A symmetric mixed finite element metod for nearly incompressible elasticity based on biortogonal systems Bisnu P. Lamicane and Ernst P. Stepan February 18, 2011 Abstract We present a symmetric version of te non-symmetric mixed finite element metod presented in [23] for nearly incompressible elasticity. Te displacement pressure formulation of linear elasticity is discretized using a Petrov Galerkin discretization for te pressure equation in [23] leading to a non-symmetric saddle point problem. A new tree-field formulation is introduced to obtain a symmetric saddle point problem wic allows us to use a biortogonal system. Working wit a biortogonal system, we can statically condense out all auxiliary variables from te saddle point problem arriving at a symmetric and positive-definite system based only on te displacement. We also derive a residual based error estimator for te mixed formulation of te problem. Keywords: Mixed finite elements, symmetric system, Petrov Galerkin discretization, biortogonal system AMS Subject Classification: 65N30, 65N15, 74B10 1 Introduction It is a well-known fact tat linear, bilinear or trilinear finite elements based on te standard displacement based formulation exibit a locking effect wen applied to nearly incompressible elasticity. A popular approac is to use a mixed formulation introducing pressure as an extra unknown. Ten one as to work wit two finite element spaces: one for discretizing te displacement field, and te oter for discretizing te pressure. Tese finite element spaces sould also be compatible in te sense tat a suitable inf- condition is satisfied. One is Scool of Matematical & Pysical Sciences, Matematics Building, University of Newcastle, University Drive, Callagan, NSW 2308, Australia, Bisnu.Lamicane@newcastle.edu.au Leibniz University Hannover, Institute for Applied Matematics, Am Welfengarten 1, 30167 Hannover, Germany, stepan@ifam.uni-annover.de 1
also interested in statically condensing out te pressure from te system arriving at a formulation based only on te displacement. Te pressure variable can be obtained as a post-processing step. Tere are many different metods to alleviate te locking effect in nearly incompressible elasticity using quadrilateral or exaedral meses. Some popular metods are enanced assumed strain [32, 31, 18, 1], mixed strain metod [19, 20] and strain gap metod [30]. Tese metods are based on a tree-field formulation, popularly known as te Hu-Wasizu formulation, wic is a mixed formulation based on stress, strain and displacement. Using te stability of te mini-element [2], te mixed enanced formulation is extended to simplicial meses in [34], were te pressure variable is assumed to be continuous. Te matematical analysis of te enanced assumed strain for linear elasticity can be found in [29, 7, 26]. We ave sown a uniform convergence of te tree-field formulation in te nearly incompressible elasticity using a modified Hu-Wasizu formulation. However, te matematical analysis in [26] is quite restrictive and only covers a class of quadrilateral meses. Moreover, we could not negate te presence of spurious pressure modes in te pressure or te volumetric part of te stress [7, 26]. One of te simplest mixed formulations for elasticity is a displacement pressure formulation [11, 6]. Tis formulation is obtained by introducing pressure as an extra variable, and writing a variational equation for te pressure. Starting wit tis simplest mixed formulation, a non-symmetric mixed finite element metod based on linear finite elements on simplicial meses is presented in [23] for linear elasticity and in [25] for nonlinear elasticity. Te stability of te formulation is sown by recourse to te mini-element formulation [2] of te Stokes problem. As te variational equation for te pressure is obtained by using a Petrov Galerkin