Finite Element Methods for Linear Elasticity

Transcription

1 Finite Element Metods for Linear Elasticity Ricard S. Falk Department of Matematics - Hill Center Rutgers, Te State University of New Jersey 110 Frelinguysen Rd., Piscataway, NJ falk@mat.rutgers.edu Key words: mixed metod, finite element, elasticity 1 Introduction Finite element metods wit strong symmetry Exterior calculus on R n Basic finite element spaces and teir properties Mixed formulation of te equations of elasticity wit weak symmetry From te de Ram complex to an elasticity complex wit weak symmetry Well-posedness of te weak symmetry formulation of elasticity Conditions for stable approximation scemes Stability of finite element approximation scemes Refined error estimates Examples of stable finite element metods for te weak symmetry formulation of elasticity References Introduction Te equations of linear elasticity can be written as a system of equations of te form Tis work supported by NSF grants DMS and DMS /8/06.

2 2 Ricard S. Falk Aσ = ε(u), div σ = f in. (1) Here te unknowns σ and u denote te stress and displacement fields caused by a body force f acting on a linearly elastic body wic occupies a region R n, wit boundary. Ten σ takes values in te space S = R n n sym of symmetric n n matrices and u takes values in V = R n. Te differential operator ε is te symmetric part of te gradient, (i.e., (ε(u)) ij = ( u i / x j + u j / x i )/2), div denotes te divergence operator, applied row-wise, and te compliance tensor A = A(x) : S S is a bounded and symmetric, uniformly positive definite operator reflecting te properties of te material at eac point. In te isotropic case, te mapping σ Aσ as te form ( ) σ, Aσ = 1 2µ λ 2µ + nλ tr(σ)i were λ(x), µ(x) are positive scalar coefficients, te Lamé coefficients, and tr denotes te trace. If te body is clamped on te boundary, ten te proper boundary condition for te system 1 is u = 0 on. For simplicity, tis boundary condition will be assumed trougout te discussion ere. However, tere are issues tat arise wen oter boundary conditions are assumed (e.g.,, traction boundary conditions σn = 0). Te modifications needed to deal wit suc boundary conditions are discussed in detail in [9]. In te case wen A is invertible, i.e., σ = A 1 ε(u) = Cε(u), ten for isotropic elasticity, Cτ = 2µ(τ +λ tr τi). We may ten formulate te elasticity system weakly in te form: Find σ L 2 (, S), u H 1 (; V) suc tat σ : τ dx Cε(u) : τ dx = 0, τ L 2 (, S), σ : ε(v) dx = f v dx, v H 1 (; V), were σ : τ = n i,j=1 σ ijτ ij. Note tat in tis case, we may eliminate σ completely to obtain te pure displacement formulation: Find u H 1 (; V) suc tat Cε(u) : ε(v) dx = f v dx, v H 1 (; V). As te material becomes incompressible, i.e., λ, tis will not be a good formulation, since te operator norm of C is also approacing infinity. Instead, we can consider a formulation involving u and a new variable p = (λ/[2µ + nλ]) tr σ. Taking te trace of te equation Aσ = ε(u), we find tat div u = λ 1 p. Ten we may write σ = 2µε(u) + pi, and tus obtain te variational formulation: Find u H 1 (; V), p L 2 0() = {p L 2 () : p dx = 0}, suc tat 2µ ε(u) : ε(v) dx + p div v dx = f v dx, v H 1 (; V), div u q dx = λ 1 p q dx, q L 2 0().

3 Finite Element Metods for Linear Elasticity 3 Tis formulation makes sense even for te limit λ and in tat case gives te stationary Stokes equations. Even in te case of nearly incompressible elasticity, one sould apply metods tat are stable for te Stokes equations. Since suc metods will be considered in oter lectures, we will not consider tem ere. Instead, we now turn to oter types of weak formulations involving bot σ and u. One of tese is to seek σ H(div, ; S), te space of squareintegrable symmetric matrix fields wit square-integrable divergence, and u L 2 (; V), satisfying (Aσ : τ + div τ u) dx = 0, τ H(div, ; S), (2) div σ v dx = f v dx, v L 2 (; V). A second weak formulation, tat enforces te symmetry weakly, seeks σ H(div, ; M), u L 2 (; V), and p L 2 (; K) satisfying (Aσ : τ + div τ u + τ : p) dx = 0, τ H(div, ; M), div σ v dx = f v dx, v L 2 (; V), (3) σ : q dx = 0, q L 2 (; K), were M is te space of n n matrices, K te subspace of skew-symmetric matrices, and te compliance tensor A(x) is now considered as a symmetric and positive definite operator mapping M into M. Stable finite element discretizations wit reasonable computational complexity based on te variational formulation 2 ave proved very difficult to construct. In particular, it is not possible to simply take multiple copies of standard finite elements for scalar elliptic problems, since te resulting stress matrix will not be symmetric. One successful approac as been to use composite elements, in wic te approximate displacement space consists of piecewise polynomials wit respect to one triangulation of te domain, wile te approximate stress space consists of piecewise polynomials wit respect to a different, more refined, triangulation [22, 30, 24, 4]. In two space dimensions, te first stable finite elements wit polynomial sape functions were presented in [10]. Te simplest and lowest order spaces in te family of spaces constructed consist of discontinuous piecewise linear vector fields for displacements and a stress space wic is locally te span of piecewise quadratic matrix fields and te cubic matrix fields tat are divergence-free. Hence, it takes 24 stress and six displacement degrees of freedom to determine an element on a given triangle. A simpler first-order element pair wit 21 stress and tree displacement degrees of freedom per triangle is also constructed in [10]. All of tese elements require vertex degrees of freedom. To obtain simpler elements, te same autors also considered nonconforming elements in [12]. One

4 4 Ricard S. Falk element constructed tere approximates te stress by a nonconforming piecewise quadratic wit 15 degrees of freedom and approximates te displacement field by discontinuous linear vectors (6 local degrees of freedom). A second element reduces te number of degrees of freedom to 12 and 3, respectively. See also [11] for an overview. In tree dimensions, a piecewise quartic stress space is constructed wit 162 degrees of freedom on eac tetraedron in [1]. Because of te lack of suitable mixed elasticity elements tat strongly impose te symmetry of te stresses, a number of autors ave developed approximation scemes based on te weak symmetry formulation 3: see [22], [2], [3], [27], [28], [29], [5], [25], [26], [21]. Altoug 2 and 3 are equivalent on te continuous level, an approximation sceme based on 3 may not produce a symmetric approximation to te stress tensor, depending on te coices of finite element spaces. Tese notes will mainly concentrate on te development and analysis of finite element approximations of te equations of linear elasticity based on te mixed formulation 3 wit weak symmetry. Using a generalization of an approac first developed in [8] in two dimensions and [6] in tree dimensions, and ten expanded furter in [9], we establis a systematic way to obtain stable finite element approximation scemes. Te families of metods developed in [8] and [6] are te prototype examples and we sow tat tey satisfy te conditions we develop for stability. However, te somewat more general approac we present ere allows us to analyze some of te previously proposed scemes discussed above in te same systematic manner and also leads to a new sceme. Before considering weakly symmetric scemes, we first discuss some metods based on te strong symmetry formulation 2. 2 Finite element metods wit strong symmetry In tis section, we consider finite element metods based on te variational formulation 2. Tus, we let Σ H(div, ; S) and V L 2 (; V) and seek σ Σ and u V satisfying (Aσ : τ+div τ u ) dx = 0, τ Σ, div σ v dx = f v dx, v V. Tis is in a form to wic one may apply te standard analysis of mixed finite element teory (e.g., [14, 15, 20, 18]. We note tat in te case of isotropic elasticity, if we write σ = σ D + (1/n) tr σi, were tr σ D = 0, ten σ 2 0 = σ D (1/n) tr σ 2 0 and so Aσ : σ dx = [ ] 1 2µ σ 1 D : σ D + (tr σ)2 dx. 2µ + nλ Tus, tis form is not uniformly coercive as λ. However, for all σ satisfying

