MIXED FINITE ELEMENT METHODS FOR LINEAR ELASTICITY WITH WEAKLY IMPOSED SYMMETRY

 Simon Pearson
 11 months ago
 Views:
Transcription
1 MATHEMATICS OF COMPUTATION Volume 76, Number 260, October 2007, Pages S (07) 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 rowwise 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, 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 Te work o te secon autor was supporte in part by NSF grant DMS Te work o te tir autor was supporte by te Norwegian Researc Council c 2007 American Matematical Society Reverts to public omain 28 years rom publication
2 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 squareintegrable ivergence, an all v L 2 (Ω; V), te space o squareintegrable 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 twoimensional 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
3 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 skewsymmetric 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 socalle 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
4 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 treeimensional 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 stressstrain 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
5 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 3vectors, 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 vectorvalue 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 (vectorvalue) unctions in L 2 (Ω; V) wose ivergence belongs to L 2 (Ω). Similarly, H(iv, Ω; M) enotes te subspace o (matrixvalue) 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
6 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 wellknown 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
7 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 ivergenceree 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 siximensional 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 treeimensional 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,
8 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 coorinateree 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 korms on Ω, i.e. Λ k =Λ k (Ω) = C (Ω; Alt k V), were Alt k V enotes te vector space o alternating klinear 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.
9 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 korms or wic te norm ω m := ( α ω 2) 1/2 α m is inite (were te sum is over multiinices o egree at most m). Tus, 0orms are scalar unctions an 1orms are covector iels. We will not empasize te istinction between vectors an covectors, since, given te inner prouct in V, wemayientiya1ormω wit te vector iel v or wic ω(p) = v p, p V. In te treeimensional case, we can ientiy a 2orm ω wit a vector iel v an a 3orm 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 korm ω on Ω may be restricte to a ierential korm 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 korms wit values in X, i.e., Λ k (X) =C (Ω; Alt k (V; X)), were Alt k (V; X) are alternating klinear 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 siximensional 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 )].
10 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 Wvalue 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).
11 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 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 rowwise 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 rowwise ivergence C (M) C (V).
12 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 initeimensional 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
13 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 3orms 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 Kvalue 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),
14 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
15 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 welleine 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 vectorvalue moments o te orm (6.1)
16 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 anvvalue 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.
17 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 scalarvalue spaces. Te extensions to vectorvalue 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.
18 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 1orms (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 vectorvalue 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
