ L2 (Ω; M2×2 ) : div σh ∈ U, σh|T ∈ Pk (T ; M2×2 )}. The higher order methods BDMSk are defined for k ≥ 2 by Uh = {vh ∈ U : vh|T ∈ Pk−1 (T ; R2 ) ∀ T ∈ Th }, Wh = {γh ∈ W : γh|T ∈ Pk (T ; M2×2 skew ) ∀ T ∈ Th },

Σh = {σh ∈ L2 (Ω; M2×2 ) : div σ ∈ U, σh|T ∈ Pk (T ; M2×2 ) ⊕ Bk−1 (T )}, Σt,h = {σh ∈ Σh : σh n = t˜ on ΓN }, where t˜ is the orthogonal projection of t in L2 (E) onto Pk (E; R2 ), and Bk−1 (T ) = {σ ∈ L2 (T ; M2×2 ) : σ = Curl(bT w), w ∈ Pk−1 (T ; R2 )}. Using the interpolation operators for RT0 and BDMk (see [BF], Sect. III.3.3) we can construct an interpolation operator Πh : W 1,2 (Ω; M2×2 ) → Σh such that for all τ ∈ W 1,2 (Ω; M2×2 ) Z (2.1) div(Πh τ − τ )vh dx = 0 ∀ vh ∈ Uh , and Ω

(2.2)

kΠh τ − τ k2;T ≤ c6 hT |τ |1,2;T .

The projection Πh is defined in such a way that Z (2.3) (Πh τ − τ )∇h vh dx = 0 ∀ vh ∈ Uh , and Ω

(2.4)

τ n = 0 on ΓN

⇒

Πh τ n = 0 on ΓN .

If Ph0 denotes the orthogonal projection in L2 onto L00 ⊂ Uh , L00 , the space of piecewise constant functions, we have the estimate kv − Ph0 vk2;T ≤ c7 hT |v|1,2;T

∀ v ∈ W 1,2 (T ) ∀ T ∈ Th .

Finally we use Cl´ement’s interpolation operator [Cl] Rh : W 1,2 (Ω) → L11 onto the space of continuous, piecewise linear functions, which satisfies the interpolation estimates kv − Rh vk2;T ≤ c8 hT kvk1,2;ωT , 1/2

kv − Rh vk2;E ≤ c9 hE kvk1,2;ωE ,

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where ωT = ∪{T 0 ∈ Th : T¯ ∩ T¯0 6= ∅} and ωE = ∪{T ∈ Th : E ⊂ T¯}. Notice that Rh satisfies v = ci on Γi

(2.5)

⇒

Rh v = ci on Γi .

The number of triangles in ωT is uniformly bounded by some constant c10 , which depends only on the shape of the triangles. Throughout the paper we write Sym(σ) and Skw(σ) for the symmetric and the skew-symmetric part of a matrix σ and use little Greek letters for matrices, little Latin letters of vectors and capital Greek letters for scalars. We use the symbols ∇h and curlh if we apply the corresponding differential operators on each triangle to a function that is globally not smooth. 3. A Helmholtz decompostion for symmetric tensor fields The following two results on the Helmholtz decomposition are essential for the subsequent proofs. We add a sketch of their proofs for the convenience of the reader. Lemma 3.1. Assume that A is a symmetric, positive definite tensor of fourth order. Let ρ ∈ L2 (Ω; M2×2 ). Then there exists q ∈ W 1,2 (Ω; R2 ) with q = 0 on ΓD and f ∈ W 1,2 (Ω; R2 ) with f = ci ∈ R2 on Γi , c0 = 0, such that ρ = ∇q + A−1 Curlf. Proof. The classical proof for the existence of a Helmholtz decomposition for vector fields u ∈ L2 (Ω; R2 ) can be modified to yield the existence of Φ, Ψ ∈ L2 (Ω) such that Ψ = 0 on ΓD , Φ = ci on Γi and u = ∇Ψ + Curl Φ. To do so, consider  Z  1 A∇p : ∇p − Aρ : ∇p dx. I(p) = Ω 2 It follows from the direct method in the calculus of variations that there exists a unique q ∈ W 1,2 (Ω; R2 ) with q = 0 on ΓD such that I(q) = min{I(p) : p ∈ W 1,2 (Ω; R2 ), p = 0 on ΓD } and q satisfies the Euler–Lagrange equation Z (A∇q − Aρ)∇φdx = 0 for all φ ∈ W 1,2 (Ω; R2 ) Ω

with

φ = 0 on ΓD .