5 Finite Element Metods for Linear Elasticity 5 tr σ dx = 0, div σ = 0, (4) one can sow (cf. [15]) tat tr σ 0 C σ D 0, and ence (Aσ, σ) α σ 2 H(div) for all σ satisfying 4, wit α independent of λ. Tis is wat is needed to satisfy te first Brezzi condition wit a constant independent of λ. A simple result of mixed finite element teory, giving conditions under wic te second Brezzi condition is satisfied, and tat fits te metods tat we will consider ere, is te following. Teorem 2.1 Suppose tat for every τ H 1 (), tere exists Π τ Σ satisfying div(τ Π τ) v dx = 0, v V, Π τ H(div) C τ H(div). Furter suppose tat for all τ Σ satisfying div τ v dx = 0, v V, tat div τ = 0. Ten for all v V, σ σ 0 C σ Π σ 0, u u 0 C( u v 0 + σ σ 0 ). To describe some finite element metods based on te strong symmetry formulation, we let P k (X, Y ) denote te space of polynomial functions on X of degree at most k and taking values in Y. 2.1 Composite elements One of te first metods based on te symmetric formulation was te metod of [30] analyzed in [24]. We describe below only te triangular element (tere was also a similar quadrilateral element). Te basic idea is to approximate te stresses by a composite finite element. Starting from a mes T of triangles, one connects te barycenter of eac triangle K to te tree vertices to form a composite element made up of tree triangles, i.e., K = T 1 T 2 T 3. We ten define Σ = {τ H(div, ; S) : τ Ti P 1 (T i, S)}, V = {v L 2 () : v K P 1 (K, R 2 }. Composite Element Tus te displacements are defined on te coarse mes T. By te definition of Σ K, we start from a space of 27 degrees of freedom, on wic we impose at most 12 constraints tat require tat τn be continuous across eac of te tree internal edges of K. In fact, tese constraints are all independent. Ten, a key point is to sow tat on eac K, τ is uniquely determined by

6 6 Ricard S. Falk te following 15 degrees of freedom (i) te values of τ n at two points on eac edge of K and (ii) K τ ij dx, i, j = 1, 2. It is ten easy to ceck tat if K div τ v dx = 0 for v P 1(K, R 2 ), ten div τ = 0. If we define Π to correspond to te degrees of freedom, ten it is also easy to ceck tat K div(τ Π τ) v dx = 0 for v P 1 (K, R 2 ). After establising te H(div, ) norm bound on Π σ, one easily obtains te error estimates: σ σ 0 C 2 σ 2, u u 0 C 2 ( σ 2 + u 2 ). Te use of composite finite elements to approximate te stress tensor was extended to a family of elements in [4]. For k 2, Σ = {τ H(div, ; S) : τ Ti P k (T i, S)}, V = {v L 2 () : v K P k 1 (K, R 2 )}. Te space Σ is constructed so tat if τ Σ K, ten τn will be continuous across internal edges, and in addition div τ P k 1 (K, R 2 ), i.e., it is a vector polynomial on K, not just on eac of te T i. Te degrees of freedom for an element τ Σ on te triangle K are cosen to be (τn) p ds, p P k (e, R 2 ), for eac edge e, e τ : ϱ dx, ϱ ε(p k 1 (K, R 2 )) + airy(λ 2 1λ 2 1λ 2 3P k 4 (K, R)), K were te λ i are te barycentric coordinates of K and ( ) Jφ airy φ = 2 φ/ y 2 2 φ/ x y 2 φ/ x y 2 φ/ x 2. One can sow tat dim Σ K = (3/2)k 2 + (9/2)k + 6. In te lowest order case k = 2, tere are 18 edge degrees of freedom and 3 interior degrees of freedom on eac macro-triangle K. For te general case k 2, it is sown tat u u 0 C r u r, 2 r k, σ σ 0 C r u r+1, 1 r k + 1, div(σ σ ) 0 C r div σ r, 0 r k. 2.2 Non-composite elements of Arnold and Winter We now turn to te more recent metods tat produce approximations to bot stresses and displacements tat are polynomial on eac triangle T T (since tere are no macro triangles, we no longer use K to denote a generic triangle). Te approac of [10] is based on te use of discrete differential complexes and te close relation between te construction of stable mixed

7 Finite Element Metods for Linear Elasticity 7 finite element metods for te approximation of te Laplacian and discrete versions of te de Ram complex R C () curl C (; R 2 ) div C () 0. If we assume tat is simply-connected, tis sequence is exact (i.e., te range of eac map is te kernel of te following one). As discussed later in tis paper, many of te standard spaces leading to stable mixed finite element metods for Laplace s equation ave te property tat te following diagram commutes id R C curl () C (, R 2 div ) C () 0 I Π P, (5) R curl div Q Σ V 0 were I, Π, P are te natural interpolation operators into te corresponding finite element spaces Q, Σ, and V. For example, te simplest case is wen Q is te space of continuous piecewise linear functions, Σ te space of lowest order Raviart-Tomas elements, and V te space of piecewise constants. Te rigt alf of te commuting diagram, involving Π and P is a key result in establising te second Brezzi stability condition. See [7],and [9] for furter discussion of tis idea. Te starting point of [10] is tat tere is also an elasticity differential complex, wic summarizes important aspects of te structure of te plane elasticity system, i.e., P 1 () C () J C (, S) div C (, R 2 ) 0. (6) Again assuming tat is simply-connected, tis sequence is also exact. Tus tis sequence encodes te fact tat every smoot vector-field is te divergence of a smoot symmetric matrix-field, tat te divergence-free symmetric matrix-fields are precisely tose tat can be written as te Airy stress-field associated to some scalar potential, and tat te only potentials for wic te corresponding Airy stress vanises are te linear polynomials. Te result stated above is in terms of smoot functions, but analogous results old wit less smootness. For example, te sequence P 1 () H 2 () J H(div, ; S) div L 2 (, R 2 ) 0 (7) is also exact. Te well-posedness of te continuous problem, i.e., tat for every f L 2 (, R 2 ), tere exists a unique (σ, u) H(div, ; S) L 2 (, R 2 ) wic is a critical point of (1.1), follows from tis. Just as tere is a close relation between te construction of stable mixed finite element metods for te approximation of te Laplacian and discrete versions of te de Ram complex, tere is also a close relation between mixed finite elements for linear elasticity and discretization of te elasticity complex,