19 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.
20 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 twoimensional 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 siximensional 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 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 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,
21 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 }.
22 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 siximensional 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,
Differentiation Rules c 2002 Donald Kreider and Dwight Lahr
Dierentiation Rules c 00 Donal Kreier an Dwigt Lar Te Power Rule is an example o a ierentiation rule. For unctions o te orm x r, were r is a constant real number, we can simply write own te erivative rater
More information1. Filling an initially porous tube under a constant head imposed at x =0
Notes on Moving Bounary problems, Voller U o M, volle00@umn.eu. Filling an initially porous tube uner a constant hea impose at x =0 Governing equation is base on calculating the water volume lux by the
More informationarxiv: v1 [math.na] 18 Nov 2011
MIMETIC FRAMEWORK ON CURVILINEAR QUADRILATERALS OF ARBITRARY ORDER JASPER KREEFT, ARTUR PALHA, AND MARC GERRITSMA arxiv:1111.4304v1 [mat.na] 18 Nov 2011 Abstract. In tis paper iger orer mimetic iscretizations
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationPreconditioning in H(div) and Applications
1 Preconditioning in H(div) and Applications Douglas N. Arnold 1, Ricard S. Falk 2 and Ragnar Winter 3 4 Abstract. Summarizing te work of [AFW97], we sow ow to construct preconditioners using domain decomposition
More informationDifferential Calculus Definitions, Rules and Theorems
Precalculus Review Differential Calculus Definitions, Rules an Teorems Sara Brewer, Alabama Scool of Mat an Science Functions, Domain an Range f: X Y a function f from X to Y assigns to eac x X a unique
More informationDerivatives of trigonometric functions
Derivatives of trigonometric functions 2 October 207 Introuction Toay we will ten iscuss te erivates of te si stanar trigonometric functions. Of tese, te most important are sine an cosine; te erivatives
More informationFINITE ELEMENT EXTERIOR CALCULUS: FROM HODGE THEORY TO NUMERICAL STABILITY
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 47, Number 2, April 2010, Pages 281 354 S 02730979(10)012784 Article electronically publised on January 25, 2010 FINITE ELEMENT EXTERIOR
More informationON THE CONSISTENCY OF THE COMBINATORIAL CODIFFERENTIAL
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 366, Number 10, October 2014, Pages 5487 5502 S 00029947(2014)061345 Article electronically publised on February 26, 2014 ON THE CONSISTENCY OF
More informationLecture Notes Di erentiating Trigonometric Functions page 1
Lecture Notes Di erentiating Trigonometric Functions age (sin ) 7 sin () sin 8 cos 3 (tan ) sec tan + 9 tan + 4 (cot ) csc cot 0 cot + 5 sin (sec ) cos sec tan sec jj 6 (csc ) sin csc cot csc jj c Hiegkuti,
More informationNONCONFORMING MIXED ELEMENTS FOR ELASTICITY
Mathematical Models and Methods in Applied Sciences Vol. 13, No. 3 (2003) 295 307 c World Scientific Publishing Company NONCONFORMING MIXED ELEMENTS FOR ELASTICITY DOUGLAS N. ARNOLD Institute for Mathematics
More informationLinear FirstOrder Equations
5 Linear FirstOrer Equations Linear firstorer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More informationNOTES ON EULERBOOLE SUMMATION (1) f (l 1) (n) f (l 1) (m) + ( 1)k 1 k! B k (y) f (k) (y) dy,
NOTES ON EULERBOOLE SUMMATION JONATHAN M BORWEIN, NEIL J CALKIN, AND DANTE MANNA Abstract We stuy a connection between EulerMacLaurin Summation an Boole Summation suggeste in an AMM note from 196, which
More informationChapter 6: EnergyMomentum Tensors
49 Chapter 6: EnergyMomentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energymomentum tensors, then applies these ieas to the case of Bohm's moel.
More informationLINEARIZATION METHODS FOR VARIATIONAL INTEGRATORS AND EULERLAGRANGE EQUATIONS
LINEARIZATION METHODS FOR VARIATIONAL INTEGRATORS AND EULERLAGRANGE EQUATIONS Tesis by AnilN.Hirani Avisors James R. Arvo, Jerrol E. Marsen In Partial Fulfillment of te Requirements for te Degree of Master
More informationFINITE ELEMENT EXTERIOR CALCULUS FOR PARABOLIC EVOLUTION PROBLEMS ON RIEMANNIAN HYPERSURFACES
