MIXED FINITE ELEMENT METHODS FOR LINEAR ELASTICITY WITH WEAKLY IMPOSED SYMMETRY

MATHEMATICS OF COMPUTATION Volume 76, Number 260, October 2007, Pages 1699 1723 S 0025-5718(07)01998-9 Article electronically publise on May 9, 2007 MIXED FINITE ELEMENT METHODS FOR LINEAR ELASTICITY WITH WEAKLY IMPOSED SYMMETRY DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER Abstract. In tis paper, we construct new inite element metos or te approximation o te equations o linear elasticity in tree space imensions tat prouce irect approximations to bot stresses an isplacements. Te metos are base on a moiie orm o te Hellinger Reissner variational principle tat only weakly imposes te symmetry conition on te stresses. Altoug tis approac as been previously use by a number o autors, a key new ingreient ere is a constructive erivation o te elasticity complex starting rom te e Ram complex. By mimicking tis construction in te iscrete case, we erive new mixe inite elements or elasticity in a systematic manner rom known iscretizations o te e Ram complex. Tese elements appear to be simpler tan te ones previously erive. For example, we construct stable iscretizations wic use only piecewise linear elements to approximate te stress iel an piecewise constant unctions to approximate te isplacement iel. 1. Introuction Te equations o linear elasticity can be written as a system o equations o te orm (1.1) Aσ = ɛu, iv σ = in Ω. Here te unknowns σ an u enote te stress an isplacement iels engenere by a boy orce acting on a linearly elastic boy wic occupies a region Ω R 3. Ten σ takesvaluesintespaces := R 3 3 sym o symmetric matrices an u takes values in V := R 3. Te ierential operator ɛ is te symmetric part o te graient, te iv operator is applie row-wise to a matrix, an te compliance tensor A = A(x) : S S is a boune an symmetric, uniormly positive einite operator relecting te properties o te boy. I te boy is clampe on te bounary Ω o Ω, ten te proper bounary conition or te system (1.1) is u =0on Ω. For simplicity, tis bounary conition will be assume ere. Te issues tat arise wen oter bounary conitions are assume (e.g., te case o pure traction bounary conitions σn = g) are iscusse in [9]. Receive by te eitor October 31, 2005 an, in revise orm, September 11, 2006. 2000 Matematics Subject Classiication. Primary 65N30; Seconary 74S05. Key wors an prases. Mixe meto, inite element, elasticity. Te work o te irst autor was supporte in part by NSF grant DMS-0411388. Te work o te secon autor was supporte in part by NSF grant DMS03-08347. Te work o te tir autor was supporte by te Norwegian Researc Council. 1699 c 2007 American Matematical Society Reverts to public omain 28 years rom publication

1700 DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER Te pair (σ, u) can alternatively be caracterize as te unique critical point o te Hellinger Reissner unctional (1 (1.2) J (τ,v)= Ω 2 Aτ : τ +ivτ v v) x. Te critical point is sougt among all τ H(iv, Ω; S), te space o squareintegrable symmetric matrix iels wit square-integrable ivergence, an all v L 2 (Ω; V), te space o square-integrable vector iels. Equivalently, (σ, u) H(iv, Ω; S) L 2 (Ω; V) is te unique solution to te ollowing weak ormulation o te system (1.1): (Aσ : τ +ivτ u) x = 0, τ H(iv, Ω; S), Ω (1.3) Ω iv σ vx = Ω vx, v L2 (Ω; V). A mixe inite element meto etermines an approximate stress iel σ an an approximate isplacement iel u as te critical point o J over Σ V were Σ H(iv, Ω; S) anv L 2 (Ω; V) are suitable piecewise polynomial subspaces. Equivalently, te pair (σ,u ) Σ V is etermine by te weak ormulation (1.3), wit te test space restricte to Σ V. As is well known, te subspaces Σ an V cannot be cosen arbitrarily. To ensure tat a unique critical point exists an tat it provies a goo approximation o te true solution, tey must satisy te stability conitions rom Brezzi s teory o mixe metos [12, 13]. Despite our ecaes o eort, no stable simple mixe inite element spaces or elasticity ave been constructe. For te corresponing problem in two space imensions, stable inite elements were presente in [10]. For telowest orer element, te space Σ is compose o piecewise cubic unctions, wit 24 egrees o reeom per triangle, wile te space V consists o piecewise linear unctions. Anoter approac wic as been iscusse in te two-imensional case is te use o composite elements, in wic V consists o piecewise polynomials wit respect to one triangulation o te omain, wile Σ consists o piecewise polynomials wit respect to a ierent, more reine, triangulation [5, 21, 23, 31]. In tree imensions, a partial analogue o te element in [10] as been propose an sown to be stable in [1]. Tis element uses piecewise quartic stresses wit 162 egrees o reeom per tetraeron, an piecewise linear isplacements. Because o te lack o suitable mixe elasticity elements, several autors ave resorte to te use o Lagrangian unctionals wic are moiications o te Hellinger Reissner unctional given above [2, 4, 6, 27, 28, 29, 30], in wic te symmetry o te stress tensor is enorce only weakly or abanone altogeter. In orer to iscuss tese metos, we consier te compliance tensor A(x) as a symmetric an positive einite operator mapping M into M, werem is te space o 3 3 matrices. In te isotropic case, or example, te mapping σ Aσ as te orm Aσ = 1 ( λ σ 2µ 2µ +3λ tr(σ)i), were λ(x),µ(x) are positive scalar coeicients, te Lamé coeicients. A moiication o te variational principle iscusse above is obtaine i we consier te extene Hellinger Reissner unctional (1.4) J e (τ,v,q)=j(τ,v)+ τ : qx Ω

MIXED METHODS WITH WEAKLY IMPOSED SYMMETRY 1701 over te space H(iv, Ω; M) L 2 (Ω; V) L 2 (Ω; K), were K enotes te space o skew symmetric matrices. We note tat te symmetry conition or te space o matrix iels is now enorce troug te introuction o te Lagrange multiplier, q. Acriticalpoint(σ, u, p) o te unctional J e is caracterize as te unique solution o te system (Aσ : τ +ivτ u + τ : p) x = 0, τ H(iv, Ω; M), Ω (1.5) Ω iv σ vx = Ω vx, v L2 (Ω; V), Ω σ : qx = 0, q L2 (Ω; K). Clearly, i (σ, u, p) is a solution o tis system, ten σ is symmetric, i.e., σ H(iv, Ω; S), an tereore te pair (σ, u) H(iv, Ω; S) L 2 (Ω; V) solveste corresponing system (1.3). On te oter an, i (u, p) solves (1.3), ten u H 1 (Ω; V) an, i we set p to te skew-symmetric part o gra u, ten(σ, u, p) solves (1.5). In tis respect, te two systems (1.3) an (1.5) are equivalent. However, te extene system (1.5) leas to new possibilities or iscretization. Assume tat we coose inite element spaces Σ V Q H(iv, Ω; M) L 2 (Ω; V) L 2 (Ω; K) an consier a iscrete system corresponing to (1.5). I (σ,u,p ) Σ V Q is a iscrete solution, ten σ will not necessarily inerit te symmetry property o σ. Instea,σ will satisy te weak symmetry conition σ : qx=0, or all q Q. Ω Tereore, tese solutions in general will not correspon to solutions o te iscrete system obtaine rom (1.3). Discretizations base on te system (1.5) will be reerre to as mixe inite element metos wit weakly impose symmetry. Suc iscretizations were alreay introuce by Fraejis e Veubeke in [21] an urter evelope in [4]. In particular, te so-calle PEERS element propose in [4] or te corresponing problem in two space imensions use a combination o piecewise linear unctions an cubic bubble unctions, wit respect to a triangulation o te omain, to approximate te stress σ, piecewise constants to approximate te isplacements, an continuous piecewise linear unctions to approximate te Lagrange multiplier p. Prior to te PEERS paper, Amara an Tomas [2] evelope metos wit weakly impose symmetry using a ual ybri approac. Te lowest orer meto tey iscusse approximates te stresses wit quaratic polynomials plus bubble unctions an te multiplier by iscontinuous constant or linear polynomials. Te isplacements are approximate on bounary eges by linear unctions. Generalizations o te iea o weakly impose symmetry to oter triangular elements, rectangular elements, an tree space imensions were evelope in [28], [29], [30] an [24]. In [29], a amily o elements is evelope in bot two an tree imensions. Te lowest orer element in te amily uses quaratics plus te curls o quartic bubble unctions in two imensions or quintic bubble unctions in tree imensions to approximate te stresses, iscontinuous linears to approximate te isplacements, an iscontinuous quaratics to approximate te multiplier. In aition, a lower orer meto is introuce tat approximates te stress by piecewise linear unctions augmente by te curls o cubic bubble unctions plus a cubic bubble times te graient o local rigi motions. Te multiplier is approximate by iscontinuous piecewise linear unctions an te isplacement by local rigi motions. Morley [24] extens