8 8 Ricard S. Falk given above. Te stable pairs of finite element spaces (Σ, V ) introduced in [10] ave te property tat div Σ = V, i.e., te sort sequence Σ div V 0 (8) is exact. Moreover, if tere are projections P : C (, R 2 ) V and Π : C (, S) Σ defined by te degrees of freedom tat determine te finite element spaces, it can be sown tat te following diagram commutes: C (, S) Π div C (, R 2 ) P (9) Σ div Te stability of te mixed metod follows from te exactness of 8, te commutativity of 9, and te well-posedness of te continuous problem. Information about te construction of suc finite element spaces can be gained by completing te sequence 8 to a sequence analogous to 6. For tis purpose, we set Q = {q H 2 () : Jq Σ }. Note Q is a finite element approximation of H 2 (). Moreover, tere is a natural interpolation operator I : C () Q so tat te following diagram wit exact rows commutes: V P 1 () C J () C div (, S) C (, R 2 ) 0 id I Π P P 1 () Q J Σ div V 0 For a description of tis construction, see [10]. As discussed tere, under quite general conditions, te existence of a stable pair of spaces (Σ, V ) approximating H(div, ; S) L 2 (, R 2 ), implies te existence of a finite element approximation Q of H 2 () related to Σ and V troug te diagram above. Te fact tat te space Q requires C 1 () finite elements represents a substantial obstruction to te construction of stable mixed elements, and in part accounts for teir slow development. In fact, te lowest order element proposed in [10] corresponds to coosing Q to be te Argyris space of C 1 piecewise quintic polynomials (te simplest coice). Since JQ Σ, one ten sees tat Σ must be a piecewise cubic space, and since te Argyris space as second derivative degrees of freedom at te vertices, te degrees of freedom for Σ will include vertex degrees of freedom, not usually expected for subspaces of H(div; ). Te family of elements developed in [10] cooses for k 1, te local degrees of freedom for Σ to be Σ T = P k+1 (T, S) + {τ P k+2 (T, S) : div τ = 0} = {τ P k+2 (T, S) : div τ P k (T, R 2 )}, V T = P k (T, R 2 ).

9 Finite Element Metods for Linear Elasticity 9 Now dim V T = (k + 2)(k + 1) and it is sown in [10] tat dim Σ T = (3k k + 28)/2 and tat a unisolvent set of local degrees of freedom is given by te values of 3 components of τ(x) at eac vertex x of T (9 degrees of freedom) te values of te moments of degree at most k of te two normal components of τ on eac edge e of T (6k + 6 degrees of freedom) te value of te moments T τ : φ dx, φ P k(t, R 2 ) + airy(b 2 T P k 2(T, R)). For tis family of elements, it is sown in [10] tat σ σ 0 C r σ r, 1 r k + 2, div(σ σ ) 0 C r div σ r, 0 r k + 1, u u 0 C r u r+1, 1 r k + 1. Tere is a variant of te lowest degree (k = 1) element involving fewer degrees of freedom. In tis element, one cooses V T to be te space of infinitesimal rigid motions on T, i.e., vector functions of te form (a by, c + bx). Ten Σ T = {τ P 3 (T, S) : div τ V T }. Te element diagram for te coice k = 1 and a simplified element are depicted below. Fig. 1. k = 1 and simplified Arnold-Winter elements In [12], te autors obtain simpler elements wit fewer degrees of freedom, and also avoid te use of vertex degrees of freedom by developing nonconforming elements. Corresponding to te coice V T = P 1 (T, R 2 ), one cooses for te stress sape functions Σ T = {τ P 2 (T, S) : n τn P 1 (e, R), for eac edge e of T }. Te space Σ T as dimension 15, wit degrees of freedom given by te values of te moments of degree 0 and 1 of te two normal components of τ on eac edge e of T (12 degrees of freedom), te value of te tree components of te moment of degree 0 of τ on T (3 degrees of freedom). Note tat tis element is a nonconforming approximation of H(div, ; S), since altoug t τn may be quadratic on an edge, only its two lowest order

10 10 Ricard S. Falk moments are determined on eac edge. Hence, τn may not be continuous across element boundaries. Tis space may be simplified in a manner similar to te lowest order conforming element, i.e., te displacement space may be cosen to be piecewise rigid motions and te stress space ten reduced by requiring tat te divergence be a rigid motion on eac triangle. Te local dimension of te resulting space is 12 and te first two moments of te normal traction on eac edge form a unisolvent set of degrees of freedom. Fig. 2. Two nonconforming Arnold-Winter elements As noted earlier, for k = 1, te corresponding space Q is te Argyris space consisting of C 1 piecewise quintic polynomials. Tere is also an analogous relationsip for te composite elements discussed earlier. For te element of [24], te space Q is te Cloug-Tocer composite H 2 element and for te element family of [4], te Q spaces are te iger order composite elements of [17]. Fig. 3. Q spaces for k = 1 conforming element, nonconforming element, and composite element of [24] Te remainder of tese notes will be devoted to te development and analysis of mixed finite element metods based on te formulation 3 of te equations of elasticity wit weak symmetry. An important advantage of suc an approac is tat it allows us to approximate te stress matrix by n copies of standard finite element approximations of H(div, ) used to discretize scalar second order elliptic problems. In fact, to develop our approximation scemes for 3, we will eavily exploit te many close connections between tese two problems. Altoug tere is some overead to te development, muc of te structure of tese connections is most clearly seen in te language of differential forms. Tus, we devote te next section to a brief overview of te necessary background material.

11 3 Exterior calculus on R n Finite Element Metods for Linear Elasticity 11 To simplify matters, we will consider exterior calculus on R n, and summarize only te specific results we will need. 3.1 Differential forms Suppose tat is an open subset of R n. For 0 k n, we let Λ k denote te space of smoot differential k-forms of, i.e., Λ k = Λ k () = C (; Alt k V), were Alt k V denotes te vector space of alternating k-linear maps on V. If ω Λ k (), tis means tat at eac point x, tere is a map ω x Alt k V, i.e, ω x assigns to eac k tuple of vectors v 1,..., v k of V, a real number ω x (v 1,..., v k ) wit te mapping linear in eac argument and reversing sign wen two arguments are intercanged. A general element of Λ k () may be written ω x = a σ dx σ(1) dx σ(k), 1 σ(1)< <σ(k) n were te a σ C (). If we allow instead a σ C p (), a σ L 2 (), a σ H s (), etc., we obtain te spaces C p Λ(), L 2 Λ(), H s Λ(), etc. Tus, wen n = 2, for k = 0, 1, 2, ω Λ k () will ave te respective forms w, w 1 dx 1 + w 2 dx 2, w dx 1 dx 2. To see te connection between differential forms and scalar and vector-valued functions, we may identify w Λ 0 () and w dx 1 dx 2 Λ 2 () wit te function w C () and w 1 dx 1 + w 2 dx 2 Λ 1 () wit te vector (w 1, w 2 ) or te vector ( w 2, w 1 ) C (; R 2 ). Te associated fields are called proxy fields for te forms. Wen n = 3, for k = 0, 1, 2, 3, ω Λ k () will ave te respective forms w, w 1 dx 1 + w 2 dx 2 + w 3 dx 3, w 1 dx 2 dx 3 w 2 dx 1 dx 3 + w 3 dx 1 dx 2, w dx 1 dx 2 dx 3. In tis case, we may identify w Λ 0 () and w dx 1 dx 2 dx 3 Λ 3 () wit te function w C () and w 1 dx 1 + w 2 dx 2 + w 3 dx 3 or w 1 dx 2 dx 3 w 2 dx 1 dx 3 + w 3 dx 1 dx 2 wit te vector (w 1, w 2, w 3 ) C (; R 2 ). Te correspondences are listed in Table 1. To evaluate ω x (v 1,, v k ), we need a formula for evaluating te k-form dx σ(1) dx σ(k) (v 1,..., v k ). Rater tan presenting te general case, we note tat for v, w, z R n, dx i (v) = v i, dx i dx j (v, w) = v i w j v j w i, and for n = 3, dx 1 dx 2 dx 3 (v, w, z) = det(v w z).