It follows that π = A∇q−Aρ is a divergence free vectorfield and by Green’s formula Z Z hπn, φids = (divπφ + π∇φ)dx ∀φ ∈ W 1,2 (Ω; R2 ). ∂Ω

Ω

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In view of the Euler-Lagrange equations we conclude Z hπn, φids = 0 for all φ ∈ W 1,2 (Ω; R2 ) with ∂Ω

φ = 0 on ΓD .

With φ ≡ 1 in a neighourhood of one component of the Neumann boundary and φ ≡ 0 in a neighbourhood of all the other components as well as on a neighbourhood of the Dirichlet boundary, we infer that πn has mean value zero on all connected components of the Neumann boundary. With φ ≡ 1 we deduce the same property on the Dirichlet boundary and thus there exists an f ∈ W 1,2 (Ω; R2 ) such that π = Curlf (see [GR], Chapter I, Theorem 3.1). Since πn = Curlf n = ∇f t, where t is a tangential vector, this concludes the proof. u t Furthermore, we also need a symmetric variant and define X1 = {v ∈ W 1,2 (Ω; R2 ) : v = 0 on ΓD }, X2 = {Φ ∈ W 2,2 (Ω) : Z Φdx = 0, Curl Φ = ci on Γi , ci ∈ R2 , c0 = 0}. Ω

Lemma 3.2. Let σ ∈ L2 (Ω; M2×2 sym ). Then there exists v ∈ X1 and Φ ∈ X2 such that σ = CE(v) + Curl Curl Φ. Proof. In view of Korn’s inequality there exists, by the direct method of the calculus of variations, a unique minimiser v ∈ X1 of Z Z 1 CE(v) : E(v)dx − I(v) = σ : E(v)dx. Ω 2 Ω In particular, v satisfies the corresponding Euler–Lagrange equations Z Z CE(v) : ∇w dx = σ : ∇w dx ∀ w ∈ X1 . Ω

Ω

L2 (Ω; M2×2 sym ).

Let τ = σ − CE(v) ∈ The classical Helmholtz decomposition applied to the rows of τ yields the existence of q ∈ X1 and h ∈ W 1,2 (Ω; R2 ), h = ci on Γi with c0 = 0 such that τ = ∇q + Curl h (we refer to Lemma 3.1 for details). If we use q as a test function in the Euler Lagrange equations we deduce in view of the orthogonality of ∇q and Curl h in L2 Z Z Z 2 0= (CE(v) − σ) : ∇q dx = |∇q| dx + Curl h : ∇q dx Ω

Ω

Ω

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and therefore q ≡ 0 and τ = Curl h. From the symmetry of τ we deduce −h1,1 = h2,2 , i.e., div h = 0. Since h is constant on the connected components of ΓN which are by assumption closed Lipschitz curves, we conclude that the mean value of hh, ni vanishes on Γi for i = 0, . . . , M . Green’s formula then implies that the mean value vanishes also on ΓD and hence there exists a stream function Φ ∈ W 2,2 (Ω) with Curl Φ = h. Subtracting from Φ a suitable constant if necessary we obtain the assertion of the lemma. t u 4. An estimate for the trace of a tensor field The following technical results are needed in the estimates below. The first estimate is a modification of well-established estimates of the trace of a tensor field by its divergence and the deviatoric part (see, e.g., [BF, Proposition 3.1 in Sect. IV.3]). Lemma 4.1. Let Σ0 be a closed subspace of H(div; Ω) which does not contain the constant tensor Id. Then there exists a constant c11 (which depends only on Σ0 ) such that   ∀τ ∈ Σ0 . ktr τ k2;Ω ≤ c11 kτ D k2;Ω + kdiv τ k2;Ω Proof. Assume the contrary. Then there exists a sequence (τj ) ∈ Σ0 satisfying ktr τj k2;Ω = 1,

kτjD k2;Ω + kdiv τj k2;Ω → 0.