1702 DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER PEERS to a amily o triangular elements, to rectangular elements, an to tree imensions. In aition, te multiplier is approximate by nonconorming rater tan continuous piecewise polynomials. Tere is a close connection between mixe inite elements or linear elasticity an iscretization o an associate ierential complex, te elasticity complex, wic will be introuce in 3 below. In act, te importance o tis complex was alreay recognize in [10], were mixe metos or elasticity in two space imensions were iscusse. Te new ingreient ere is tat we utilize a constructive erivation o te elasticity complex starting rom te e Ram complex. Tis construction is escribe in Eastwoo [18] an is base on te te Bernstein Gelan Gelan resolution; c. [11] an also [14]. By mimicking te construction in te iscrete case, we are able to erive new mixe inite elements or elasticity in a systematic manner rom known iscretizations o te e Ram complex. As a result, we can construct new elements in bot two an tree space imensions wic are signiicantly simpler tan tose erive previously. For example, we will construct stable iscretizations o te system (1.5) wic only use piecewise linear an piecewise constant unctions, as illustrate in te igure below. For simplicity, te entire iscussion o te present paper will be given in te tree-imensional case. A etaile iscussion in two space imensions can be oun in [8]. Besies te metos iscusse ere, we note tat by sligtly generalizing te approac o tis paper, one can also analyze some o te previously known metos mentione above tat are also base on te weak symmetry ormulation (see [19] or etails). Figure 1. Elements or te stress, isplacement, an multiplier in te lowest orer case in two imensions an tree imensions. An alternative approac to construct inite element metos or linear elasticity is to consier a pure isplacement ormulation. Since te coeicient A in (1.1) is invertible, te stress σ can be eliminate using te irst equation in (1.1), te stress-strain relation. Tis leas to te secon orer equation (1.6) iv A 1 ɛu= in Ω or te isplacement u. A weak solution o tis equation can be caracterize as te global minimizer o te energy unctional (1 E(u) = Ω 2 A 1 ɛu: ɛu+ u ) x over te Sobolev space H0 1 (Ω; V). Here H0 1 (Ω; V) enotes te space o all square integrable vector iels on Ω, wit square integrable erivatives, an wic vanis on te bounary Ω. A inite element approac base on tis ormulation, were we seek a minimum over a inite element subspace o H0 1 (Ω; V) is stanar an iscusse in textbooks, (e.g., [16]). However, or more general moels, arising, or example, in viscoelasticity an plasticity (c. [15]), te stress strain relation is not local an an elimination o te stress σ is impossible. For suc moels, a pure

MIXED METHODS WITH WEAKLY IMPOSED SYMMETRY 1703 isplacement moel is exclue, an a mixe approac seems to be an obvious alternative. Te construction o stable mixe elements or linear elasticity is an important step in te construction o mixe metos or tese more complicate moels. Anoter avantage o te mixe approac is tat we automatically obtain scemes wic are uniormly stable in te incompressible limit, i.e., as te Lamé parameter λ tens to ininity. Since tis beavior o mixe metos is well known, we will not ocus urter on tis property ere. A more etaile iscussion in tis irection can, or example, be oun in [5]. An outline o te paper is as ollows. In 2, we escribe te notation to be use, state our main result, an provie some preliminary iscussion on te relation between stability o mixe inite element metos an iscrete exact complexes. In 3, we present two complexes relate to te two mixe ormulations o elasticity given by (1.3) an (1.5). In 4, we introuce te ramework o ierential orms an sow ow te elasticity complex can be erive rom te e Ram complex. In 5, we erive iscrete analogues o te elasticity complex beginning rom iscrete analogues o te e Ram complex an ientiy te require properties o te iscrete spaces necessary or tis construction. Tis proceure is our basic esign principle. In 6, we apply te construction o te preceing section to speciic iscrete analogues o te e Ram complex to obtain a amily o iscrete elasticity complexes. In 7 we use tis amily to construct stable inite element scemes or te approximation o te mixe ormulation o te equations o elasticity wit weakly impose symmetry. Finally, in 8, we sow ow a sligtly more complicate proceure leas to a simpliie elasticity element. 2. Notation, statement o main results, an preliminaries We begin wit some basic notation an ypoteses. We continue to enote by V = R 3 te space o 3-vectors, by M te space o 3 3 real matrices, an by S an K te subspaces o symmetric an skew symmetric matrices, respectively. Te operators sym : M S an skw : M K enote te symmetric an skew symmetric parts, respectively. Note tat an element o te space K can be ientiie wit its axial vector in V givenbytemapvec:k V: vec 0 v 3 v 2 v 3 0 v 1 = v 1 v 2, v 2 v 1 0 v 3 i.e., vec 1 (v)w = v w or any vectors v an w. We assume tat Ω is a omain in R 3 wit bounary Ω. We sall use te stanar unction spaces, like te Lebesgue space L 2 (Ω) an te Sobolev space H s (Ω). For vector-value unctions, we inclue te range space in te notation ollowing a semicolon, so L 2 (Ω; X) enotes te space o square integrable unctions mapping Ω into a norme vector space X. Te space H(iv, Ω; V) enotes te subspace o (vector-value) unctions in L 2 (Ω; V) wose ivergence belongs to L 2 (Ω). Similarly, H(iv, Ω; M) enotes te subspace o (matrix-value) unctions in L 2 (Ω; M) wose ivergence (by rows) belongs to L 2 (Ω; V). Assuming tat X is an inner prouct space, ten L 2 (Ω; X) as a natural norm an inner prouct, wic will be enote by an (, ), respectively. For a Sobolev space H s (Ω; X), we enote te norm by s an or H(iv, Ω; X), te norm is enote by v iv := ( v 2 + iv v 2 ) 1/2. Te space P k (Ω) enotes te

1704 DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER space o polynomial unctions on Ω o total egree k. Usually we abbreviate tis to just P k. In tis paper we sall consier mixe inite element approximations erive rom (1.5). Tese scemes take te orm: Fin (σ,u,p ) Σ V Q suc tat Ω (Aσ : τ +ivτ u + τ : p ) x = 0, τ Σ, (2.1) Ω iv σ vx = Ω vx, v V, Ω σ : qx = 0, q Q, were now Σ H(iv, Ω; M), V L 2 (Ω; V) anq L 2 (Ω; K). Following te general teory o mixe inite element metos (c. [12, 13]) te stability o te sale point system (2.1) is ensure by te ollowing conitions: (A1) τ 2 iv c 1(Aτ, τ) wenever τ Σ satisies (iv τ,v) =0 v V, an (τ,q)=0 q Q, (A2) or all nonzero (v, q) V Q, tere exists nonzero τ Σ wit (iv τ,v)+(τ,q) c 2 τ iv ( v + q ), were c 1 an c 2 are positive constants inepenent o. Te main result o tis paper, given in Teorem 7.1, is to construct a new amily o stable inite element spaces Σ, V, Q tat satisy te stability conitions (A1) an (A2). We sall sow tat or r 0, te coices o te Néélec secon amily o H(iv) elements o egree r +1orΣ (c. [26]) an o iscontinuous piecewise polynomials o egree r or V an Q provie a stable inite element approximation. In contrast to te previous work escribe in te introuction, no stabilizing bubble unctions are neee; nor is interelement continuity impose on te multiplier. In 8 we also iscuss a somewat simpler lowest orer element (r =0)inwicte local stress space is a strict subspace o te ull space o linear matrix iels. Our approac to te construction o stable mixe elements or elasticity is motivate by te success in eveloping stable mixe elements or steay eat conuction (i.e., te Poisson problem) base on iscretizations o te e Ram complex. We recall (see, e.g., [7]) tat tere is a close connection between te construction o suc elements an iscretizations o te e Ram complex (2.2) R C (Ω) gra C (Ω; V) curl C (Ω; V) iv C (Ω) 0. More speciically, a key to te construction an analysis o stable mixe elements is a commuting iagram o te orm (2.3) gra curl R C (Ω) C (Ω; V) C (Ω; V) C (Ω) 0 Π 1 Π c Π Π 0 gra curl iv R W U V Q 0. Here, te spaces V H(iv) an Q L 2 are te inite element spaces use to iscretize te lux an temperature iels, respectively. Te spaces U H(curl) an W H 1 are aitional inite element spaces, wic can be oun or all well-known stable element coices. Te bottom row o te iagram is a iscrete e Ram complex, wic is exact wen te e Ram complex is (i.e., wen te omain is contractible). Te vertical operators are projections etermine by te iv