12 12 Ricard S. Falk Table 1. Correspondence between alternating algebraic forms on R 3 and scalars/vectors Alt 0 R 3 = R c c Alt 1 R 3 = R 3 u 1 dx 1 + u 2 dx 2 + u 3 dx 3 u Alt 2 R 3 = R 3 u 3 dx 1 dx 2 u 2 dx 1 dx 3 +u 1 dx 2 dx 3 u Alt 3 R 3 = R c dx 1 dx 2 dx 3 c For ω Alt j V and η Alt k V, te exterior product or wedge product ω η Alt j+k V is bilinear and associative, and satisfies te anti-commutativity condition η ω = ( 1) jk ω η, ω Alt j V, η Alt k V. Tus, dx i dx j = dx j dx i and so dx i dx i = 0. If ω = n i=1 w idx i Λ 1 () and η Λ 0 (), ten ω η simply multiplies eac of te coefficients w i by η. If η = n i=1 η idx i Λ 1 (), ten from te bilinearity and antisymmetry, we ave ω η = w 1 η 1 dx 1 dx 1 + w 1 η 2 dx 1 dx 2 + w 2 η 1 dx 2 dx 1 + w 2 η 1 dx 2 dx 2 = (w 1 η 2 w 2 η 1 )dx 1 dx 2, n = 2, ω η = (w 1 η 2 w 2 η 1 )dx 1 dx 2 + (w 1 η 3 w 3 η 1 )dx 1 dx 3 + (w 2 η 3 w 3 η 2 )dx 2 dx 3, n = 3. Finally, if η Λ 2 () = η 1 dx 2 dx 3 η 2 dx 1 dx 3 + η 3 dx 1 dx 2, ten ω η = (w 1 η 1 + w 2 η 2 + w 3 η 3 )dx 1 dx 2 dx 3. One can give a general formula for te wedge product, wic we omit ere. If ω x and η x Λ k () are given by a σ dx σ(1) dx σ(k), b σ dx σ(1) dx σ(k), 1 σ(1)< <σ(k) n 1 σ(1)< <σ(k) n 1 σ(1)< <σ(k) n respectively, we can define te inner products ω x, η x = a σ b σ, ω, η = ω x, η x dx 1 dx n, were dx 1 dx n is te volume form. A key object in our presentation is te exterior derivative d = d k : Λ k () Λ k+1 (), defined by d a σ dx σ(1) dx σ(k) = σ n i=1 a σ x i dx i dx σ(1) dx σ(k).

13 Finite Element Metods for Linear Elasticity 13 Table 2. Correspondences between differential forms ω on R 3 and scalar/vector fields w on. k Λ k () HΛ k () dω 0 C () H 1 () grad w 1 C (; R 3 ) H(curl, ; R 3 ) curl w 2 C (; R 3 ) H(div, ; R 3 ) div w 3 C () L 2 () 0 As we sall see below, te exterior derivative operator d corresponds to te standard differential operators grad, curl, div, and rot. Wen n = 2, if ω Λ 0 (), ten d 0 ω = w/ x 1 dx 1 + w/ x 2 dx 2 Λ 1 (). Identifying w/ x 1 dx 1 + w/ x 2 dx 2 wit te vector ( w/ x 1, w/ x 2 ), d 0 corresponds to grad. If instead, we identify w/ x 1 dx 1 + w/ x 2 dx 2 wit te vector ( w/ x 2, w/ x 1 ), ten d 0 corresponds to curl. If µ = w 1 dx 1 + w 2 dx 2 Λ 1 (), ten d 1 µ = ( w 2 / x 1 w 1 / x 2 )dx 1 dx 2 Λ 2 (). If we identify w 1 dx 1 + w 2 dx 2 wit te vector (w 1, w 2 ), ten d 1 corresponds to rot. If instead, we identify w 1 dx 1 + w 2 dx 2 wit te vector( w 2, w 1 ), ten d 1 corresponds to div. Wen n = 3, if ω Λ 0 (), ten d 0 ω = w/ x 1 dx 1 + w/ x 2 dx 2 + w/ x 3 dx 3 Λ 1 (). Identifying w/ x 1 dx 1 + w/ x 2 dx 2 + w/ x 3 dx 3 wit ( w/ x 1, w/ x 2, w/ x 3 ), d 0 corresponds to grad. If µ = w 1 dx 1 + w 2 dx 2 + w 3 dx 3 Λ 1 (), ten d 1 µ = ( w 3 / x 2 w 2 / x 3 )dx 2 dx 3 ( w 1 / x 3 w 3 / x 1 )dx 1 dx 3 + ( w 2 / x 1 w 1 / x 2 )dx 1 dx 2 Λ 2 (). Identifying w 1 dx 1 + w 2 dx 2 + w 3 dx 3 wit te vector (w 1, w 2, w 3 ), d 1 corresponds to curl. Finally, if µ = w 1 dx 2 dx 3 w 2 dx 1 dx 3 + w 3 dx 1 dx 2 Λ 2 (), ten d 2 µ = ( w 1 / x 1 + w 2 / x 2 + w 3 / x 3 )dx 1 dx 2 dx 3 Λ 3 (). Identifying µ wit (w 1, w 2, w 3 ), d 2 corresponds to div. Table 2 summarizes correspondences between differential forms and teir proxy fields in te case R 3. An important role in our analysis is played by te de Ram sequence, te sequence of spaces and mappings given in te notation of differential forms by: R Λ 0 () d0 Λ 1 () d1 d Λ n () 0. By introducing proxy fields and te usual differential operators, te de Ram complex (and its L 2 version) take te following forms. For R 3, R C () grad C (; R 3 ) curl C (; R 3 ) div C () 0, R H 1 () grad H(curl, ; R 3 ) curl H(div, ; R 3 ) div L 2 () 0. For R 2, te de Ram complex becomes or R C () grad C (; R 2 ) rot C () 0,