FINITE ELEMENT EXTERIOR CALCULUS FOR PARABOLIC EVOLUTION PROBLEMS ON RIEMANNIAN HYPERSURFACES MICHAEL HOLST AND CHRIS TIEE ABSTRACT. Over te last ten years, te Finite Element Exterior Calculus (FEEC) as
More informationIntroduction to finite element exterior calculus
Introduction to finite element exterior calculus Ragnar Winther CMA, University of Oslo Norway Why finite element exterior calculus? Recall the de Rham complex on the form: R H 1 (Ω) grad H(curl, Ω) curl
More informationMathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative
Matematics 5 Workseet 11 Geometry, Tangency, and te Derivative Problem 1. Find te equation of a line wit slope m tat intersects te point (3, 9). Solution. Te equation for a line passing troug a point (x
More informationAnalysis of A Continuous Finite Element Method for H(curl, div)elliptic Interface Problem
Analysis of A Continuous inite Element Metod for Hcurl, div)elliptic Interface Problem Huoyuan Duan, Ping Lin, and Roger C. E. Tan Abstract In tis paper, we develop a continuous finite element metod for
More informationA MixedHybridDiscontinuous Galerkin Finite Element Method for ConvectionDiffusion Problems
A MixedHybridDiscontinuous Galerkin Finite Element Metod for ConvectionDiffusion Problems Herbert Egger Joacim Scöberl We propose and analyse a new finite element metod for convection diffusion problems
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More informationChapter 9 Method of Weighted Residuals
Chapter 9 Metho of Weighte Resiuals 9 Introuction Metho of Weighte Resiuals (MWR) is an approimate technique for solving bounary value problems. It utilizes a trial functions satisfying the prescribe
More informationEssential Microeconomics : EQUILIBRIUM AND EFFICIENCY WITH PRODUCTION Key ideas: Walrasian equilibrium, first and second welfare theorems
Essential Microeconomics  5.2: EQUILIBRIUM AND EFFICIENCY WITH PRODUCTION Key ideas: Walrasian equilibrium, irst and second welare teorems A general model 2 First welare Teorem 7 Second welare teorem
More informationA SHORT INTRODUCTION TO BANACH LATTICES AND
CHAPTER A SHORT INTRODUCTION TO BANACH LATTICES AND POSITIVE OPERATORS In tis capter we give a brief introduction to Banac lattices and positive operators. Most results of tis capter can be found, e.g.,
More informationMATH745 Fall MATH745 Fall
MATH745 Fall 5 MATH745 Fall 5 INTRODUCTION WELCOME TO MATH 745 TOPICS IN NUMERICAL ANALYSIS Instructor: Dr Bartosz Protas Department of Matematics & Statistics Email: bprotas@mcmasterca Office HH 36, Ext
More informationCopyright c 2008 Kevin Long
Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closedform solutions to initial value problems. A closedform solution is an explicit algebriac formula
More informationLATTICEBASED DOPTIMUM DESIGN FOR FOURIER REGRESSION
The Annals of Statistics 1997, Vol. 25, No. 6, 2313 2327 LATTICEBASED DOPTIMUM DESIGN FOR FOURIER REGRESSION By Eva Riccomagno, 1 Rainer Schwabe 2 an Henry P. Wynn 1 University of Warwick, Technische
More informationLower Bounds for the Smoothed Number of Pareto optimal Solutions
Lower Bouns for the Smoothe Number of Pareto optimal Solutions Tobias Brunsch an Heiko Röglin Department of Computer Science, University of Bonn, Germany brunsch@cs.unibonn.e, heiko@roeglin.org Abstract.
More informationFINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES
MATHEMATICS OF COMPUTATION Volume 83, Number 288, July 2014, Pages 1551 1570 S 00255718(2013)027834 Article electronically publishe on October 17, 2013 FINITE ELEMENT DIFFERENTIAL FORMS ON CUBICAL MESHES
More informationPOLYNOMIAL AND SPLINE ESTIMATORS OF THE DISTRIBUTION FUNCTION WITH PRESCRIBED ACCURACY
APPLICATIONES MATHEMATICAE 36, (29), pp. 2 Zbigniew Ciesielski (Sopot) Ryszard Zieliński (Warszawa) POLYNOMIAL AND SPLINE ESTIMATORS OF THE DISTRIBUTION FUNCTION WITH PRESCRIBED ACCURACY Abstract. Dvoretzky
More informationChapter 5 FINITE DIFFERENCE METHOD (FDM)
MEE7 Computer Modeling Tecniques in Engineering Capter 5 FINITE DIFFERENCE METHOD (FDM) 5. Introduction to FDM Te finite difference tecniques are based upon approximations wic permit replacing differential
More informationQuantum Numbers and Rules
OpenStaxCNX module: m42614 1 Quantum Numbers and Rules OpenStax College Tis work is produced by OpenStaxCNX and licensed under te Creative Commons Attribution License 3.0 Abstract Dene quantum number.