MIXED METHODS WITH WEAKLY IMPOSED SYMMETRY 1705 natural egrees o reeom o te inite element spaces. As pointe out in [7], tere are many suc iscretizations o te e Ram complex. A iagram analogous to (2.3), but wit te e Ram complex replace by te elasticity complex eine just below, will be crucial to our construction o stable mixe elements or elasticity. Discretization o te elasticity complex also gives insigt into te iiculties o constructing inite element approximations o te mixe ormulation o elasticity wit strongly impose symmetry; c. [8]. 3. Te elasticity complex We now procee to a escription o two elasticity complexes, corresponing to strongly or weakly impose symmetry o te stress tensor. In te case o strongly impose symmetry, relevant to te mixe elasticity system (1.3), te caracterization o te ivergence-ree symmetric matrix iels will be neee. In orer to give suc a caracterization, eine curl : C (Ω; M) C (Ω; M) to be te ierential operator eine by taking curl o eac row o te matrix. Ten eine a secon orer ierential operator J : C (Ω; S) C (Ω; S) by (3.1) Jτ =curl(curlτ) T, τ C (Ω; S). It is easy to ceck tat iv J =0antatJ ɛ =0. Inoterwors, (3.2) T C (V) ɛ C (S) J C (S) iv C (V) 0 is a complex. Here te epenence o te omain Ω is suppresse, i.e., C (S) = C (Ω; S), an T = T(Ω) enotes te six-imensional space o ininitesimal rigi motions on Ω, i.e., unctions o te orm x a + Bx wit a V an B K. In act, wen Ω is contractible, ten (3.2) is an exact sequence, a act wic will ollow rom te iscussion below. Te complex (3.2) will be reerre to as te elasticity complex. A natural approac to te construction o stable mixe inite elements or elasticity woul be to exten te complex (3.2) to a complete commuting iagram o te orm (2.3), were (3.2) is te top row an te bottom row is a iscrete analogue. However, ue to te pointwise symmetry requirement on te iscrete stresses, tis construction requires piecewise polynomials o ig orer. For te corresponing problem in two space imensions, suc a complex was propose in [10] wit a piecewise cubic stress space; c. also [8]. An analogous complex was erive in te tree-imensional case in [3]. It uses a piecewise quartic space, wit 162 egrees o reeom on eac tetraeron or te stresses. We consier te ormulation base on weakly impose symmetry o te stress tensor, i.e., te mixe system (1.5). Ten te relevant complex is, instea o (3.2), (3.3) T C (gra, I) (V K) C J (M) C (iv,skw) (M) T C (V K) 0. Here, T = { (v, gra v) v T }, an J : C (Ω; M) C (Ω; M) enotes te extension o te operator eine on C (Ω; S) by (3.1) suc tat J 0onC (Ω; K). We remark tat J may be written (3.4) Jτ =curlξ 1 curl τ,

1706 DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER were Ξ : M M is te algebraic operator (3.5) Ξµ = µ T tr(µ)δ, Ξ 1 µ = µ T 1 2 tr(µ)δ, wit δ te ientity matrix. Inee, i τ is symmetric, ten curl τ is trace ree, an tereore te einition (3.4) reuces to (3.1) on C (Ω; S). On te oter an, i τ is skew wit axial vector u, ten curl τ = Ξgrau, ansocurlξ 1 curl τ =0. Observe tat tere is a close connection between (3.2) an (3.3). In act, (3.2) can be erive rom (3.3) by perorming a projection step. To see tis, consier te iagram (3.6) (gra, I) J (iv,skw) T T C (V K) C (M) C (M) C (V K) 0 π 0 π 1 π 2 π 3 T C ɛ (V) C J (S) C (S) were te projection operators π k are eine by iv C (V) 0, π 0 (u, q) =u, π 1 (σ) =π 2 (σ) =sym(σ), π 3 (u, q) =u iv q. We may ientiy C (V) wit a subspace o C (V K), namely, {(u, q) : u C (V),q =skw(grau)}. Uner tis ientiication, T C (V) correspons to T C (V K). We ientiy te C (V) on te rigt wit a ierent subspace o C (V K), namely, {(u, q) : u C (V),q =0}. Wit tese ientiications, te bottom row is a subcomplex o te top row, an te operators π k are all projections. Furtermore, te iagram commutes. It ollows easily tat te exactness o te upper row implies exactness o te bottom row. In te next section, we sall iscuss tese complexes urter. In particular, we sow te elasticity complex wit weakly impose symmetry, i.e., (3.3) ollows rom te e Ram complex (2.2). Hence, as a consequence o te iscussion above, bot (3.2) an (3.3) will ollow rom (2.2). 4. From te e Ram to te elasticity complex Te main purpose o tis section is to emonstrate te connection between te e Ram complex (2.2) an te elasticity complex (3.3). In particular, we sow tat wenever (2.2) is exact, (3.3) is exact. Tis section serves as an introuction to a corresponing construction o a iscrete elasticity complex, to be given in te next section. In te ollowing section, te iscrete complex will be use to construct stable inite elements or te system (1.5). Te e Ram complex (2.2) is most clearly state in terms o ierential orms. Here we briely recall te einitions an properties we will nee. We use a completely coorinate-ree approac. For a sligtly more expane iscussion an te expressions in coorinates see, e.g., [7, 4]. We let Λ k enote te space o smoot ierential k-orms on Ω, i.e. Λ k =Λ k (Ω) = C (Ω; Alt k V), were Alt k V enotes te vector space o alternating k-linear maps on V. I ω Λ k we let ω x Alt k V enote ω evaluate at x, i.e., we use subscripts to inicate te spatial epenence.

MIXED METHODS WITH WEAKLY IMPOSED SYMMETRY 1707 Using te inner prouct on Alt k V inerite rom te inner prouct on V (see equation (4.1) o [7, 4]), we may also eine te Hilbert space L 2 Λ k (Ω) = L 2 (Ω; Alt k V) o square integrable ierential orms wit norm enote by, an also te mt orer Sobolev space H m Λ k (Ω) = H m (Ω; Alt k V), consisting o square integrable k-orms or wic te norm ω m := ( α ω 2) 1/2 α m is inite (were te sum is over multi-inices o egree at most m). Tus, 0-orms are scalar unctions an 1-orms are covector iels. We will not empasize te istinction between vectors an covectors, since, given te inner prouct in V, wemayientiya1-ormω wit te vector iel v or wic ω(p) = v p, p V. In te tree-imensional case, we can ientiy a 2-orm ω wit a vector iel v an a 3-orm µ witascalarielc by ω(p, q) =v p q, µ(p, q, r) =c(p q r), p,q,r V. Te exterior erivative = k :Λ k Λ k+1 is eine by ω x (v 1,...,v k+1 ) k+1 (4.1) = ( 1) j+1 vj ω x (v 1,...,ˆv j,...,v k+1 ), ω Λ k,v 1,...,v k+1 V, j=1 were te at is use to inicate a suppresse argument an v enotes te irectional erivative in te irection o te vector v. It is useul to eine HΛ k = { ω L 2 (Ω; Alt k V) ω L 2 (Ω; Alt k+1 V) }, wit norm given by ω 2 HΛ = ω 2 + ω 2. Using te ientiications given above, te k correspon to gra, curl, an iv or k =0, 1, 2, respectively, an te HΛ k correspon to H 1, H(curl), H(iv), an, or k =3,L 2. Te e Ram complex (2.2) can ten be written (4.2) R Λ 0 Λ 1 Λ 2 Λ 3 0. It is a complex since =0. A ierential k-orm ω on Ω may be restricte to a ierential k-orm on any submaniol M Ω; at eac point o M te restriction o ω is an alternating linear orm on tangent vectors. Moreover, i im M = k, te integral ω is eine. M I X is a vector space, ten Λ k (X) =Λ k (Ω; X) reers to te k-orms wit values in X, i.e., Λ k (X) =C (Ω; Alt k (V; X)), were Alt k (V; X) are alternating k-linear orms on V wit values in X. Given an inner prouct on X, we obtain an inner prouct on Λ k (X). Obviously te corresponing complex (4.3) X Λ 0 (X) Λ 1 (X) Λ 2 (X) Λ 3 (X) 0, is exact wenever te e Ram complex is. We now construct te elasticity complex as a subcomplex o a complex isomorpic to te e Ram complex wit values in te six-imensional vector space W := K V. First, or any x R 3 we eine K x : V K by K x v =2skw(xv T ). We ten eine an operator K :Λ k (Ω; V) Λ k (Ω; K) by (4.4) (Kω) x (v 1,...,v k )=K x [ω x (v 1,...,v k )].