14 14 Ricard S. Falk R C () curl C (; R 2 ) div C () 0, depending on weter we identify w 1 dx 1 + w 2 dx 2 Λ 1 () wit te vector (w 1, w 2 ) or te vector ( w 2, w 1 ). Tere are also analogous L 2 complexes. 4 Basic finite element spaces and teir properties We now turn to te definition of te finite element spaces we sall use in our approximation scemes and teir properties. For tis we follow te approac developed in [9]. We begin by defining P r as te space of polynomials in n variables of degree at most r and P r Λ k as te space of differential k-forms wit coefficients belonging to P r. Let T be a triangulation of by n + 1 simplices T and set P r Λ k (T ) = {ω HΛ k () : ω T P r Λ k (T ) T T }, r 0 P r Λ k (T ) = {ω HΛ k () : ω T P r Λ k (T ) T T }, r 1, were P r Λ k (T ) := P r 1 Λ k (T ) + κp r 1 Λ k+1 (T ) and κ = κ k+1 : Λ k+1 (T ) Λ k (T ) is te Koszul differential defined for ω = σ a σdx σ(1) dx σ(k+1) Λ k+1 by κω = σ k+1 ( 1) i+1 a σ x σ(i) dx σ(1) dx ˆ σ(i) dx σ(k+1), i=1 were te notation ˆ dx σ(i) means tat te term is omitted in te sum. Note tat κ k κ k+1 = 0, and one can sow tat te Koszul complex 0 P r n Λ n () κn P r n+1 Λ () κ κ1 P r Λ 0 () 0, is exact. For R 3, tis complex becomes 0 P r 3 () x P r 2 (; R 3 ) x P r 1 (; R 3 ) x P r () 0. Comparing to te corresponding polynomial de Ram complex 0 P r () grad P r 1 (; R 3 ) curl P r 2 (; R 3 ) div P r 3 () 0, we see tat te Koszul differential increases polynomial degree and decreases te order of te differential form, wile exterior differentiation does exactly te opposite. We note tat P r Λ 0 (T ) = P r Λ 0 (T ), r 1 and P r Λ n (T ) = P r+1 Λn (T ), r 0. Using proxy fields, we can identify tese spaces of finite element differential forms wit finite element spaces of scalar and vector functions. In Tables 3 and 4, we summarize te correspondences between spaces of finite

15 Finite Element Metods for Linear Elasticity 15 Table 3. Correspondences between finite element differential forms and te classical finite element spaces for n = 2. k Λ k () Classical finite element space 0 P rλ 0 (T ) Lagrange elements of degree r 1 P rλ 1 (T ) Brezzi Douglas Marini H(div) elements of degree r 2 P rλ 2 (T ) discontinuous elements of degree r 0 P r Λ 0 (T ) Lagrange elements of degree r 1 P r Λ 1 (T ) Raviart Tomas H(div) elements of order r 1 2 P r Λ 2 (T ) discontinuous elements of degree r 1 Table 4. Correspondences between finite element differential forms and te classical finite element spaces for n = 3. k Λ k () Classical finite element space 0 P rλ 0 (T ) Lagrange elements of degree r 1 P rλ 1 (T ) Nédélec 2nd-kind H(curl) elements of degree r 2 P rλ 2 (T ) Nédélec 2nd-kind H(div) elements of degree r 3 P rλ 3 (T ) discontinuous elements of degree r 0 P r Λ 0 (T ) Lagrange elements of degree r 1 P r Λ 1 (T ) Nédélec 1st-kind H(curl) elements of order r 1 2 P r Λ 2 (T ) Nédélec 1st-kind H(div) elements of order r 1 3 P r Λ 3 (T ) discontinuous elements of degree r 1 element differential forms and classical finite element spaces in two and tree dimensions. Degrees of freedom for tese spaces are given as follows. For te space P r Λ k (T ), we use Tr f ω ν, ν P r j+k Λj k (f), f j (T ), f for k j min(n, r + k 1), were Tr f ω denotes te trace of ω on te face f and j (T ) is te set of all j-dimensional subsimplices generated by T. For example, wen n = 3, j (T ) is te set of vertices, edges, faces, or tetraedra in te mes T for j = 0, 1, 2, 3. In tis case, wen j = 0, i.e., f is a vertex, f Tr f ω means w(f), were w is te function associated wit ω Λ 0 (). Wen j = 1, i.e., f is an edge of a tetraedron, f Tr f ω = w t dµ, were f w is te vector associated to ω Λ 1 () and t is te unit tangent vector to f. Wen j = 2, i.e., f is a face of a tetraedron, f Tr f ω = w n dµ, were w f is te vector associated to ω Λ 2 () and n is te unit outward normal to f. Finally, wen j = 3, i.e., f is a tetraedron, f Tr f ω = f w dµ, were w is te function associated to ω Λ 3 ().

16 16 Ricard S. Falk Analogously, te degrees of freedom for te space Pr Λ k (T ) are given by Tr f ω ν, ν P r j+k 1 Λ j k (f), f j (T ), f for k j min(n, r + k 1). Note te key property tat te degrees of freedom for eac space are defined in terms of wedge products wit elements of te oter space. An important property of tese finite element spaces is tat tey form discrete de Ram sequences. In fact, as sown in [9], in n dimensions, tere are exactly 2 distinct sequences. Wen n = 2 and r 0, tese are 0 P r+2 Λ 0 (T ) d0 P r+1 Λ 1 (T ) d1 P r Λ 2 (T ) 0, 0 P r+1 Λ 0 (T ) d0 Pr+1 Λ1 (T ) d1 P r Λ 2 (T ) 0. Wen n = 3 and r 0, we ave te four sequences 0 P r+3 Λ 0 (T ) d0 P r+2 Λ 1 (T ) d1 P r+1 Λ 2 (T ) d2 P r Λ 3 (T ) 0, 0 P r+2 Λ 0 (T ) d0 P r+1 Λ 1 (T ) d1 Pr+1 Λ2 (T ) d2 P r Λ 3 (T ) 0, 0 P r+2 Λ 0 (T ) d0 Pr+2 Λ1 (T ) d1 P r+1 Λ 2 (T ) d2 P r Λ 3 (T ) 0, 0 P r+1 Λ 0 (T ) d0 Pr+1 Λ1 (T ) d1 Pr+1 Λ2 (T ) d2 P r Λ 3 (T ) 0. Te first and last of tese are exact sequences involving only te P r Λ k (T ) or Pr Λ k (T ) spaces alone, wile te middle two mix te two spaces. As we sall see, to obtain mixed finite element metods for elasticity wen n = 3, it is one of tese middle sequences tat will play a key role. To eac of te spaces P r Λ k (T ), we may associate a canonical projection operator Π(= Π T ) : C 0 Λ k () P r Λ k (T ) defined by te equations: Tr f Πω ν = Tr f ω ν, ν P r j+k Λj k (f), f j (T ), f f for k j min(n, r+k 1). Similarly, to eac of te spaces Pr Λ k (T ), we may associate a canonical projection operator Π(= Π T ) : C 0 Λ k () Pr Λ k (T ) defined by te equations Tr f Πω ν = Tr f ω ν, ν P r j+k 1 Λ j k (f), f j (T ), f f for k j min(n, r + k 1). A key property of tese projection operators is tat tey commute wit te exterior derivative, i.e., te following four diagrams commute.