Thus we may choose a subsequence (again denoted by τj ) such that τj * τ in L2 (Ω; Md×d ) and div τj * div τ in L2 (Ω; Rd ). Clearly τ ∈ Σ0 with τ D = 0 and therefore τ = α · Id with α ∈ L2 (Ω). On the other hand we have div τ = ∇α = 0 and hence α is constant. Since Id 6∈ Σ0 we conclude τ = 0. It follows from the weak convergence of the sequence τj that Z 1 cj = tr τj dx → 0 |Ω| Ω and thus σj = τj − (4.1)

cj d

· Id (d being the dimension) satisfies by assumption lim ktr σj k2;Ω = 1.

j→∞

We now adapt the arguments from [BF], p. 199, to obtain a contradiction. Since the integral mean of tr σj is zero we can solve the equation div wj = −tr σj for some wj ∈ W01,2 (Ω; Rd ) which satisfies the a priori estimate kwj k1,2;Ω ≤ c12 ktr σj k2;Ω ≤ c12 ktr τj k2;Ω = c12 .

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Using the above identities, we calculate Z Z Z tr σj div wj dx = − σj : ∇wj dx + σjD : ∇wj dx ktr σj k22;Ω = − Ω

Ω

Ω

≤ (kdiv σj k2;Ω + kσjD k2;Ω )kwj k1,2;Ω → 0 as j → ∞. This contradicts (4.1) and proves the lemma.

t u

Moreover, we will use the following estimate. Lemma 4.2.R Assume that Φ ∈ W 2,2 (Ω) satisfies Curl Φ = 0 on ΓN if ΓN 6= ∅ or Ω tr CurlCurlΦ = 0 if ΓN = ∅. Then there exists a constant c12 which depends only on Ω and ΓN such that k∆Φk2;Ω ≤ c12 k(Curl Curl Φ)D k2;Ω . Furthermore, kCurl Curl Φk22;Ω ≤ c13 kCurl Curl Φk2C−1 ;Ω , where the constant c13 depends only on Ω, ΓN and µ. Proof. Assume first that ΓN 6= ∅. Let Γ0 be a maximal line segment contained in ΓN 6= ∅, and define Z n o Σ0 = σ ∈ H(div; Ω) : σnds = 0 . Γ0

Clearly Σ0 is a weakly closed subspace of H(div; Ω) and Id 6∈ Σ0 . From div Curl Curl Φ = 0 and Z Curl Curl Φnds = 0 Γ0

we R have Curl Curl Φ ∈ Σ0 . If ΓN = 0, let Σ0 := {σ ∈ H(div; Ω): Ω trσ dx = 0} and, by assumption, Curl Curl Φ ∈ Σ0 . Hence, the first inequality follows from Lemma 4.1. From this, Z |Curl Curl Φ|2 dx Ω Z Z 1 = |∆Φ|2 dx + |(Curl Curl Φ)D |2 dx 2 Ω Ω Z c212 + 1) C−1 Curl Curl Φ : Curl Curl Φdx. t u ≤ 2µ( 2 Ω

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5. Proof of the upper bound We begin with the estimate for the error ε := σ − σh in the stress variable. Lemma 3.2 implies the existence of v ∈ X1 and Φ ∈ X2 such that (5.1)

ε = CE(v) + Curl Curl Φ + φ,

where φ = Skw(σh ) is the skew-symmetric part of σh . Since Z Z (5.2) E(v) : Curl Curl Φdx = ∇v : Curl Curl Φdx Ω ZΩ = hv, (∇Curl φ)tidx = 0, ∂Ω

E(v) and Curl Curl Φ are orthogonal in L2 (Ω; M2×2 ). Therefore, we obtain the decomposition (5.3) kεk2C−1 ;2;Ω = kCurl Curl Φk2C−1 ;2;Ω + kE(v)k2C;2;Ω + kφk2C−1 ;2;Ω , where we used for A = C and A = C−1 the notation Z kτ k2A;2;Ω = Aτ : τ dx. Ω

√ √ For√C as in (1.3) we have with c14 = 1/ 2µ and c15 = max{1/ 2µ, d/ dλ + 2µ} kτ k2;Ω ≤ c14 kτ kC;2;Ω ,

kτ kC−1 ;2;Ω ≤ c15 kτ k2;Ω .

In particular these constants are independent of λ for λ → ∞. In the next lemmas we estimate the three terms on the right hand side of (5.3). All constants are independent of λ and h and depend only on µ, Ω and the shape of the triangles. Lemma 5.1. There exists a constant c16 such that we have n kCurl Curl ΦkC−1 ;2;Ω ≤ c16 khT curlh (C−1 σh + γh )k22;Ω 1/2

+ khE J((C−1 σh + γh )t)k22;E 0

o1/2

h

Proof: We deduce from (5.1), (5.2) and C−1 σ = E(u) Z 2 Curl Curl Φ : C−1 εdx kCurl Curl ΦkC−1 ;2;Ω = Ω Z =− Curl Curl Φ : (C−1 σh + γh )dx. Ω

.