1708 DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER Next, we eine an isomorpism Φ : Λ k (W) Λ k (W) by Φ(ω, µ) =(ω + Kµ,µ), wit inverse given by Φ 1 (ω, µ) =(ω Kµ,µ). Next, eine te operator A :Λ k (W) Λ k+1 (W) bya =ΦΦ 1. Inserting te isomorpisms Φ in te W-value e Ram sequence, we obtain a complex (4.5) Φ(W) Λ 0 A (W) Λ 1 (W) A Λ 2 (W) A Λ 3 (W) 0, wic is exact wenever te e Ram complex is. Te operator A as a simple orm. Using te einition o Φ, we obtain or (ω, µ) Λ k (W), A(ω, µ) =Φ (ω Kµ,µ)=Φ(ω Kµ, µ) =(ω Sµ,µ), were S = S k :Λ k (V) Λ k+1 (K), k =0, 1, 2isgivenbyS = K K. Usingte einition (4.1) o te exterior erivative, te einition (4.4) o K, an te Leibniz rule (4.6) (ω µ) =ω µ +( 1) k ω µ, ω Λ k, µ Λ l, we obtain (4.7) k+1 (Sω)(v 1,...,v k+1 )= ( 1) j+1 K vj [ω(v 1,...,ˆv j,...v k+1 )], ω Λ k (Ω; V). j=1 Note tat te operator S is purely algebraic, an inepenent o x. Since 2 =0,weave S = 2 K K = (K K) or (4.8) S = S. Noting tat (S 1 µ)(v 1,v 2 )=K v1 [µ(v 2 )] K v2 [µ(v 1 )] = 2 skw[v 1 µ(v 2 ) T v 2 µ(v 1 ) T ], µ Λ 1 (Ω; V), v 1,v 2 V, we in, using te ientity (4.9) a b = 2vecskwab T, tat S 1 is invertible wit (S1 1 ω)(v 1) v 2 v 3 = 1 2 [vec( ω(v 2,v 3 ) ) v 1 vec ( ω(v 1,v 2 ) ) v 3 +vec ( ω(v 1,v 3 ) ) v 2 ], ω Λ 2 (Ω; K),v 1,v 2,v 3 V. We now eine te esire subcomplex. Deine Γ 1 = { (ω, µ) Λ 1 (Ω; W) ω = S 1 µ }, Γ 2 = { (ω, µ) Λ 2 (Ω; W) ω =0}, wit projections π 1 :Λ 1 (Ω; W) Γ 1 an π 2 :Λ 2 (Ω; W) Γ 2 given by π 1 (ω, µ) =(ω, S1 1 ω), π2 (ω, µ) =(0,µ+ S1 1 ω).

MIXED METHODS WITH WEAKLY IMPOSED SYMMETRY 1709 Using (4.8), it is straigtorwar to ceck tat A maps Λ 0 (W) intoγ 1 an Γ 1 into Γ 2, an tat te iagram (4.10) A Φ(W) Λ 0 (W) Λ 1 (W) i A Λ 2 (W) π 1 π 2 A Λ 3 (W) 0 i Φ(W) Λ 0 A (W) Γ 1 A Γ 2 A Λ 3 (W) 0 commutes, an tereore te subcomplex in te bottom row is exact wen te e Ram complex is. Tis subcomplex is, essentially, te elasticity complex. Inee, by ientiying elements (ω, µ) Γ 1 wit ω Λ 1 (K), an elements (0,µ) Γ 2 wit µ Λ 2 (V), te subcomplex becomes (4.11) Φ(W) Λ 0 (K V) ( 0, S 0 ) Λ 1 (K) 1 S 1 1 1 Λ 2 (V) ( S 2, 2 ) T Λ 3 (K V) 0. Tis complex may be ientiie wit (3.3). As an initial step o tis ientiication we observe tat te algebraic operator Ξ : C (M) C (M) appearing in (3.3) via (3.4) an te operator S 1 :Λ 1 (V) Λ 2 (K) are connecte by te ientity (4.12) Ξ = Υ 1 2 S 1Υ 1, were Υ 1 : C (M) Λ 1 (V) anυ 2 : C (M) Λ 2 (K) aregivenbyυ 1 F (v) =Fv an Υ 2 F (v 1,v 2 )=vec 1 F (v 1 v 2 )orf C (M). In act, using (4.9), we ave or any v 1,v 2 V, S 1 Υ 1 F (v 1,v 2 )=2skw[v 1 (Fv 2 ) T v 2 (Fv 1 ) T ] =vec 1 (v 2 Fv 1 v 1 Fv 2 ). On te oter an, Υ 2 ΞF (v 1,v 2 )=vec 1 [ΞF (v 1 v 2 )], an ence (4.12) ollows rom te algebraic ientity ΞF (v 1 v 2 )=v 2 Fv 1 v 1 Fv 2, wic ols or any F M. We may urter ientiy te our spaces o iels in (3.3) wit te corresponing spaces o orms in (4.11) in a natural way: (u, p) C (V K) (vec 1 u, vec p) Λ 0 (K V). F C (M) ω Λ 1 (K) givenbyω(v) =vec 1 (Fv). F C (M) µ Λ 2 (V) givenbyµ(v 1,v 2 )=F(v 1 v 2 ). (u, p) C (V K) (ω, µ) Λ 3 (K V) given by ω(v 1,v 2,v 3 ) = p(v 1 v 2 v 3 ), µ(v 1,v 2,v 3 )=u(v 1 v 2 v 3 ). Uner tese ientiications, we in tat 0 : Λ 0 (K) Λ 1 (K) correspons to te row-wise graient C (V) C (M). S 0 :Λ 0 (V) Λ 1 (K) correspons to te inclusion o C (K) C (M). 1 S1 1 1 :Λ 1 (K) Λ 2 (V) correspons to J =curlξ 1 curl : C (M) C (M). 2 :Λ 2 (V) Λ 3 (V) correspons to te row-wise ivergence C (M) C (V).