17 Finite Element Metods for Linear Elasticity 17 Λ k () Π d k Λ k+1 () Λ k () Π Π d k Λ k+1 () Π P r Λ k (T ) d k Pr 1 Λ k+1 (T ) P r Λ k (T ) d k P r Λ k+1 (T ) Λ k () Π d k Λ k+1 () Λ k () Π Π d k Λ k+1 () Π P r Λ k (T ) d k P r Λ k+1 (T ) P r Λ k (T ) d k Pr 1 Λ k+1 (T ). Tese commuting diagrams will also play an essential role in te construction of stable mixed finite element approximation scemes for te equations of elasticity. 4.1 Differential forms wit values in a vector space To study te equations of linear elasticity in te language of differential forms, we will need to use differential forms wit values in a vector space. Let V and W be finite dimensional vector spaces. We ten define te space Λ k (V ; W ) of differential forms on V wit values in W. Te two examples we ave in mind are wen V = V = R n and W = V or W = K, te set of anti-symmetric matrices. Wen n = 2, ω Λ k (V; V), k = 0, 1, 2 will ave te respective forms ( w1 w 2 ), ( w11 w 21 ) dx 1 + ( w12 w 22 ) dx 2, wile ω Λ k (V; K) will ave te respective forms ( w1 w 2 ) dx 1 dx 2, wχ, w 1 χdx 1 + w 2 χdx 2, wχdx 1 dx 2, were χ = ( ) Recalling tat te 1-form w 1 dx 1 + w 2 dx 2 can be identified eiter wit te vector (w 1, w 2 ) or te vector ( w 2, w 1 ), we will ave te analogous possibilities in te case of vector or matrix-valued forms. Since we will be interested in de Ram sequences involving te operator div, we coose te second identification. Hence, ( w11 w 21 ) dx 1 + ( w12 w 22 te matrix ( ) W11 W 12 W 21 W 22 ) dx 2 Λ 1 (V; V) will be identified wit ( w12 w = 11 w 22 w 21 ), (10) and w 1 χdx 1 + w 2 χdx 2 Λ 1 (V; K) wit te vector ( w 2, w 1 ). Wen n = 3, ω Λ k (V; V) will ave te respective forms

18 18 Ricard S. Falk w 1 w 2, w 11 w 21 dx 1 + w 12 w 22 dx 2 + w 13 w 23 dx 3, w 3 w 31 w 32 w 33 w 11 w 21 dx 2 dx 3 w 12 w 22 dx 1 dx 3 + w 13 w 23 dx 1 dx 2, w 31 w 32 w 33 w 1 w 2 dx 1 dx 2 dx 3. w 3 Hence, Λ 0 (V; V) and Λ 3 (V; V) ave obvious identifications wit te space of 3 dimensional vectors and Λ 1 (V; V) and Λ 2 (V; V) ave obvious identifications wit te space of 3 3 matrices (i.e, W ij = w ij in bot cases). In fact, in treating te equations of elasticity on a domain R n, we sall represent te stress as an element of Λ (, V). To describe Λ k (V; K), it will be convenient to introduce te operator Skw taking a 3-vector to a skew-symmetric matrix. i.e., 0 w 3 w 2 Skw(w 1, w 2, w 3 ) = w 3 0 w 1. w 2 w 1 0 Ten ω Λ k (V; K) will ave te respective forms Skw(w 1, w 2, w 3 ), Skw(w 11, w 21, w 31 )dx 1 + Skw(w 12, w 22, w 32 )dx 2 + Skw(w 13, w 23, w 33 )dx 3, Skw(w 11, w 21, w 31 )dx 2 dx 3 Skw(w 12, w 22, w 32 )dx 1 dx 3 + Skw(w 13, w 23, w 33 )dx 1 dx 2, Skw(w 1, w 2, w 3 )dx 1 dx 2 dx 3. Note tat from te above formulas, tere is an obvious identification of Λ 0 (V; K) and Λ 3 (V; K) wit te space of 3-dimensional vectors and of Λ 1 (V; K) and Λ 2 (V; K) wit 3 3 matrices (again wit W ij = w ij in bot cases). In te mixed formulation of elasticity, we sall need a special operator S = S k : Λ k (V, V) Λ k+1 (V, K) defined as follows: First define K k : Λ k (; V) Λ k (; K) by K k ω = Xω T ωx T, were X = (x 1,, x n ) T. Ten define S k = d k K k K k+1 d k : Λ k (; V) Λ k+1 (; K). Using te definition of te exterior derivative, te definition of K, and te Leibniz rule, one can sow tat for any vector (v 1,..., v k+1 ),

19 (S k ω) x (v 1,..., v k+1 ) Finite Element Metods for Linear Elasticity 19 k+1 = ( 1) j+1 [v j ω T (v 1,, ˆv j,, v k+1 ) ω(v 1,, ˆv j,, v k+1 )vj T ], j=1 were te notation ˆv j means tat tis argument is omitted. Tus, S k is a purely algebraic operator. More specifically, we sall need tis operator wen k = n 2 and k =. We examine tese cases below for n = 2 and n = 3. Wen n = 2, we get for ω = (w 1, w 2 ) T, K 0 ω = (w 1 x 2 w 2 x 1 )χ, and after a simple computation, S 0 ω = (d 0 K 0 K 1 d 0 )ω = w 2 χdx 1 + w 1 χdx 2. Note tat S 0 is invertible wit If ω Λ 1 (V; V) is given by: S 1 0 [µ 1χdx 1 + µ 2 χdx 2 ] = (µ 2, µ 1 ) T. ω = w 1 dx 1 + w 2 dx 2, w 1 = (w 11, w 21 ) T, w 2 = (w 12, w 22 ) T, ten S 1 ω = (w 11 + w 22 )χdx 1 dx 2. If we identity ω wit a matrix W by ( ) ( ) W11 W 12 w12 w = 11, W 21 W 22 w 22 w 21 ten we can identify S 1 ω wit te matrix ( ) 0 W 12 W 21 = W W T 2 skw W. W 21 W 12 0 Wen n = 3, we get for ω = w 1 dx 1 + w 2 dx 2 + w 3 dx 3, wit w j = (w 1j, w 2j, w 3j ) T, S 1 ω = Skw( w 33 w 22, w 12, w 13 )dx 2 dx 3 Skw(w 21, w 11 w 33, w 23 )dx 1 dx 3 + Skw(w 31, w 32, w 11 w 22 )dx 1 dx 2. If we identify ω Λ 1 (V; V) wit a matrix W by W ij = w ij, and identify S 1 ω Λ 2 (V; K) wit te matrix U given by w 33 w 22 w 21 w 31 U = w 12 w 11 w 33 w 32, w 13 w 23 w 11 w 22 ten, W and U are related by te equations U = ΞW W T tr(w )I, W = Ξ 1 U U T 1 2 tr(u)i.

20 20 Ricard S. Falk Hence, S 1 is invertible. If ω = w 1 dx 2 dx 3 w 2 dx 1 dx 3 + w 3 dx 1 dx 2, ten 0 w 21 w 12 w 31 w 13 S 2 ω = w 12 w 21 0 w 32 w 23 dx 1 dx 2 dx 3. w 13 w 31 w 23 w 32 0 If we identify ω wit te matrix W given by W ij = w ij, ten by te above, S 2 ω may be identified wit te matrix 2 skw W. We easily obtain from te fact tat d k+1 d k = 0 and te definition S k = d k K k K k+1 d k tat d k+1 S k + S k+1 d k = 0. Tis identify, for k = n 2, i.e., d S n 2 + S d n 2 = 0 is te key identity in establising stability of continuous and discrete variational formulations of elasticity wit weak symmetry. Note tat tis formula is muc more complicated and also different in different dimensions wen stated in terms of proxy fields (wic are reasons wy we ave introduced differential forms). Wen n = 2 and k = 0, if we identify ω = (w 1, w 2 ) T Λ 0 (; V) wit te vector W, ten te formula (d 1 S 0 + S 1 d 0 )ω = 0 becomes (div W )χ + 2 skw curl W = 0. Wen n = 3 and k = 1, if we identify ω Λ 1 (; V) wit te matrix W, ten te formula (d 2 S 1 + S 2 d 1 )ω = 0 becomes Skw div(ξw ) 2 skw curl W = 0. 5 Mixed formulation of te equations of elasticity wit weak symmetry In order to write 3 in te language of exterior calculus, we will use te spaces of vector-valued differential forms presented in te previous section. We assume tat is a contractible domain in R n, V = R n, and K is again te space of skew-symmetric matrices. We sowed in te last section tat te operator S = S : Λ (; V) Λ n (; K) corresponds (up to a factor of ±2) to taking te skew-symmetric part of its argument. Setting d = d, te elasticity problem 3 becomes: Find (σ, u, p) HΛ (; V) L 2 Λ n (; V) L 2 Λ n (; K) suc tat Aσ, τ + dτ, u Sτ, p = 0, τ HΛ (; V), (11) dσ, v = f, v, v L 2 Λ n (; V), Sσ, q = 0, q L 2 Λ n (; K). Tis problem is well-posed in te sense tat, for eac f L 2 Λ n (; V), tere exists a unique solution (σ, u, p) HΛ (; V) L 2 Λ n (; V) L 2 Λ n (; K), and te solution operator is a bounded operator