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Let b = Curl Φ ∈ W 1,2 (Ω; R2 ) and define bh := Rh b ∈ L11 . Since Φ ∈ X2 we deduce b = Curl Φ = ci on Γi and therefore (Curl b)n = 0 on ΓN . In view of (2.5), Curl bh ∈ RT0 is an admissible test tensor in the discrete equation (1.8) and we obtain Z (5.4) Curl bh : (C−1 σh + γh )dx = 0. Ω

Therefore, by an integration by parts on each triangle, Z 2 kCurl Curl ΦkC−1 ;2;Ω = − Curl(Curl Φ − bh ) : (C−1 σh + γh )dx Ω Z = hCurl Φ − bh , curl(C−1 σh + γh )idx Ω Z hCurl Φ − bh , J((C−1 σh + γh )t)ids + ≤

X

Eh

kCurl Φ − bh k2;T kcurl(C−1 σh + γh )k2;T

T ∈Th

+

X

kCurl Φ − bh k2;E kJ((C−1 σh + γh )t)k2;E

E∈Eh0

 X 1/2 √ ≤ c8 c10 kCurl Φk1,2;Ω h2T kcurl(C−1 σh + γh )k22;T T ∈Th

 X 1/2 √ + 2c9 kCurl Φk1,2;Ω hE kJ((C−1 σh + γh )t)k22;E . E∈Eh0

If ΓN 6= ∅ we conclude with Lemma 4.2P since Curl Φ = 0 on ΓRN . Otherwise we deduce from (1.8) with τh := Id ∈ 0,h and γh = 0 that Ω tr σh dx = 0. Thus we obtain from (5.1) R R R dx = Ω tr(ε − CE(v))dx = Ω tr CE(u − v)dx Ω tr CurlCurlΦ R R = (2λ + 2µ) Ω div(u − v)dx = (2λ + 2µ) ∂Ω n(u − v)ds = 0 since u, v ∈ W01,2 (Ω; R2 ). In view of Poincar´e’s inequality and Lemma 4.2 we obtain kCurl Φk1,2;Ω ≤ c17 k∇Curl Φk2;Ω ≤ c17 c13 kCurl Curl ΦkC−1 ;Ω . √ √ = 2c17 c13 max{c8 c10 , The assertion of the lemma follows with c 16 √ 2c9 }. u t Lemma 5.2. There exists a constant c18 such that we have kE(v)k2C;2;Ω + kφk2C−1 ;2;Ω ≤ c218 khT div εk22;Ω + kSkw(σh )k22;Ω 1/2

+khE εnk22;ΓN .

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C. Carstensen, G. Dolzmann

Proof. It follows from (5.1) and (5.2) since ε ∈ H(div; Ω) and ε+Skw(σh ) is symmetric by an integration by parts that Z kE(v)k2C;2;Ω = E(v) : (ε + Skw(σh ))dx Ω Z Z = ∇v : εdx + ∇v : Skw(σh )dx Ω Ω Z Z Z = − hv, div εidx + hv, εnids + ∇v : Skw(σh )dx. Ω

∂Ω

Ω

The definition of the continuous and the discrete problem implies Z hdiv ε, vh idx = 0 ∀ vh ∈ Uh Ω

and therefore, when c19 is the constant in Korn’s inequality, Z Z hv, div εidx = hv − Ph0 v, div εidx Ω

Ω

0 ≤ khT div εk2;Ω kh−1 T (v − Ph v)k2;Ω ≤ c7 |v|1,2;Ω khT div εk2;Ω ≤ c7 c19 kE(v)k2;Ω khT div εk2;Ω . 1/2

We use the trace inequality kvk2;E ≤ c20 hE (h−1 T kvk2;T + k∇vk2;T ) to estimate the boundary integral. By definition of Σg,h Z Z X hv, εnids = hv − Ph0 v, εnids ≤ kv − Ph0 vkE,2 kεnk2;E ΓN

ΓN

E∈Eh,N 1/2

≤ (c7 + 1)c20 |v|1,2;Ω khE εnk2;ΓN √ and the proof of the lemma follows with c18 = 3c14 c19 (c7 + 1).