More informationTopological Sensitivity Analysis for Threedimensional Linear Elasticity Problem
Topological Sensitivity Analysis for Threeimensional Linear Elasticity Problem A.A. Novotny, R.A. Feijóo, E. Taroco Laboratório Nacional e Computação Científica LNCC/MCT, Av. Getúlio Vargas 333, 25651075
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationConductance from Transmission Probability
Conductance rom Transmission Probability Kelly Ceung Department o Pysics & Astronomy University o Britis Columbia Vancouver, BC. Canada, V6T1Z1 (Dated: November 5, 005). ntroduction For large conductors,
More informationMIXED DISCONTINUOUS GALERKIN APPROXIMATION OF THE MAXWELL OPERATOR. SIAM J. Numer. Anal., Vol. 42 (2004), pp
MIXED DISCONTINUOUS GALERIN APPROXIMATION OF THE MAXWELL OPERATOR PAUL HOUSTON, ILARIA PERUGIA, AND DOMINI SCHÖTZAU SIAM J. Numer. Anal., Vol. 4 (004), pp. 434 459 Abstract. We introduce and analyze a
More informationTractability results for weighted Banach spaces of smooth functions
Tractability results for weighte Banach spaces of smooth functions Markus Weimar Mathematisches Institut, Universität Jena ErnstAbbePlatz 2, 07740 Jena, Germany email: markus.weimar@unijena.e March
More informationEVALUATING HIGHER DERIVATIVE TENSORS BY FORWARD PROPAGATION OF UNIVARIATE TAYLOR SERIES
MATHEMATICS OF COMPUTATION Volume 69, Number 231, Pages 1117 1130 S 00255718(00)011200 Article electronically publishe on February 17, 2000 EVALUATING HIGHER DERIVATIVE TENSORS BY FORWARD PROPAGATION
More informationProduct and Quotient Rules and HigherOrder Derivatives. The Product Rule
330_003.q 11/3/0 :3 PM Page 119 SECTION.3 Prouct an Quotient Rules an HigherOrer Derivatives 119 Section.3 Prouct an Quotient Rules an HigherOrer Derivatives Fin the erivative o a unction using the Prouct
More informationBasic Strategy for CardCounters: An Analytic Approach. by N. Richard Werthamer
Basic Strategy or CarCounters: An Analytic Approach by N. Richar Werthamer Abstract We evelop an eicient metho, similar to that o Marcus (7) who calls it Counter Basic Strategy, or casino blackack car
More informationPermanent vs. Determinant
Permanent vs. Determinant Frank Ban Introuction A major problem in theoretical computer science is the Permanent vs. Determinant problem. It asks: given an n by n matrix of ineterminates A = (a i,j ) an
More informationTHE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225
THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:
More informationUsing the definition of the derivative of a function is quite tedious. f (x + h) f (x)
Derivative Rules Using te efinition of te erivative of a function is quite teious. Let s prove some sortcuts tat we can use. Recall tat te efinition of erivative is: Given any number x for wic te limit
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationG j dq i + G j. q i. = a jt. and
Lagrange Multipliers Wenesay, 8 September 011 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine
More informationCombining functions: algebraic methods
Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)
More informationTHE GENUINE OMEGAREGULAR UNITARY DUAL OF THE METAPLECTIC GROUP
THE GENUINE OMEGAREGULAR UNITARY DUAL OF THE METAPLECTIC GROUP ALESSANDRA PANTANO, ANNEGRET PAUL, AND SUSANA A. SALAMANCARIBA Abstract. We classify all genuine unitary representations of the metaplectic
More informationPure Further Mathematics 1. Revision Notes
Pure Further Mathematics Revision Notes June 20 2 FP JUNE 20 SDB Further Pure Complex Numbers... 3 Definitions an arithmetical operations... 3 Complex conjugate... 3 Properties... 3 Complex number plane,
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationCOMPOSITION OF BINARY QUADRATIC FORMS: FROM GAUSS TO BHARGAVA. 1. Introduction
COMPOSITION OF BINARY QUADRATIC FORMS: FROM GAUSS TO BHARGAVA FRANÇOIS SÉGUIN Abstract. In is 004 article [], Barava introuces a new way to unerstan te composition law o interal binary quaratic orms trou
More informationCounting Lattice Points in Polytopes: The Ehrhart Theory
3 Counting Lattice Points in Polytopes: The Ehrhart Theory Ubi materia, ibi geometria. Johannes Kepler (1571 1630) Given the profusion of examples that gave rise to the polynomial behavior of the integerpoint
More informationFinite element exterior calculus: A new approach to the stability of finite elements
Finite element exterior calculus: A new approach to the stability of finite elements Douglas N. Arnold joint with R. Falk and R. Winther Institute for Mathematics and its Applications University of Minnesota
More informationand from it produce the action integral whose variation we set to zero:
Lagrange Multipliers Monay, 6 September 01 Sometimes it is convenient to use reunant coorinates, an to effect the variation of the action consistent with the constraints via the metho of Lagrange unetermine
More informationarxiv: v1 [math.na] 17 Jul 2014
Div FirstOrder System LL* FOSLL* for SecondOrder Elliptic Partial Differential Equations Ziqiang Cai Rob Falgout Sun Zang arxiv:1407.4558v1 [mat.na] 17 Jul 2014 February 13, 2018 Abstract. Te firstorder
More informationRobustness and Perturbations of Minimal Bases
Robustness an Perturbations of Minimal Bases Paul Van Dooren an Froilán M Dopico December 9, 2016 Abstract Polynomial minimal bases of rational vector subspaces are a classical concept that plays an important
More informationThe Effects of Mutual Coupling and Transformer Connection Type on Frequency Response of Unbalanced Three Phase Electrical Distribution System
Energy an Power Engineering, 00,, 847 oi:0.46/epe.00.405 Publise Online November 00 (ttp://www.scirp.org/journal/epe) Te Effects of Mutual Coupling an Transformer Connection Type on Frequency Response
More informationTHE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS
MATHEMATICS OF COMPUTATION Volume 00, Number 0, Pages 000 000 S 00255718XX00000 THE DISCRETE PLATEAU PROBLEM: CONVERGENCE RESULTS GERHARD DZIUK AND JOHN E. HUTCHINSON Abstract. We solve te problem of
More informationAnalytic Functions. Differentiable Functions of a Complex Variable
Analytic Functions Differentiable Functions of a Complex Variable In tis capter, we sall generalize te ideas for polynomials power series of a complex variable we developed in te previous capter to general
More informationAn Optimal Algorithm for Bandit and ZeroOrder Convex Optimization with TwoPoint Feedback
Journal of Machine Learning Research 8 07)  Submitte /6; Publishe 5/7 An Optimal Algorithm for Banit an ZeroOrer Convex Optimization with wopoint Feeback Oha Shamir Department of Computer Science an
More information3 The variational formulation of elliptic PDEs
Chapter 3 The variational formulation of elliptic PDEs We now begin the theoretical stuy of elliptic partial ifferential equations an bounary value problems. We will focus on one approach, which is calle
More informationLIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION
LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION LAURA EVANS.. Introduction Not all differential equations can be explicitly solved for y. Tis can be problematic if we need to know te value of y
More informationA SPLITTING LEASTSQUARES MIXED FINITE ELEMENT METHOD FOR ELLIPTIC OPTIMAL CONTROL PROBLEMS
INTERNATIONAL JOURNAL OF NUMERICAL ANALSIS AND MODELING Volume 3, Number 4, Pages 6 626 c 26 Institute for Scientific Computing and Information A SPLITTING LEASTSQUARES MIED FINITE ELEMENT METHOD FOR
More information1 Introuction In the past few years there has been renewe interest in the nerson impurity moel. This moel was originally propose by nerson [2], for a
Theory of the nerson impurity moel: The Schrieer{Wol transformation re{examine Stefan K. Kehrein 1 an nreas Mielke 2 Institut fur Theoretische Physik, uprecht{karls{universitat, D{69120 Heielberg, F..
More informationInterconnected Systems of Fliess Operators
Interconnecte Systems of Fliess Operators W. Steven Gray Yaqin Li Department of Electrical an Computer Engineering Ol Dominion University Norfolk, Virginia 23529 USA Abstract Given two analytic nonlinear
More informationSYNCHRONOUS SEQUENTIAL CIRCUITS
CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents
More informationRank, Trace, Determinant, Transpose an Inverse of a Matrix Let A be an n n square matrix: A = a11 a1 a1n a1 a an a n1 a n a nn nn where is the jth col
Review of Linear Algebra { E18 Hanout Vectors an Their Inner Proucts Let X an Y be two vectors: an Their inner prouct is ene as X =[x1; ;x n ] T Y =[y1; ;y n ] T (X; Y ) = X T Y = x k y k k=1 where T an
More informationDedicated to the 70th birthday of Professor Lin Qun
Journal of Computational Matematics, Vol.4, No.3, 6, 4 44. ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS Guangwei Yuan Xudeng Hang Laboratory of Computational Pysics,
More information12.11 Laplace s Equation in Cylindrical and
SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering
More informationICES REPORT Multirate Undrained Splitting for Coupled Flow and Geomechanics in Porous Media
ICES REPORT 1613 May 2016 Multirate Undrained Splitting or Coupled Flow and Geomecanics in Porous Media by Tameem Almani, Kundan Kumar, Mary F. Weeler Te Institute or Computational Engineering and Sciences
More informationCHAPTER (A) When x = 2, y = 6, so f( 2) = 6. (B) When y = 4, x can equal 6, 2, or 4.