21 Finite Element Metods for Linear Elasticity 21 L 2 Λ n (; V) HΛ (; V) L 2 Λ n (; V) L 2 Λ n (; K). Tis will follow from te general teory of suc saddle point problems [14] once we establis two conditions: (W1) τ 2 HΛ c 1 Aτ, τ wenever τ HΛ (; V) satisfies dτ, v = 0 v L 2 Λ n (; V) and Sτ, q = 0 q L 2 Λ n (; K), (W2) for all nonzero (v, q) L 2 Λ n (; V) L 2 Λ n (; K), tere exists nonzero τ HΛ (; V) wit dτ, v Sτ, q c 2 τ HΛ ( v + q ), for some positive constants c 1 and c 2. Te first condition is obvious (and does not even utilize te ortogonality of Sτ). However, te second condition is more subtle. We will verify it in Teorem 7.2 in a subsequent section. We next consider a finite element discretizations of 11. For tis, we coose families of finite-dimensional subspaces Λ (V) HΛ (; V), Λ n (V) L 2 Λ n (; V), Λ n (K) L 2 Λ n (; K), indexed by, and seek te discrete solution (σ, u, p ) Λ (V) Λ n (V) (K) suc tat Λ n Aσ, τ + dτ, u Sτ, p = 0, τ Λ (V), (12) dσ, v = f, v v Λ n (V), Sσ, q = 0, q Λ n (K). In analogy wit te well-posedness of te problem 11, te stability of te saddle point system 12 will be ensured by te Brezzi stability conditions: (S1) τ 2 HΛ c 1(Aτ, τ) wenever τ Λ (V) satisfies dτ, v = 0 v Λ n (V) and Sτ, q = 0 q Λn (K), (S2) for all nonzero (v, q) Λ n (V) Λn (K), tere exists nonzero τ Λ (V) wit dτ, v Sτ, q c 2 τ HΛ ( v + q ), were now te constants c 1 and c 2 must be independent of. Te difficulty is, of course, to design finite element spaces satisfying tese conditions. We ave seen previously tat tere is a close relation between te construction of stable mixed finite element metods for te approximation of te equations of linear elasticity and discretization of te associated elasticity complex 6. Tis relationsip extends an analogous relationsip between te construction of stable mixed finite element metods for Poisson s equation and discretization of te de Ram complex. It turns out tat tere is also a close, but non-obvious, connection between te elasticity complex and te de Ram complex. Tis connection is described in [19] and is related to a general construction given in [13], called te BGG resolution (see also [16]). Te elasticity complex 6 is related to te formulation of te equations of elasticity wit strong symmetry. It is also possible to derive an elasticity complex tat is related to te equations of elasticity wit weak symmetry, again starting from te de Ram complex. In [8] (two dimensions) and [6]

22 22 Ricard S. Falk (tree dimensions), suc an elasticity complex is derived and a discrete version of te BGG construction also developed. Tis was ten used to derive stable mixed finite element metods for elasticity in a systematic manner based on te finite element versions of te de Ram sequence described earlier. Te resulting elements in bot two and tree space dimensions are simpler tan any derived previously. For example, te simple coice of P 1 Λ (T ; V) for stress, P 0 Λ n (T ; V) for displacement, and P 0 Λ n (T ; K) for te multiplier results in a stable discretization of te problem 12. In Figure 4, tis element is depicted in two dimensions. For stress, te degrees of freedom are te first two moments of its trace on te edges, and for te displacement and multiplier, teir integrals on te triangle (two components for displacement, one for te multiplier). Moreover, tis element is te lowest order of a family of stable elements in n dimensions utilizing P r Λ (T ; V) for stress, P r 1 Λ n (T ; V) for displacement, and P r 1 Λ n (T ; K) for te multiplier. In fact, te lowest order element may be simplified furter, so tat only a subset of linear vectors is needed to approximate te stress. More details of tis simplified element are presented in Section 11. Fig. 4. Approximation of stress, displacement, and multiplier for te simplest element in two dimensions. In te next section, we follow te approac in [9] and outline ow an elasticity complex wit weakly imposed symmetry can be derived from te de Ram complex. Since tis derivation produces a sequence in te notation of differential forms, we ten translate our results to te more classical notation for elasticity in two and tree dimensions. In Section 7, we give a proof of te well-posedness of te mixed formulation of elasticity wit weak symmetry for te continuous problem, as a guide for establising a similar result for te discrete problem. Based on tis proof, we develop in Section 8 te conditions tat we will need for stable approximation scemes. Tese results are ten used to establis te main stability result for weakly symmetric mixed finite element approximations of te equations of elasticity in Section 9 and some more refined estimates in Section 10. Te results presented in tis paper are for te case of displacement boundary conditions. An extension to te equations of elasticity wit traction boundary conditions can be found in [9].

23 Finite Element Metods for Linear Elasticity 23 6 From te de Ram complex to an elasticity complex wit weak symmetry In tis section, we discuss te connection of te elasticity complex in n dimensions wit te de Ram complex. Details of te derivation can be found in [6] and [9] and follow te ideas in a a derivation of elasticity from te de Ram sequence in te case of strongly imposed symmetry given in [19] in tree dimensions. We start wit te two vector-valued de Ram sequences, one wit values in V and one wit values in K, i.e., Λ n 2 (; K) dn 2 Λ (; K) d Λ n (K) 0, Λ n 3 (; V) dn 3 Λ n 2 (; V) dn 2 Λ (; V) d Λ n (V) 0, Using te fact tat tese sequences are exact, one is able to sow tat te sequence Λ n 3 (W) (dn 3, Sn 3) Λ n 2 (; K) dn 2 S 1 n 2 dn 2 Λ (; V) ( S,d ) T Λ n (W) 0 (13) is exact, were W = K V. We refer to te sequence 13 as te elasticity sequence wit weak symmetry. Crucial to tis construction is te fact tat te operator S n 2 : H 1 Λ n 2 (; V) H 1 Λ (; K) is an isomorpism. We next interpret tis sequence in te language of differential operators in two and tree dimensions. Wen n = 2, we ave te sequence Λ 0 (; K) d0 S 1 0 d0 Λ 1 (; V) ( S1,d1)T Λ 2 (W) 0. Hence, if we begin wit an element wχ Λ 0 (; K) tat we identify wit te scalar function w, ten d 0 (wχ) = w χdx 1 + w χdx 2, x 1 x 2 d 0 S 1 0 [d 0(wχ)] = ( 2 w/ x 1 x 2 2 w/ x 2 1 S 1 ( w, w x 2 x 1 0 [d 0(wχ)] = ) ( dx w/ x w/ x 1 x 2 We ten identity tis vector-valued 1-form wit te matrix ( ) 2 w/ x w/ x 1 x 2 2 w/ x 1 x 2 2 w/ x 2 Jw. 1 w 21 ) dx 2. ) T, To ( translate ) ( te second ) part of te sequence, we begin wit an element ω = w11 w12 dx 1 + dx 2 Λ 1 (V; V) tat we identify (as in 10) wit te matrix w 22