t u

Throughout the rest of the section we use the notation ρh := C−1 σh + γh , ρ := C−1 σ+γ = ∇u. Since Lemma 5.1 and Lemma 5.2 provide an estimate for kσ − σh kC−1 ;2;Ω it suffices to bound kρ − ρh k2;Ω in order to obtain an estimate for kγ − γh k2;Ω . Lemma 5.3. There exists a constant c21 such that we have  ||ρ − ρh ||C;Ω ≤ c21 ||hT curlh ρh ||22;Ω + ||hT div ε||22;Ω (5.5)

1/2

1/2

+||Skw(σh )||22;Ω + ||hE J(ρh t)||22;E 0 + ||hE εn||22;EN h

1/2

.

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201

Proof. In view of Lemma 3.1 there exist f ∈ W 1,2 (Ω; R2 ), f = ci on Γi , c0 = 0 and q ∈ W 1,2 (Ω; R2 ), q = 0 on ΓD such that ρ − ρh = C−1 Curl f + ∇q, Z C(ρ − ρh ) : (ρ − ρh )dx = C−1 Curl f : Curl f dx Ω ΩZ + C∇q : ∇qdx.

(5.6)

Z

Ω

The first term on the right hand side can be estimated by Z Z C−1 Curl f : Curl f dx = (ρ − ρh ) : Curl f dx Ω ΩZ =− ρh Curl f dx. Ω

Let Rh f ∈ L11 be the Cl´ement interpolation of f . Since Curl Rh f is an admissible test tensor we deduce in view of (5.4) with an integration by parts Z Z − ρh : Curl f dx = − ρh : Curl(f − Rh f )dx ΩZ Ω Z = hcurl ρh , f − Rh f idx − hJ(ρh t), f − Rh f ids Ω

≤

X

Eh

kcurl ρh k2;T kf − Rh f k2;T +

T ∈Th

X

kJ(ρh t)k2;E kf − Rh f k2;E

E∈Eh0

 X 1/2 √ ≤ c8 c10 kf k1,2;Ω h2T kcurl ρh k22;T +

√

T ∈Th

2c9 kf k1,2;Ω

≤ c22 k∇f k2;Ω

 X

 X T ∈Th

with c22 = we deduce

√

hE kJ(ρh t)k22;E

E∈Eh0

h2T kcurl ρh k22;T +

1/2

X

hE kJ(ρh t)k22;E

1/2

E∈Eh0

√ √ 2c17 max{c8 c10 , 2c9 }. Since k∇f k2;Ω = kCurl f k2;Ω

kCurl f kC−1 ,2;Ω  X 1/2 X ≤ c22 h2T kcurl ρh k22;T + hE kJ(ρh t)k22;E . T ∈Th

E∈Eh0

202

C. Carstensen, G. Dolzmann

Taking the symmetric part in (5.1) and (5.6) we get Sym(ε) = CE(v) + Curl Curl Φ, SymC(ρ − ρh ) = Sym(ε) = CE(q) + Sym(Curl f ), hence CE(v − q) = Sym(Curl f ) − Curl Curl Φ. Thus we may estimate kE(v − q)k2C;2;Ω Z Z = CE(v − q) : E(v − q)dx = Sym(Curl f ) : E(v − q)dx Ω Ω Z = Curl f : E(v − q)dx ≤ kCurl f kC−1 ;Ω kE(v − q)kC;2;Ω , Ω

and hence kE(v − q)kC;Ω ≤ kCurl f kC−1 ;Ω . By Korn’s inequality we have Z CSym(∇q) : Sym(∇q)dx k∇qkC;Ω = ΩZ + CSkw(∇q) : Skw(∇q)dx Ω Z 2 ≤ kE(q)kC;Ω + 2µ |∇q|2 dx ≤ (1 + 2µc219 )kE(q)k2C;Ω Ω

and therefore we obtain by the triangle inequality, the estimates above, and Lemma 5.2 kρ − ρh k2C;Ω = k∇qk2C;Ω + kCurl f k2C−1 ;Ω ≤ (1 + 2µc219 )kE(q)k2C;Ω + kCurl f k2C−1 ;Ω ≤ 2(1 + 2µc219 )(kE(q − v)k2C;Ω + kE(v)k2C;Ω ) + kCurl f k2C−1 ;Ω ≤ (2(1 + 2µc219 ) + 1)kCurl f k2C−1 ;Ω + 2(1 + 2µc219 )kE(v)k2C;Ω   X X ≤ c222 (4µc219 + 3) h2T kcurl ρh k22;T + hE kJ(ρh t)k22;E +

2(2µc219

+

T ∈Th 1)c218

E∈Eh

 1/2 · khT div εk22;Ω + kSkw(σh )k22;Ω + khE εnk22;ΓN . 