formulation, were te trial and test spaces for te pressure are different, te discrete system is non-symmetric. Te trial and test spaces for te pressure form a biortogonal system, and ence te associated Gram matrix is diagonal. Tis allows an easy static condensation of te pressure from te system leading to a reduced system. As te non-symmetricity is a main drawback of tis approac, in tis paper, we present a symmetric version of tis approac. Te symmetric version is obtained by introducing a Lagrange multiplier variable as in te case of te biarmonic equation [14]. Te continuous formulation is given in Section 2, and te discretization is presented in Section 3. Te numerical analysis of te discrete formulation is presented in Section 4. Te formulation is extended to nonlinear yperelasticity in Section 5. We ave also derived residual based error estimator for our formulation in Section 6. Finally, we ave sown some numerical experiments in te last section. 2
2 Te boundary value problem of linear elasticity Tis section is devoted to te introduction of te boundary value problem of linear elasticity. We consider a omogeneous isotropic linear elastic material body occupying a bounded domain in R d, d {2, 3} wit Lipscitz boundary Γ. For a prescribed body force f [L 2 ()] d, te governing equilibrium equation in reads div σ = f, (1) were σ is te symmetric Caucy stress tensor. Te stress tensor σ is defined as a function of te displacement u by te Saint-Venant Kircoff constitutive law σ = 1 2 C( u + [ u]t ), (2) were C is te fourt-order elasticity tensor. Te action of te elasticity tensor C on te strain tensor ε(u) := 1 2 ( u + [ u]t ) is defined as σ = Cε(u) = (tr ε(u))1 + 2µ ε(u). (3) Here, 1 is te identity tensor, and and µ are te Lamé parameters, wic are constant in view of te assumption of a omogeneous body, and wic are assumed positive. Te displacement is assumed to satisfy te omogeneous Diriclet boundary condition u = 0 on Γ. (4) We are interested in te nearly incompressible case, wic corresponds to being very large. Here we use standard notations L 2 (), H 1 () and H 1 0 () for Sobolev spaces, see [8, 15] for details. Let V := [H 1 0 ()] d be te vector Sobolev space wit inner product (, ) 1, and norm 1, defined in te standard way: (u, v) 1, := d i=1 (u i, v i ) 1,, and te norm being induced by tis inner product. We define te bilinear form A(, ) and te linear functional l( ) by A : V V R, A(u, v) := Cε(u) : ε(v) dx, l : V R, l(v) := f v dx. Ten te standard weak form of te linear elasticity problem is as follows: given l V, find u V tat satisfies A(u, v) = l(v), v V, (5) were V is te space of continuous linear functionals on V. Te assumptions on C guarantee tat A(, ) is symmetric, continuous, and V -elliptic. Hence by using standard arguments it can be sown tat (5) as a unique solution u V. 3
Furtermore, we assume tat te domain is convex and polygonal or polyedral suc tat u [H 2 ()] d V, and tere exists a constant C independent of suc tat u 2, + div u 1, C f 0,. (6) Te a priori estimate (6) as been sown in [9] for te two-dimensional linear elasticity posed in a convex domain wit polygonal boundary, see [21] for te tree-dimensional case wit convex domain and polyedral boundary. As pointed out in te introduction, te linear elasticity problem can be recast into different mixed formulations. Te easiest mixed formulation is given by introducing pressure as an extra variable, wic leads to penalized Stokes equations. Defining p := div u and R := {q L 2 () : q dx = 0}, a mixed variational formulation of linear elastic problem (5) is given by: find (u, p) V R suc tat were ã(u, v) := 2µ ã(u, v) + b(v, p) = l(v), v V, b(u, q) 1 c(p, q) = 0, q R, (7) ε(u) : ε(v) dx, b(v, q) := div v q dx, and c(p, q) := p q dx. We note tat p R because of te omogeneous Diriclet boundary condition. Te