1710 DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER S 2 : Λ 2 (V) Λ 3 (K) correspons to te operator 2skw : C (M) C (K). Tus, moulo tese ientiications an te (unimportant) constant actor in te last ientiication, (3.3) an (4.11) are ientical. Hence we ave establise te ollowing result. Teorem 4.1. Wen te e Ram complex (2.2) is exact, (i.e., te omain is contractible), ten so is te elasticity complex (3.3). To en tis section, we return to te operator S :Λ k (V) Λ k+1 (K) eine by S = K K. Let K :Λ k (K) Λ k (V) beteajointok (wit respect to te Eucliean inner prouct on V an te Frobenius inner prouct on K), wic is given by (K ω) x (v 1,...,v k )= 2ω x (v 1,...,v k )x. Deine S :Λ k (K) Λ k+1 (V) by S = K K. Recall tat te wege prouct :Λ k Λ l Λ k+l is given by (ω µ)(v 1,...,v k+l ) = (sign σ)ω(v σ1,...,v σk )µ(v σk+1,...,v σk+l ), ω Λ k,µ Λ l,v i V, were te sum is over te set o all permutations o {1,...,k+ l}, orwicσ 1 < σ 2 < <σ k an σ k+1 <σ j+2 < <σ k+l. Tis extens as well to ierential orms wit values in an inner prouct space, using te inner prouct to multiply te terms insie te summation. Using te Leibniz rule (4.6), we ave (4.13) (Sω) µ =( 1) k ω S µ, ω Λ k (V), µ Λ l (K). We tus ave Kω µ =( 1) k+1 Kω µ + (Kω µ) =( 1) k+1 ω K µ + (ω K µ), an Kω µ = ω K µ =( 1) k+1 ω K µ + (ω K µ). Subtracting tese two expressions gives (4.13). For later reerence, we note tat, analogously to (4.7), we ave (4.14) k+1 (S ω)(v 1,...,v k+1 )= 2 ( 1) j+1 ω(v 1,...,ˆv j,...v k+1 )v j, ω Λ k (Ω; K). j=1 5. Te iscrete construction In tis section we erive a iscrete version o te elasticity sequence by aapting te construction o te previous section. To carry out te construction, we will use two iscretizations o te e Ram sequence. For k =0, 1, 2, 3, let Λ k enote a inite-imensional space o HΛ k or wic Λ k Λk+1, an or wic tere exist projections Π =Π k :Λk Λ k wic make te ollowing iagram commute: (5.1) R Λ 0 Λ 1 Λ 2 Λ 3 0 Π Π Π Π R Λ 0 Λ 1 Λ 2 Λ 3 0 Tis is simply te iagram (2.3) written in te language o ierential orms. We o not make a speciic coice o te iscretization yet, but, as recalle in 2, tere exist many suc iscrete e Ram complexes base on piecewise polynomials. In

MIXED METHODS WITH WEAKLY IMPOSED SYMMETRY 1711 act, as explaine in [7], or eac polynomial egree r 0wemaycooseΛ 3 to be te space o all piecewise polynomial 3-orms wit respect to some simplicial ecomposition o Ω, an construct our suc iagrams. We make te assumption tat P 1 (Ω) Λ 0, wic is true in all te cases mentione. Let Λ k be a secon set o inite imensional spaces wit corresponing projection operators Π enjoying te same properties, giving us a secon iscretization o te e Ram sequence. Supposing a compatibility conition between tese two iscretizations, wic we escribe below, we sall construct a iscrete elasticity complex. We start wit te complex (5.2) K V Λ 0 (K) Λ 0 (V) Λ 3 (K) Λ 3 (V) 0 were Λ k (K) enotes te K-value analogue o Λk an similarly or Λ k (V). For brevity, we enceort write Λ k (W) orλk (K) Λ k (V). As a iscrete analogue o te operator K, we eine K : Λ k (V) Λk (K) byk =Π K were Π is te interpolation operator onto Λ k (K). Next eine S = S k, : Λ k (V) Λk+1 (K) bys = K K,ork =0, 1, 2. Observe tat te iscrete version o (4.8), (5.3) S = S, ollows exactly as in te continuous case. From te commutative iagram (5.1), we see tat S = Π K Π K =Π (K K)=Π S. Continuing to mimic te continuous case, we eine te automorpism Φ on Λ k (W) by Φ (ω, µ) =(ω + K µ, µ), an te operator A :Λ k (W) Λk+1 (W) bya =Φ Φ 1,wicleasto A (ω, µ) =(ω S µ, µ). Tanks to te assumption tat P 1 Λ 0,weaveΦ (W) =Φ(W). Hence, inserting te isomorpisms Φ into (5.2), we obtain (5.4) Φ(W) Λ 0 (W) A Λ 1 (W) A Λ 2 (W) A Λ 3 (W) 0. In analogy to te continuous case, we eine Γ 1 = { (ω, µ) Λ 1 (W) ω = S 1, µ }, Γ 2 = { (ω, µ) Λ 2 (W) ω =0}. As in te continuous case, we can ientiy Γ 2 wit Λ 2 (V), but, unlike in te continuous case, we cannot ientiy Γ 1 wit Λ1 (K),sinceweonotrequiretatS 1, be invertible (it is not in te applications). Hence, in orer to erive te analogue o te iagram (4.10) we require a surjectivity assumption: (5.5) Te operator S 1, maps Λ 1 (V) onto Λ 2 (K). Uner tis assumption, te operator S = S 1, as a rigt inverse S mapping Λ 2 (K) intoλ1 (V). Tis allows us to eine iscrete counterparts o te projection operators π 1 an π 2 by π 1 (ω, µ) =(ω, µ S S µ + S ω), π2 (ω, µ) =(0,µ+ S ω),

1712 DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER an obtain te iscrete analogue o (4.10): (5.6) Φ(W) Λ 0 (W) i A Λ 1 (W) π 1 A Λ 2 (W) π 2 A Λ 3 (W) 0 i Φ(W) Λ 0 (W) A Γ 1 A Γ 2 A Λ 3 (W) 0 It is straigtorwar to ceck tat tis iagram commutes. For example, i (ω, µ) Λ 0 (W), ten πa 1 (ω, µ) =π(ω 1 S µ, µ) =(ω S µ, µ S S µ + S [ω S µ]) =(ω S µ, µ S [S µ + S µ]) = A (ω, µ), were te last equality ollows rom (5.3). Tus te bottom row o (5.6) is a subcomplex o te top row, an te vertical maps are commuting projections. In particular, wen te top row is exact, so is te bottom. Tus we ave establise te ollowing result. Teorem 5.1. For k =0,...,3, letλ k be a inite imensional subspace o HΛk or wic Λ k Λk+1 an or wic tere exist projections Π =Π k :Λk Λ k tat make te iagram (5.1) commute. Let Λ k be a secon set o inite imensional spaces wit corresponing projection operators Π k wit te same properties. I S 1, := Π 1 K Π2 K maps Λ 1 (V) onto Λ2 (K), an te bottom row o (5.1) is exact or bot sequences Λ k an Λ k, ten te iscrete elasticity sequence given by te bottom row o (5.6) is also exact. Te exactness o te bottom row o (5.6) suggests tat te ollowing coice o inite element spaces will lea to a stable iscretization o (2.1): Σ Λ 2 (V), V Λ 3 (V), Q Λ 3 (K). In te next section we will make speciic coices or te iscrete e Ram complexes, an ten veriy te stability in te ollowing section. For use in te next section, we state te ollowing result, giving a suicient conition or te key requirement tat S 1, be surjective. Proposition 5.2. I te iagram (5.7) Λ 1 (V) Π 1 S 1 Λ 2 (K) Λ 1 (V) S 1, Λ 2 (K) commutes, ten S 1, is surjective. 6. A amily o iscrete elasticity complexes In tis section, we present a amily o examples o te general iscrete construction presente in te previous section by coosing speciic iscrete e Ram complexes. Tese urnis a amily o iscrete elasticity complexes, inexe by an integer egree r 0. In te next section we use tese complexes to erive inite elements or elasticity. In te lowest orer case, te meto will require only piecewise linear unctions to approximate stresses an piecewise constants to approximate te isplacements an multipliers. Π 2