The assertion of the lemma follows with c21 = max{c222 (4µc219 + 3), 2(2µc219 + 1)c218 }1/2 . u t The next step in the proof of Theorem 1.1 is an estimate for the displacement error e = u − uh . The proof requires a duality argument and relies on the regularity assumption (1.4).

Error estimates in elasticity

203

Lemma 5.4. If the regularity assumption (1.4) holds, then there exists a constant c23 such that  kek2;Ω ≤ c23 khT div εk22;Ω + khT Skw(σh )k22;Ω 1/2 1/2 + inf khT (C−1 σh + γh − ∇vh )k22;Ω + khE εnk22;ΓN . vh ∈Uh

Proof. Let z ∈ W 2,2 (Ω) be the solution of the problem div CE(z) = e

in Ω,

z = 0 on ΓD ,

and

CE(z)n = 0 on ΓN ,

and let τ := CE(z). By assumption (1.4), kzk2,2;Ω + kτ k1,2;Ω ≤ c1 kek2;Ω . Consequently, by (2.1), (1.8), (2.5) and an integration by parts Z Z Z 2 kek2;Ω = hu − uh , div τ idx = − ∇u : τ dx − huh , div Πh τ idx Ω Z Ω ZΩ (∇vh − ∇u) : τ dx + (C−1 σh + γh − ∇vh ) : Πh τ dx = Ω Z Ω + ∇vh : (Πh τ − τ )dx. Ω

The last term on the right hand side vanishes according to (2.3). By the definition of τ and (2.2) we deduce Z 2 kek2;Ω = (∇vh − ∇u) : τ dx Ω Z + (C−1 σh + γh − ∇vh ) : (Πh τ − τ )dx ZΩ + (C−1 σh + γh − ∇vh ) : τ dx Z Ω = (C−1 σh + γh − ∇vh ) : (Πh τ − τ )dx Ω Z + (C−1 σh + γh − ∇u) : CE(z)dx Ω

≤ c6 khT (C−1 σh + γh − ∇vh )k2;Ω |τ |1,2;Ω Z + (C−1 σh + γh − ∇u) : CE(z)dx. Ω

The second term on the right hand side can be rewritten as Z Z −1 (C σh + γh − E(u)) : CE(z)dx = − (σ − σh ) : E(z)dx. Ω

Ω

204

C. Carstensen, G. Dolzmann

Writing E(z) = ∇z − Skw(∇z) we obtain by an integration by parts Z (C−1 σh + γh − E(u)) : CE(z)dx Ω Z Z = hdiv(σ − σh ), zidx − h(σ − σh )n, zids Ω ΓN Z + (Skw(σh ) : Skw(∇z))dx. Ω

The orthogonal projection Ph0 z of z onto L00 is well defined and we deduce Z Z hdiv ε, zidx = hdiv ε, z − Ph0 zidx Ω

Ω

1 (z − Ph0 z)k2;Ω hT ≤ c7 khT div εk2;Ω |z|1,2;Ω ≤ c7 c1 khT div εk2;Ω kek2;Ω . ≤ khT div εk2;Ω k

The boundary term can be estimated as in Lemma 5.2 and we obtain Z 1/2 hεn, zidx ≤ c1 c20 (c7 + 1)kek2;Ω khE εnk2;ΓN . Γh

In order to bound the last term we define ξh = Rh Skw(∇z) ∈ Wh and infer with (1.9) Z Z Skw(σh ) : Skw(∇z)dx = Skw(σh ) : (Skw(∇z) − ξh )dx Ω

Ω

1 ≤ khT Skw(σh )k2;Ω k (Skw(∇z) − ξh )k2;Ω hT ≤ c8 c2 khT Skw(σh )k2;Ω kek2;Ω .