24 24 Ricard S. Falk ( ) ( ) W11 W W = 12 w12 w = 11. W 21 W 22 w 22 w 21 We ave seen previously tat S 1 ω corresponds to 2 skw W. Now ( ) w12 / x d 1 ω = 1 w 11 / x 2 dx w 22 / x 1 w 21 / x 1 dx 2 = div W dx 1 dx 2. 2 Hence, modulo some constants, we obtain te elasticity sequence C () J C (; M) (skw,div)t C (, K V) 0. Wen n = 3, we ave te sequence Λ 0 (W) (d0, S0) Λ 1 (; K) d1 S 1 1 d1 Λ 2 (; V) ( S2,d2)T Λ 3 (W) 0. Hence, if we begin wit a pair (Skw w, µ) Λ 0 (W) = Λ 0 (, K) Λ 0 (, V) tat we identify wit te pair (w, Skw µ) C (, V) C (, K), ten d 0 corresponds to te row-wise gradient and S 0 to te inclusion of C (, K) C (, M). We ave discussed previously natural identifications of Λ 1 (; K) and Λ 2 (; V) wit C (; M). Wit tese identifications, d 1 corresponds to te row-wise curl and S 1 to te operator Ξ. Finally, we ave also seen ow S 2 corresponds to te operator 2 skw. Since d 2 corresponds to te row-wise divergence, we obtain (modulo some unimportant constants), te elasticity sequence wit weak symmetry C (V K) (grad,i) C (M) curl Ξ 1 curl C (M) (skw,div)t C (K V) 0. More details, and te extension of tese ideas to more general domains, can be found in [9]. 7 Well-posedness of te weak symmetry formulation of elasticity As discussed in Section 5, to establis well-posedness of te elasticity problem wit weakly imposed symmetry 11, it suffices to verify condition (W2) of tat section. Tis may be deduced from te following teorem, wic says tat te map HΛ (; V) ( S,d)T HΛ n (; K) HΛ n (; V) is surjective. We present te proof in detail, since it will give us guidance as we construct stable discretizations. Te proof will make use of te following well-known result from partial differential equations.

25 Finite Element Metods for Linear Elasticity 25 Lemma 7.1 Let be a bounded domain in R n wit a Lipscitz boundary. Ten, for all µ L 2 Λ n (), tere exists η H 1 Λ () satisfying d η = µ. If, in addition, µ = 0, ten we can coose η H 1 Λ (). Teorem 7.2 Given (ω, µ) L 2 Λ n (; K) L 2 Λ n (; V), tere exists σ HΛ (; V) suc tat d σ = µ, S σ = ω. Moreover, we may coose σ so tat σ HΛ c( ω + µ ), for a fixed constant c. Proof. Te second sentence follows from te first by Banac s teorem, (i.e., if a continuous linear operator between two Banac spaces as an inverse, ten tis inverse operator is continuous), so we need only prove te first. (1) By Lemma 7.1, we can find η H 1 Λ (; V) wit d η = µ. (2) Since ω + S η HΛ n (; K), we can apply Lemma 7.1 a second time to find τ H 1 Λ (; K) wit d τ = ω + S η. (3) Since S n 2 is an isomorpism from H 1 Λ n 2 (; V) onto H 1 Λ (; K), we ave ϱ H 1 Λ n 2 (; V) wit S n 2 ϱ = τ. (4) Define σ = d n 2 ϱ + η HΛ (; V). (5) From steps (1) and (4), it is immediate tat d σ = µ. (6) From (4), S σ = S d n 2 ϱ S η. But, since d S n 2 = S d n 2, so S σ = ω. S d n 2 ϱ = d S n 2 ϱ = d τ = ω + S η, We note a few points from te proof. (i) Altoug te elasticity problem 11 only involves te tree spaces HΛ (; V), L 2 Λ n (; V), and L 2 Λ n (; K), te proof brings in two additional spaces from te BGG construction: HΛ n 2 (; V) and HΛ (; K). (ii) Altoug S is te only S operator arising in te formulation, S n 2 plays a role in te proof. (iii) We do not fully use te fact tat S n 2 is an isomorpism from Λ n 2 (V; V) to Λ (V; K), only te fact tat it is a surjection. Tis will prove important in te next section, wen we derive conditions for stable approximation scemes for elasticity. (iv) Oter sligtly weaker conditions can be used in some places in te proof (a fact we also exploit in discrete versions for some coices of finite element spaces). 8 Conditions for stable approximation scemes To obtain stable approximation scemes, we now mimic te key structural elements present for te continuous problem. In particular, we see tat to

26 26 Ricard S. Falk establis stability of te continuous problem, we do not use te complete exact sequences, but only te last two spaces in te top sequence and te last tree spaces in te bottom sequence, connected by te operators S n 2 and S. Λ (K) d Λ n (K) 0 S n 2 S (14) Λ n 2 (V) dn 2 Λ (V) d Λ n (V) 0. Tus, we look for five finite dimensional spaces tat are connected by a similar structure, i.e., in addition to te spaces Λ n (K) HΛn (K), Λ (V) HΛ (V), and Λ n (V) HΛn (V) used in te finite element metod, we also seek spaces Λ (K) HΛ (K) and Λ n 2 (V) HΛ n 2 (V). To mimic te structure of te continuous problem, but taking into account te comments made following Teorem 7.2, we require tat te finite element spaces are also connected by exact sequences, but were we introduce some additional flexibility by inserting te L 2 projection operator Π n and using approximations of te operators S n 2 and S. Λ (K) Πn d Λ n (K) 0 S n 2, S, (15) Λ n 2 (V) dn 2 Λ (V) d Λ n (V) 0. In anticipation of proving a stability result for te mixed finite element metod for elasticity tat mimics tat proof used in te continuous case, we need to define interpolants into eac of tese finite element spaces tat ave appropriate properties. Te reason for te coice of te specific properties will become apparent in te stability proof. We first define Π n and Π n to be te L2 projection operators into te spaces Λ n (K) and Λn (V), respectively. We ten define Π and Π to be interpolation operators mapping H 1 Λ (K) to Λ (K) and H 1 Λ (V) to (V), respectively, and satisfying Λ Π n d Π τ = Π n d τ, τ ( H 1 + P 1 )Λ (K), (16) d Π τ = Π n d τ, τ H 1 Λ (V). Π τ C τ 1, τ ( H 1 + P 1 )Λ (K), (17) Π τ C τ 1, τ H 1 Λ (V). Next, we define Π n 2 mapping H 1 Λ n 2 (V) to Λ n 2 (V) satisfying d n 2 Πn 2 ϱ c ϱ 1, ϱ H 1 Λ n 2. (18)