MIXED METHODS WITH WEAKLY IMPOSED SYMMETRY 1713 We begin by recalling te two principal amilies o piecewise polynomial spaces o ierential orms, ollowing te presentation in [7]. We enceort assume tat te omain Ω is a contractible polyeron. Let T be a triangulation o Ω. Let T be a triangulation o Ω by tetraera, an set P r Λ k (T )={ ω HΛ k (Ω) ω T P r Λ k (T ) T T }, P r + Λ k (T )={ ω HΛ k (Ω) ω T P r + Λ k (T ) T T }. Here P r + Λ k (T ):=P r Λ k (T )+κp r Λ k+1 (T )wereκ :Λ k+1 (T ) Λ k (T )iste Koszul ierential eine by (κω) x (v 1,,v k )=ω x (x, v 1,,v k ). Te spaces P r + Λ 0 (T )=P r+1 Λ 0 (T ) correspon to te usual egree r +1 Lagrange piecewise polynomial subspaces o H 1, an te spaces P r + Λ 3 (T )=P r Λ 3 (T )correspon to te usual egree r subspace o iscontinuous piecewise polynomials in L 2 (Ω). For k = 1 an 2, te spaces P r + Λ k (T ) correspon to te iscretizations o H(curl) an H(iv), respectively, presente by Néélec in [25], an te spaces P r Λ k (T ) are te ones presente by Néélec in [26]. An element ω P r Λ k (T )is uniquely etermine by te ollowing quantities: (6.1) ω ζ, ζ P + r 1+k Λ k (), (T ), k 3. Here (T ) is te set o vertices, eges, aces, or tetraera in te mes T,or =0, 1, 2, 3, respectively, an or r<0, we interpret P r + Λ k (T )=P r Λ k (T )=0. Note tat or ω Λ k, ω naturally restricts on te ace to an element o Λ k (). Tereore, or ζ Λ k (), te wege prouct ω ζ belongs to Λ () an ence te integral o ω ζ on te -imensional ace o T is a well-eine an natural quantity. Using te quantities in (6.1), we obtain a projection operator rom Λ k to P r Λ k (T ). Similarly, an element ω P r + Λ k (T ) is uniquely etermine by (6.2) ω ζ, ζ P r +k Λ k (), (T ), k 3, an so tese quantities etermine a projection. I X is a vector space, we use te notation P r Λ k (T ; X) anp r + Λ k (T ; X) to enote te corresponing spaces o piecewise polynomial ierential orms wit values in X. Furtermore, ix is an inner prouct space, te corresponing egrees o reeom are given by (6.1) an (6.2), but were te test spaces are replace by te corresponing X value spaces. To carry out te construction escribe in te previous section we nee to coose te two sets o spaces Λ k an Λ k or k = 0, 1, 2, 3. We ix r 0an set Λ k = P+ r Λ k (T ), k = 0, 1, 2, 3, an Λ 0 = P r+2λ 0 (T ), Λ1 = P r+1 + Λ1 (T ), Λ 2 = P r+1λ 2 (T ), an Λ 3 = P rλ 3. As explaine in [7], bot tese coices give a iscrete e Ram sequence wit commuting projections, i.e., a iagram like (5.1) makessenseaniscommutative. We establis te key surjectivity assumption or our coice o spaces by veriying te commutativity o (5.7). Lemma 6.1. Let Λ 1 (V) =P+ r+1 Λ1 (T ; V) an Λ 2 (K) =P+ r Λ 2 (T ; K) wit projections Π 1, Π2 eine via te corresponing vector-value moments o te orm (6.1)

1714 DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER an (6.2). IS 1, =Π 2 S 1 ten (6.3) S 1, Π1 =Π 2 S 1, an so S 1, is surjective. Proo. We must sow tat (Π 2 S 1 S 1, Π 1 )σ =0orσ Λ1 (V). Deining ω = (I Π 1 )σ, te require conition becomes Π2 S 1ω =0. Since Π 1 ω =0,weave (6.4) ω ζ =0, ζ P r+2 Λ 1 (; V), (T ), 2 3, (in act (6.4) ols or = 1 as well, but tis is not use ere). We must sow tat (6.4) implies (6.5) S 1 ω µ =0, µ P r+2 Λ 2 (; K), (T ), 2 3. From (4.13), we ave S 1 ω µ = ω ζ, wereζ = S 2 µ P r+2 Λ 1 (; V) or µ P r+2 Λ 2 (; K), as is evient rom (4.14). Hence (6.5) ollows rom (6.4). 7. Stable mixe inite elements or elasticity Base on te iscrete elasticity complexes just constructe, we obtain mixe inite element spaces or te ormulation (2.1) o te elasticity equations by coosing Σ, V,anQ as te spaces o matrix an vector iels corresponing to appropriate spaces o orms in te K- anv-value e Ram sequences use in te construction. Speciically, tese are (7.1) Σ P r+1 Λ 2 (T ; V), V P r Λ 3 (T ; V), Q P r Λ 3 (T ; K). In oter terminology, Σ may be tougt o as te prouct o tree copies o te Néélec H(iv) space o te secon kin o egree r +1, an V an Q are spaces o all piecewise polynomials o egree at most r wit values in K an V, respectively, wit no impose interelement continuity. In tis section, we establis stability an convergence or tis inite element meto. Te stability o te meto requires te two conitions (A1) an (A2) state in 2. Te irst conition is obvious since, by construction, iv Σ V, i.e., P r+1 Λ 2 (T ; V) P r Λ 3 (T ; V). Te conition (A2) is more subtle. We will prove a stronger version, namely, (A2 ) or all nonzero (v, q) V Q, tere exists nonzero τ Σ wit iv τ = v, 2Π Q skw τ = q an τ iv c( v + q ), were Π Q is te L 2 projection into Q an c is a constant. Recalling tat Γ 2 =0 P r+1λ 2 (T ; V) ana (0,σ)=( S 2, σ, σ), an tat te operator S 2 correspons on te matrix level to 2 skw, we restate conition (A2 ) in te language o ierential orms. Teorem 7.1. Given tat (ω, µ) P r Λ 3 (T ; K) P r Λ 3 (T ; V), tereexistsσ P r+1 Λ 2 (T ; V) suc tat A (0,σ)=(ω, µ) an (7.2) σ HΛ c( ω + µ ), were te constant c is inepenent o ω, µ an.

MIXED METHODS WITH WEAKLY IMPOSED SYMMETRY 1715 Beore proceeing to te proo, we nee to establis some bouns on projection operators. We o tis or te corresponing scalar-value spaces. Te extensions to vector-value spaces are straigtorwar. First we claim tat (7.3) Π 2 η c η 1 η H 1 Λ 2, Π 3 ω c ω 0 ω H 1 Λ 3. Here te constant may epen on te sape regularity o te mes, but not on te messize. Te secon boun is obvious (wit c =1),sinceΠ 3 is just te L2 projection. Te irst boun ollows by a stanar scaling argument. Namely, let ˆT enote te reerence simplex. For any ˆβ P r+1 Λ 2 ( ˆT ), we ave (7.4) ˆβ 0, ˆT c( ˆ ˆβ ˆµ + ˆζ ˆβ ˆζ ), ˆµ ˆ ˆT were ˆ ranges over te aces o ˆT,ˆµ over a basis or P r + Λ 0 ( ˆ), an ˆζ over a basis or P r 1 + Λ1 ( ˆT ). Tis is true because te integrals on te rigt an sie o (7.4) orm a set o egrees o reeom or ˆβ P r+1 Λ 2 ( ˆT ) (see (6.1)), an so we may use te equivalence o all norms on tis inite imensional space. We apply tis result wit ˆβ = ˆΠ 2 ˆη, wereˆπ 2 is te projection eine to preserve te integrals on te rigt an sie o (7.4). It ollows tat ˆΠ 2 ˆη 0, ˆT c( ˆ ˆη ˆµ + ˆζ ˆµ ˆ ˆη ˆζ ) c ˆη 1, ˆT, ˆT were we ave use a stanar trace inequality in te last step. Next, i T is an arbitrary simplex an η H 1 Λ 2 (T ), we map te reerence simplex ˆT onto T by an aine map ˆx Bˆx + b, an eine ˆη H 1 Λ 2 ( ˆT )by ˆηˆx (ˆv 1, ˆv 2 )=η x (Bˆv 1,Bˆv 2 ), or any x = Bˆx + b T an any vectors ˆv 1, ˆv 2. It is easy to ceck tat Π2 2 η = ˆΠ ˆη, an tat Π 2 η 0,T c ˆΠ 2 ˆη 0, ˆT c ˆη 1, ˆT c( η 0,T + η 1,T ) c η 1,T. Squaring an aing tis over all te simplices in te mes T gives te irst boun in (7.3). We also nee a boun on te projection o a orm in H 1 Λ 1 into Λ 1 = P+ r+1 Λ1 (T ). However, te projection operator Π 1 is not boune on H1, because its einition involves integrals over eges. A similar problem as arisen beore (see, e.g., [10]), an we use te same remey. Namely we start by eining an operator Π 1 0 : H 1 Λ 1 P r+1 + Λ1 (T ) by te conitions (7.5) Π 1 0ω ζ = ω ζ, ζ P r 1 Λ 2 (T ), T T, T T (7.6) Π 1 0ω ζ = ω ζ, ζ P r Λ 1 (), 2 (T ), (7.7) Π 1 0ω ζ =0, ζ P r+1 Λ 0 (e), e 1 (T ). e Note tat, in contrast to Π 1, in te einition o Π 1 0,weavesettetroublesome ege egrees o reeom to zero. Let ˆΠ 1 0 : H 1 Λ 1 ( ˆT ) P + r+1 Λ1 ( ˆT ) be eine analogously on te reerence element.