The estimates above imply kek2;Ω ≤ c6 c1 inf kC−1 σh + γh − ∇vh k2;Ω + c7 c1 khT div εk2;Ω vh ∈Uh

+ c8 c2 khT Skw(σh )k2;Ω + c20 (c7 + 1)kεnk2;ΓN . This proves the lemma with c23 = 2 max{c1 c6 , c1 c7 , c1 c20 , c8 c2 }.

t u

Remark. For the higher order methods BDMSk we have the improved estimate  kek2;Ω ≤ c23 kh2T div εk2;Ω + khT Skw(σh )k2;Ω  1/2 + inf khT (C−1 σh + γh − ∇vh )k2;Ω + khE εnk2;ΓN vh ∈Uh

Error estimates in elasticity

205

since we may use the interpolation onto L11 , instead of the orthogonal projection onto L00 . Proof of Theorem 1.1: Recall from (5.3) that kεk2C−1 ;2;Ω = kCurl Curl Φk2C−1 ;2;Ω + kE(v)k2C;2;Ω + kφk2C−1 ;2;Ω . In view of Lemma 5.1 and 5.2 we obtain kεk2C−1 ;2;Ω ≤ 2 max{c216 , c218 }η 2 =: c224 η 2 . Moreover, by the triangle inequality and Lemma 5.3 1 1 kγ − γh k2C;Ω ≤ (kρ − ρh k2C;Ω + kσ − σh k2C−1 ;Ω ) 2µ µ 2 2 2 ≤ 2(c21 + c24 )η =: c225 η 2 .

kγ − γh k22;Ω =

The theorem follows with Lemma 5.4 and for c3 = c23 + c24 + c25 .

t u

6. Proof of the lower bound The lower bounds in Theorem 1.2 rely on two main ingredients: a localization technique introduced in [V] and classical inverse estimates in finite element spaces. We briefly summarize the relevant results (see [V] for more details). There exists an extension operator L : C 0 (E) → C 0 (T ), T ∈ Th , E ∈ Eh , which extends polynomials of degree k on E to polynomials of same degree on T and satisfies (Lp)|E = p|E for all p ∈ Pk (E). Finally we let ψT = (maxT bT )−1 bT and we denote by ψE the uniquely determined piecewise quadratic function on ωE which satisfies supp ψE ⊂ ωE , ψE ≥ 0 and maxE ψE = 1. Lemma 6.1. ([V],Lemma 4.1) Let k ∈ N. Then there exist constants c26 , . . . , c28 , which depend only on k and the shape of the triangles such that we have for all T ∈ Th , E ∈ Eh with E ⊂ T¯ and all u ∈ Pk (T ), v ∈ Pk (E) 1/2

(6.1)

kψT uk2;T ≤ kuk2;T ≤ c26 kψT uk2;T ,

(6.2)

kψE vk2;T ≤ kvk2;E ≤ c27 kψE vk2;E ,

(6.3)

−1 c26 hE kvk2;E ≤ kψE Lvk2;T ≤ c28 hE kvk2;E .

1/2

1/2

1/2

1/2

Lemma 6.2. ([Ci1], Lemma 3.2.6) Assume that v ∈ Pk (T ) and 0 ≤ ` ≤ m. Then there exists a constant c29 , which depends only on the shape of the triangles, k, ` and m such that (6.4)

|v|m,2;T ≤ c29 h`−m |v|2;`;T . T

206

C. Carstensen, G. Dolzmann

In Lemma 6.3 and 6.4 we give bounds on the different contributions in the error estimator η in (1.10). Recall that ρh = C−1 σh + γh , ρ = C−1 σ + γ = ∇u. Lemma 6.3. There exists a constant c30 such that for all T ∈ Th   −1 −1 hT kcurl(C σh + γh )k2;T ≤ c30 kC (σ − σh ) + γ − γh k2;T . Proof. It follows from (6.1) and an integration by parts that Z 1/2 −2 2 2 c26 kcurl ρh k2;T ≤ kψT curl ρh k2;T = − ψT hcurl(ρ − ρh ), curl ρh idx T Z = (ρ − ρh ) : Curl(ψT curl ρh )dx T

≤ kρ − ρh k2;T kCurl(ψT curl ρh )k2;T . From (6.4) and (6.1) we infer −1 kCurl(ψT curl ρh )k2;T ≤ c29 h−1 T kψT curl ρh k2;T ≤ c29 hT kcurl ρh k2;T .