SECTION 31 101 CHAPTER 3 Section 31 1. No. A correspondence between two sets is a function only if eactly one element of te second set corresponds to eac element of te first set. 3. Te domain of a function
More informationShifted Independent Component Analysis
Downloae rom orbit.tu.k on: Dec 06, 2017 Shite Inepenent Component Analysis Mørup, Morten; Masen, Kristoer Hougaar; Hansen, Lars Kai Publishe in: 7th International Conerence on Inepenent Component Analysis
More informationThe continuity equation
Chapter 6 The continuity equation 61 The equation of continuity It is evient that in a certain region of space the matter entering it must be equal to the matter leaving it Let us consier an infinitesimal
More informationMATH1901 Differential Calculus (Advanced)
MATH1901 Dierential Calculus (Advanced) Capter 3: Functions Deinitions : A B A and B are sets assigns to eac element in A eactl one element in B A is te domain o te unction B is te codomain o te unction
More informationWhy Bernstein Polynomials Are Better: FuzzyInspired Justification
Why Bernstein Polynomials Are Better: FuzzyInspire Justification Jaime Nava 1, Olga Kosheleva 2, an Vlaik Kreinovich 3 1,3 Department of Computer Science 2 Department of Teacher Eucation University of
More information4. Important theorems in quantum mechanics
TFY4215 Kjemisk fysikk og kvantemekanikk  Tillegg 4 1 TILLEGG 4 4. Important theorems in quantum mechanics Before attacking threeimensional potentials in the next chapter, we shall in chapter 4 of this
More informationCONVERGENCE OF AN IMPLICIT FINITE ELEMENT METHOD FOR THE LANDAULIFSHITZGILBERT EQUATION
CONVERGENCE OF AN IMPLICIT FINITE ELEMENT METHOD FOR THE LANDAULIFSHITZGILBERT EQUATION SÖREN BARTELS AND ANDREAS PROHL Abstract. Te LandauLifsitzGilbert equation describes dynamics of ferromagnetism,
More informationSome spaces of sequences of interval numbers defined by a modulus function
Global Journal o athematical Analysis, 2 1 2014 1116 c Science Publishing Corporation wwwsciencepubcocom/inexphp/gja oi: 1014419/gjmav2i12005 Research Paper Some spaces o sequences o interval numbers
More informationStationary Gaussian Markov Processes As Limits of Stationary Autoregressive Time Series
Stationary Gaussian Markov Processes As Limits of Stationary Autoregressive Time Series Lawrence D. Brown, Pilip A. Ernst, Larry Sepp, and Robert Wolpert August 27, 2015 Abstract We consider te class,
More informationContinuity and Differentiability Worksheet
Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet Tese were not included below! Typical problems are like problems 3, p. 6; 3, p. 7; 3334, p. 7;
More informationLATTICE EXIT MODELS S. GILL WILLIAMSON
LATTICE EXIT MODELS S. GILL WILLIAMSON ABSTRACT. We discuss a class of problems wic we call lattice exit models. At one level, tese problems provide undergraduate level exercises in labeling te vertices
More informationDiscrete Hamilton Jacobi Theory and Discrete Optimal Control
49th IEEE Conference on Decision an Control December 1517, 2010 Hilton Atlanta Hotel, Atlanta, GA, USA Discrete Hamilton Jacobi Theory an Discrete Optimal Control Tomoi Ohsawa, Anthony M. Bloch, an Melvin
More informationChapter 2 Ising Model for Ferromagnetism
Capter Ising Model for Ferromagnetism Abstract Tis capter presents te Ising model for ferromagnetism, wic is a standard simple model of a pase transition. Using te approximation of meanfield teory, te
More informationA Mixed Nonconforming Finite Element for Linear Elasticity
A Mixed Nonconforming Finite Element for Linear Elasticity Zhiqiang Cai, 1 Xiu Ye 2 1 Department of Mathematics, Purdue University, West Lafayette, Indiana 479071395 2 Department of Mathematics and Statistics,
More informationLecture 6: Calculus. In Song Kim. September 7, 2011
Lecture 6: Calculus In Song Kim September 7, 20 Introuction to Differential Calculus In our previous lecture we came up with several ways to analyze functions. We saw previously that the slope of a linear
More informationPeaked Periodic Wave Solutions to the Broer Kaup Equation