1716 DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER Now or ˆρ H 1 Λ 1 ( ˆT ), ˆˆΠ 1 0 ˆρ P r+1 Λ 2 ( ˆT ), so ˆˆΠ 1 0 ˆρ 0, ˆT c( ˆ ˆˆΠ 1 0 ˆρ ˆµ + ˆζ ˆµ ˆ ˆT ˆˆΠ 1 0 ˆρ ˆζ ), were again ˆ ranges over te aces o ˆT, µ over a basis o P r + Λ 0 ( ˆ), an ζ over a basis o P r 1 + Λ1 ( ˆT ). But ˆˆΠ 1 0 ˆρ ˆµ = ˆΠ 1 0 ˆρ ˆˆµ = ˆρ ˆˆµ, ˆ ˆ were we ave use Stokes teorem an te act tat te vanising o te ege quantities in te einition o ˆΠ 1 0 to obtain te irst equality, an te ace egrees o reeom entering te einition o ˆΠ 1 0 to obtain te secon. Similarly, ˆˆΠ 1 0 ˆρ ˆζ = ˆΠ 1 0 ˆρ ˆˆζ + ˆΠ 1 0 ˆρ ˆζ = ˆρ ˆˆζ + ˆρ ˆζ = ˆˆρ ˆζ. ˆT ˆT ˆT It ollows tat ˆΠ 1 0 ˆρ 0, ˆT c ˆρ 1, ˆT, ρ H1 Λ 1 ( ˆT ). Wen we scale tis result to an arbitrary simplex an a over te mes, we obtain Π 1 0ρ c( 1 ρ + ρ 1 ), ρ H 1 Λ 1 (Ω). To remove te problematic 1 in te last estimate, we introuce te Clement interpolant R mapping H 1 Λ 1 into continuous piecewise linear 1-orms (still ollowing [10]). Ten ρ R ρ + ρ R ρ 1 c ρ 1, ρ H 1 Λ 1. Deining Π 1 : H1 Λ 1 P r+1 + Λ1 by (7.8) Π1 = Π 1 0(I R )+R, we obtain Π 1 ρ Π 1 0(I R )ρ + R ρ c( 1 (I R )ρ + (I R )ρ 1 + R ρ ) c ρ 1. Tus we ave sown tat (7.9) Π 1 ρ c ρ 1, ρ H 1 Λ 1. Having moiie Π 1 to obtain te boune operator Π 1, we now veriy tat te key property (6.3) in Lemma 6.1 carries over to (7.10) S 1, Π1 =Π 2 S 1, were we now use te vector-value orms o te projection operators. It ollows easily rom (7.8), (7.5), an (7.6) tat (6.4) ols wit ω =(I Π 1 )σ, so tat te proo o (7.10) is te same as or (6.3). We can now give te proo o Teorem 7.1. Proo o Teorem 7.1. Given µ P r Λ 3 (T ; V) tereexistsη H 1 Λ 2 (V) suc tat η = µ wit te boun η 1 c µ (since maps H 1 Λ 2 onto L 2 Λ 3 ). Similarly, given ω P r Λ 3 (T ; K) tereexistsτ H 1 Λ 2 (K) witτ = ω + S 2, Π2 η wit te boun τ 1 c ω + S 2, Π2 η. Let ρ = S1 1 τ (recall tat S 1 is an isomorpism) an set σ = Π 1 ρ + Π 2 η. ˆT ˆ ˆT ˆT

MIXED METHODS WITH WEAKLY IMPOSED SYMMETRY 1717 We will now sow tat A (0,σ)=(ω, µ). From te einition o σ, weave S 2, σ = S 2, Π 1 ρ S 2, Π2 η. Ten, using (5.3), (7.10), an te commutativity Π =Π,wesee S 2, Π 1 ρ = S 1, Π1 ρ = Π 2 S 1 ρ = Π 2 τ = Π 3 τ = Π 3 (ω + S 2, Π2 η)= ω S 2, Π2 η. Combining, we get S 2, σ = ω as esire. Furtermore, rom te commutativity Π = Π an te einition o η, weget σ = Π 2 η = Π 3 η = Π 3 µ = µ, an so we ave establise tat A (0,σ)=(µ, ω). It remains to prove te boun (7.2). Using (7.3), we ave S 2, Π2 η = Π 3 S 2 Π2 η c S 2 Π2 η c Π 2 η c η 1 c µ. Tus ρ 1 c τ 1 c( ω + µ ). Using (7.9), we ten get Π 1 ρ c ρ 1 c( ω + µ ), an, using (7.3), tat Π 2 η c η 1 c µ. Tereore σ c( ω + µ ), wile σ = µ, an tus we ave te esire boun (7.2). We ave tus veriie te stability conitions (A1) an (A2), an so may apply te stanar teory o mixe metos (c. [12], [13], [17], [20]) an stanar results about approximation by inite element spaces to obtain convergence an error estimates. Teorem 7.2. Suppose (σ, u, p) is te solution o te elasticity system (1.5) an (σ,u,p ) is te solution o iscrete system (2.1), were te inite element spaces Σ, V,anQ are given by (7.1) or some integer r 0. Ten tere is a constant C, inepenent o, suc tat σ σ iv + u u + p p C in ( σ τ iv + u v + p q ), τ Σ,v V,q Q σ σ + p p + u Π n n 1 u C( σ Π σ + p Π n p ), n 1 u u C( σ Π σ + p Π n p + u Π n u ), (σ σ ) = σ Π n σ. I u an σ are suiciently smoot, ten σ σ + u u + p p C r+1 u r+2, iv(σ σ ) C r+1 iv σ r+1. Remark. Note tat te errors σ σ an u Π n u epen on te approximation o bot σ an p. For te coices mae ere, te approximation o p is one orer less tan te approximation o σ, an tus we o not obtain improve estimates, as one oes in te case o te approximation o Poisson s equation, were te extra variable p oes not enter.