This proves the lemma with c30 = c226 c29 .

t u

Lemma 6.4. There exists a constant c31 such that the following estimate holds for all E ∈ Eh0 1/2

hE kJ((C−1 σh + γh )t)k2;E ≤ c31 kC−1 (σ − σh ) + γ − γh k2;ωE . Proof. Let vh = J((C−1 σh + γh )t). We obtain from (6.2) Z 1/2 −1 2 2 2 2 kJ((C σh + γh )t)k2;E ≤ c27 kψE Lvh k2;E = c27 ψE |Lvh |2 ds. E

An integration by parts in each triangle of ωE yields Z Z hcurl ρh , ψE Lvh idx + ρh : Curl(ψE Lvh )dx ωE ωE Z = hJ(ρh t), ψE Lvh ids, E

and so Z 0=

ωE

(hcurl ρ, ψE Lvh i + ρ : Curl(ψE Lvh ))dx.

Error estimates in elasticity

207

Therefore we obtain 1/2

kψE vh k22;E Z Z = − hcurl(ρ − ρh ), ψE Lvh idx − (ρ − ρh ) : Curl(ψE Lvh )dx ωE ωE Z Z = hcurl ρh , ψE Lvh idx − (ρ − ρh ) : Curl(ψE Lvh )dx ωE

ωE

≤ kcurl ρh k2;ωE kψE Lvh k2;ωE + kρ − ρh k2;ωE kCurl(ψE Lvh )k2;ωE . 1/2

−1/2

Let c32 be a constant such that hE /hT ≤ c32 hE for all T ∈ Th with E ⊂ T¯. Clearly, c32 depends only on the shape of the triangles in Th . We conclude with Lemma 6.2 and 6.3 1/2

hE kvh k2;E ≤ c28 hE kcurl ρh k2;ωE + c29 c28 c32 kρ − ρh k2;ωE

≤ c28 c32 (c30 + c29 )kC−1 (σ − σh ) + γ − γh k2;ωE .

This implies the result with c31 = c27 c32 (c30 + c29 ).

t u

Lemma 6.5. There exists a constant c33 such that the following estimate holds for all E ∈ Eh,N 1/2

hE k(σ − σh )nk2;E ≤ c33 (khT div(σ − σh )k2;ωE + kσ − σh k2;ωE ). Proof. Let vh = (σ − σh )n. Then Z hdiv(σ − σh ), ψE Lvh idx T Z Z = − (σ − σh ) : ∇(ψE Lvh )dx + |(σ − σh )n|2 ψE ds T

E

and thus Z |(σ − σh )n|2 ψE ds E

≤ kdiv(σ − σh )k2;T kψE Lvh k2;T + kσ − σh k2;T k∇(ψE Lvh )k2;T .

Hence we obtain from (6.2) and Lemma 6.2 that k(σ − σh )nk22;E is bounded from above by 1/2

c226 {c28 kdiv(σ − σh )k2;T + c29 c28 h−1 T kσ − σh k2;T }hE kvh k2;E and we conclude 1/2

hE k(σ − σh )nk2;E ≤ c33 (hT kdiv(σ − σh )k2;T + kσ − σh k2;T ), where c33 = c226 c28 c32 max{c29 , 1}. This implies the assertion of the lemma. u t

208

C. Carstensen, G. Dolzmann

Lemma 6.6. There exists a constant c34 such that we have   hT kC−1 σh + γh − ∇uh k2;T ≤ c34 ku − uh k2;T + hT kρ − ρh k2;T . Proof. It follows from (6.1) and an integration by parts that Z −2 2 c26 kρh − ∇uh k2;T ≤ ψT (ρh − ∇uh ) : (ρh − ∇uh )dx T Z ψT (ρ − ρh ) : (ρh − ∇uh )dx = − T Z + ψT (ρ − ∇uh ) : (ρh − ∇uh )dx T  ≤ kρ − ρh k2;T kρh − ∇uh k2;T  + ku − uh k2;T kdiv(ψT (ρh − ∇uh ))k2;T   kρh − ∇uh k2;T . ku − u k ≤ kρ − ρh k2;T + c29 h−1 2;T h T This proves the lemma c34 = c226 max{1, c29 }.

t u

Proof of Theorem 1.2: The proof is an immediate consequence of Lemmas 6.3 - 6.6. u t Acknowledgement. Most of the work was done while GD visited the Mathematisches Seminar at the Christian–Albrechts–Universit¨at zu Kiel, whose hospitality is gratefully acknowledged. He was also partially supported by ARO and NSF though grants to the Center for Nonlinear Analysis at Carnegie Mellon University, Pittsburgh.

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