Commun. Theor. Phys. 67 (2017) 22 26 Vol. 67, No. 1, January 1, 2017 Peake Perioic Wave Solutions to the Broer Kaup Equation Bo Jiang ( 江波 ) 1, an QinSheng Bi ( 毕勤胜 ) 2 1 Department o Applie Mathematics,
More informationNotes on Lie Groups, Lie algebras, and the Exponentiation Map Mitchell Faulk
Notes on Lie Groups, Lie algebras, an the Exponentiation Map Mitchell Faulk 1. Preliminaries. In these notes, we concern ourselves with special objects calle matrix Lie groups an their corresponing Lie
More informationChapter 2 Governing Equations
Chapter 2 Governing Equations In the present an the subsequent chapters, we shall, either irectly or inirectly, be concerne with the bounarylayer flow of an incompressible viscous flui without any involvement
More informationHYBRIDIZED GLOBALLY DIVERGENCEFREE LDG METHODS. PART I: THE STOKES PROBLEM
MATHEMATICS OF COMPUTATION Volume 75, Number 254, Pages 533 563 S 00255718(05)018041 Article electronically publised on December 16, 2005 HYBRIDIZED GLOBALLY DIVERGENCEFREE LDG METHODS. PART I: THE
More informationEfficient Parsing with the ProductFree Lambek Calculus
Efficient Parsing wit te ProuctFree Lambek Calculus Timoty A. D. Fowler Department of Computer Science University of Toronto 10 King s College Roa, Toronto, ON, M5S 3G4, Canaa tfowler@cs.toronto.eu Abstract
More informationComputing Derivatives
Chapter 2 Computing Derivatives 2.1 Elementary erivative rules Motivating Questions In this section, we strive to unerstan the ieas generate by the following important questions: What are alternate notations
More informationThe proper definition of the added mass for the water entry problem
The proper efinition of the ae mass for the water entry problem Leonaro Casetta lecasetta@ig.com.br Celso P. Pesce ceppesce@usp.br LIE&MO luistructure Interaction an Offshore Mechanics Laboratory Mechanical
More informationAdjoint Transient Sensitivity Analysis in Circuit Simulation
Ajoint Transient Sensitivity Analysis in Circuit Simulation Z. Ilievski 1, H. Xu 1, A. Verhoeven 1, E.J.W. ter Maten 1,2, W.H.A. Schilers 1,2 an R.M.M. Mattheij 1 1 Technische Universiteit Einhoven; email:
More informationClosepacked dimers on nonorientable surfaces
4 February 2002 Pysics Letters A 293 2002 235 246 wwwelseviercom/locate/pla Closepacked dimers on nonorientable surfaces Wentao T Lu FYWu Department of Pysics orteastern University Boston MA 02115 USA
More informationAdaptive Finite Element Method
39 Capter 3 Adaptive inite Element Metod 31 Introduction As already pointed out in capter 2, singularities occur in interface problems Wen discretizing te problem (221) wit inite Elements, te singularities
More informationImplicit Differentiation
Implicit Differentiation Thus far, the functions we have been concerne with have been efine explicitly. A function is efine explicitly if the output is given irectly in terms of the input. For instance,
More informationReflection Symmetries of qbernoulli Polynomials
Journal of Nonlinear Matematical Pysics Volume 1, Supplement 1 005, 41 4 Birtday Issue Reflection Symmetries of qbernoulli Polynomials Boris A KUPERSHMIDT Te University of Tennessee Space Institute Tullaoma,
More informationTHE STURMLIOUVILLETRANSFORMATION FOR THE SOLUTION OF VECTOR PARTIAL DIFFERENTIAL EQUATIONS. L. Trautmann, R. Rabenstein
Worksop on Transforms and Filter Banks (WTFB),Brandenburg, Germany, Marc 999 THE STURMLIOUVILLETRANSFORMATION FOR THE SOLUTION OF VECTOR PARTIAL DIFFERENTIAL EQUATIONS L. Trautmann, R. Rabenstein Lerstul
More informationEquilibrium in Queues Under Unknown Service Times and Service Value
University of Pennsylvania ScholarlyCommons Finance Papers Wharton Faculty Research 12014 Equilibrium in Queues Uner Unknown Service Times an Service Value Laurens Debo Senthil K. Veeraraghavan University
More informationFunction Spaces. 1 Hilbert Spaces
Function Spaces A function space is a set of functions F that has some structure. Often a nonparametric regression function or classifier is chosen to lie in some function space, where the assume structure
More information