1718 DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER 8. A simpliie element Recall tat te lowest orer element in te stable amily escribe above, or a iscretization base on (1.5), is o te orm Σ P 1 Λ 2 (T ; V), V P 0 Λ 3 (T ; V), Q P 0 Λ 3 (T ; K). Te purpose o tis section is to present a stable element wic is sligtly simpler tan tis one. More precisely, te spaces V an Q are uncange, but Σ will be simpliie rom ull linears to matrix iels wose tangential normal components on eac two-imensional ace o a tetraeron are only a reuce space o linears. We will still aopt te notation o ierential orms. By examining te proo o Teorem 7.1, we realize tat we o not use te complete sequence (5.2) or te given spaces. We only use te sequences P (8.1) 0 + Λ2 (T ; K) P 0 Λ 3 (T ; K) 0, P 1 + Λ1 (T ; V) P 1 Λ 2 (T ; V) P 0 Λ 3 (T ; V) 0. Te purpose ere is to sow tat it is possible to coose subspaces o some o te spaces in (8.1) suc tat te esire properties still ol. More precisely, compare to (8.1), te spaces P 1 + Λ1 (T ; V) anp 1 Λ 2 (T ; V) are simpliie, wile te tree oters remain uncange. I we enote by P 1, + Λ1 (T ; V) anp 1, Λ 2 (T ; V) te simpliications o te spaces P 1 + Λ1 (T ; V) anp 1 Λ 2 (T ; V), respectively, ten te properties we nee are tat (8.2) P + 1, Λ1 (T ; V) P 1, Λ 2 (T ; V) P 0 Λ 3 (T ; V) 0 is a complex an tat te surjectivity assumption (5.5) ols, i.e., S = S 1, maps te space P 1, + Λ1 (T ; V) ontop 0 + Λ2 (T ; K). Note tat i P 0 + Λ2 (T ; V) P 1, Λ 2 (T ; V), ten maps P 1, Λ 2 (T ; V) ontop 0 Λ 3 (T ; V). We irst sow ow to construct P 1, + Λ1 (T ; V) as a subspace o P 1 + Λ1 (T ; V). Since te construction is one locally on eac tetraeron, we will sow ow to construct a space P 1, + Λ1 (T ; V) as a subspace o P 1 + Λ1 (T ; V). We begin by recalling tat te ace egrees o reeom o P 1 + Λ1 (T ; V) aveteorm ω µ, µ P 0 Λ 1 (,V). We ten observe tat tis six-imensional space can be ecompose into P 0 Λ 1 (; V) =P 0 Λ 1 (; T )+P 0 Λ 1 (; N ), i.e., into orms wit values in te tangent space to, T or te normal space N. Tis is a 4 + 2-imensional ecomposition. Furtermore, P 0 Λ 1 (; T )=P 0 Λ 1 sym(; T )+P 0 Λ 1 skw(; T ), were µ P 0 Λ 1 (; T )isinp 0 Λ 1 sym(; T ) i an only i µ(s) t = µ(t) s or ortonormal tangent vectors s an t. Finally, we obtain a 3 + 3-imensional ecomposition P 0 Λ 1 (; V) =P 0 Λ 1 sym(; T )+P 0 Λ 1 skw(; V), were P 0 Λ 1 skw(; V) =P 0 Λ 1 skw(; T )+P 0 Λ 1 (; N ). In more explicit terms, i µ P 0 Λ 1 (F ; V) as te orm µ(q) =(a 1 t + a 2 s + a 3 n)q t +(a 4 t + a 5 s + a 6 n)q s,

MIXED METHODS WITH WEAKLY IMPOSED SYMMETRY 1719 were t an s are ortonormal tangent vectors on, n is te unit normal to, an q is a tangent vector on, tenwecanwriteµ = µ 1 + µ 2,witµ 1 P 0 Λ 1 sym (; V) an µ 2 P 0 Λ 1 skw (; V), were ( µ 1 (q) = a 1 t + a ) ( ) 2 + a 4 a2 + a 4 s q t + t + a 5 s q s, 2 2 ( ) ( ) a2 a 4 a4 a 2 µ 2 (q) = s + a 3 n q t + t + a 6 n q s. 2 2 Te reason or tis particular ecomposition o te egrees o reeom is tat i we examine te proo o Lemma 6.1, were equation (6.3) is establise, we see tat te only egrees o reeom tat are use or an element ω P 1 + Λ1 (T ; V) are te subset o te ace egrees o reeom given by ω (S 0ν), ν P 0 Λ 0 (; K). However, or ν P 0 Λ 0 (; K), µ = S 0ν is given by µ(q) =νq. Since te general element ν P 0 Λ 0 (K) can be written in te orm b 1 (ts T st T )+b 2 (nt T tn T )+ b 3 (ns T sn T ), νq =( b 1 s + b 2 n)q t +(b 1 t + b 3 n)q s or q a tangent vector, an tus µ P 0 Λ 1 skw (; V). Hence, we ave split te egrees o reeom into tree on eac ace tat we nee to retain or te proo o Lemma 6.1 an tree on eac ace tat are not neee. Te reuce space P 1, + Λ1 (T ; V) tat we now construct as two properties. Te irst is tat it still contains te space P 1 Λ 1 (T ; V) an te secon is tat te unuse ace egrees o reeom are eliminate (by setting tem equal to zero). We can acieve tese conitions by irst writing an element ω P 1 + Λ1 (T ; V) asω =Π ω +(I Π )ω, wereπ enotes te usual projection operator into P 1 Λ 1 (T ; V) eine by te ege egrees o reeom. Ten te elements in (I Π )P 1 + Λ1 (T ; V) will satisy ω µ =0, µ P 1 Λ 0 (e; V), e 1 (T ), e i.e., teir traces on te eges will be zero. Tus, tey are completely eine by te ace egrees o reeom ω µ, µ P 0 Λ 1 (; V), 2 (T ). Since tis is te case, we enceort enote (I Π )P + 1 Λ1 (T ; V) byp + 1, Λ1 (T ; V). We ten eine our reuce space P + 1, Λ1 (T ; V) =P 1 Λ 1 (T ; V)+P + 1,, Λ1 (T ; V), were P + 1,, Λ1 (T ; V) enotes te set o orms ω P + 1, Λ1 (T ; V) satisying ω µ =0, µ P 0 Λ 1 sym(; V), i.e., we ave set te unuse egrees o reeom to be zero. Ten P + 1, Λ1 (T ; V) ={ω P + 1 Λ1 (T ; V) : ω T P + 1, Λ1 (T ; V), T T }.

1720 DOUGLAS N. ARNOLD, RICHARD S. FALK, AND RAGNAR WINTHER Te egrees o reeom or tis space are ten given by (8.3) ω µ, µ P 1 Λ 0 (e; V),e 1 (T ), ω µ, µ P 0 Λ 1 skw(; V), 2 (T ). e It is clear rom tis einition tat te space P 1, + Λ1 (T ; V) will ave 48 egrees o reeom (36 ege egrees o reeom an 12 ace egrees o reeom). Te unisolvency o tis space ollows immeiately rom te unisolvency o te spaces P 1 Λ 1 (T ; V) anp + 1,, Λ1 (T ; V). Te motivation or tis coice o te space P 1, + Λ1 (T ; V) is tat it easily leas to a einition o te space P 1, Λ 2 (T ; V) tat satisies te property tat (8.2) is a complex. We begin by eining P 1, Λ 2 (T ; V) =P + 0 Λ2 (T ; V)+P + 1,, Λ1 (T ; V). It is easy to see tat tis space will ave 24 ace egrees o reeom. Note tis is a reuction o te space P 1 Λ 2 (T ; V), since We ten eine P 1 Λ 2 (T ; V) =P + 0 Λ2 (T ; V)+P + 1, Λ1 (T ; V). P 1, Λ 2 (T ; V) ={ω P 1 Λ 2 (T ; V) : ω T P 1, Λ 2 (T ; V), T T }. It is clear tat P + 0 Λ2 (T ; V) P 1, Λ 2 (T ; V). Te act tat te complex (8.2) is exact now ollows irectly rom te act tat te complex (8.4) P 1 Λ 1 (T ; V) is exact an te einition P + 0 Λ2 (T ; V) P 0 Λ 3 (T ; V) 0 P 1, Λ 1 (T ; V) =P + 0 Λ1 (T ; V)+P + 1,, Λ1 (T ; V). We will eine appropriate egrees o reeom or te space P 1, Λ 2 (T ; V)byusing a subset o te 36 egrees o reeom or P 1 Λ 2 (T ; V), i.e., o ω µ, µ P 1Λ 0 (; V). In particular, we take as egrees o reeom or P 1, Λ 2 (T ; V), ω µ, µ P 1,skw Λ 0 (; V), 2 (T ), were P 1,skw Λ 0 (; V) enotes te set o µ P 1 Λ 0 (; V) tat satisy µ P 0 Λ 1 skw (; V). It is easy to ceck tat suc µ will ave te orm (8.5) µ = µ 0 + α 1 (x t)n + α 2 (x s)n + α 3 [(x t)s (x s)t], were µ 0 P 0 Λ 0 (; V). Since P 1,skw Λ 0 (; V) is a six-imensional space on eac ace, te above quantities speciy 24 egrees o reeom or te space P 1, Λ 2 (T ; V). To see tat tese are a unisolvent set o egrees o reeom or P 1, Λ 2 (T ; V), we let ω = ω 0 + ω 1,were ω 0 P 0 + Λ2 (T ; V) anω 1 P 1,, Λ 1 (T ; V) an set all egrees o reeom equal to zero. Ten or µ P 0 Λ 0 (; V), since (ω 0 + ω 1 ) µ = ω 0 µ,