existence, uniqueness and te stability of te mixed formulation (7) can be establised by using te standard saddle point teory [11, 5]. Using a pair of finite element bases forming a biortogonal system to discretize te pressure equation, an efficient numerical sceme arises [23, 25]. Since a Petrov Galerkin discretization is applied to te pressure equation, were te trial and test spaces are different, te arising linear system is non-symmetric even after eliminating te pressure. Terefore, we consider a new formulation, wic allows us to use a biortogonal system and leads to a symmetric formulation. To tis end, we write te standard weak formulation of te linearly elastic problem as a minimization problem: Since min u V 1 A(u, u) l(u). (8) 2 A(u, u) = 2µ ε(u) : ε(u) dx + div u div u dx, we introduce a pressure-like variable φ := div u R, and write te minimization problem (8) as a constrained minimization problem: ( ) 1 min 2µε(u) : ε(u) dx + φ 2 dx l(u). (9) (u,φ) V R φ= 2 div u 4
We write a weak variational equation for te equation φ = div u in terms of te Lagrange multiplier space R to obtain te saddle point problem of te minimization problem (9). Te saddle point formulation is to find (u, φ, ξ) V R R suc tat a((u, φ), (v, ψ)) + b((v, ψ), ξ) = l(v), (v, ψ) V R, b((u, φ), η) = 0, η R, (10) were a((u, φ), (v, ψ)) = 2µ b((u, φ), η) = ε(u) : ε(v) dx + (div u φ ) η dx. φψ dx, and Here te Lagrange multiplier plays te role of pressure of te formulation (7), i.e., ξ = φ = div u. Tis idea can be easily generalized to nonlinear elasticity if te problem can be written as a minimization problem (9). In order to sow tat te saddle point problem (10) as a unique solution, we want to apply a standard saddle point teory [11, 6, 8]. To tis end, we need to sow te following tree conditions of well-posedness. 1. Te linear form l( ), te bilinear forms a(, ) and b(, ) are continuous on te spaces on wic tere are defined. 2. Te bilinear form a(, ) is coercive on te space K defined as K = {(v, ψ) V L 2 0() : b((v, ψ), η) = 0, η L 2 0()}. 3. Te bilinear form b(, ) satisfies te inf- condition: b((v, ψ), η) β η 0,, η L 2 (v,ψ) V L 2 0 () v 1, + ψ 0() (11) 0, for a constant β > 0 independent of. It is easy to sow tat te linear form l( ), and te bilinear forms a(, ) and b(, ) are continuous. Te coercivity of te bilinear form a(, ) follows from Korn s inequality: a((u, φ), (u, φ)) = 2µ ε(u) 2 0, + φ 2 0, C( u 2 1, + φ 2 0,). (12) Tus te bilinear form a(, ) is coercive on te wole space V L 2 0(). Now we sow tat te bilinear form b(, ) satisfies te inf- condition uniformly in. 5
Lemma 1 Tere exists β > 0 independent of suc tat Proof b((v, ψ), η) β η 0,, η L 2 (v,ψ) V L 2 0 () v 1, + ψ 0(). (13) 0, b((v, ψ), η) (v,ψ) V L 2 0 () v 1, + ψ 0, ( ) = 1 b((v, ψ), η) b((v, ψ), η) +. 2 (v,ψ) V L 2 0 () v 1, + ψ 0, (v,ψ) V L 2 0 () v 1, + ψ 0, Now we substitute ψ = 0 in te remum for te first term, and v = 0 in te remum for te second term to get b((v, ψ), η) 1 (v,ψ) V L 2 0 () v 1, + ψ 0, 2 div v η dx + 1 v V v 1, 2 ψ η. ψ L 2 0 () ψ 0, Te result follows from te fact tat [17] v V div v η dx v 1, β 1 η 0,. Summarizing we ave proved te following teorem. Teorem 2 Te saddle point problem (10) as a unique solution (u, φ, ξ) V L 2 0() L 2 0() and u 1, + φ 0, + ξ 0, f 0,. 3 Finite element discretization We consider a quasi-uniform triangulation T of te polygonal or polyedral domain, were T consists of simplices, eiter triangles or tetraedra. Making use of te standard linear finite element space S defined on te triangulation T S := { v H 1 () : v T P 1(T ), T T } and te space of bubble functions { B := b T P d+1 (T ) : b T T = 0, and T } b T dx > 0, T T, we introduce our finite element space for te displacement as V = (S B ) d V. Te bubble function on an element T is most often defined as b T (x) = c b Π d+1 i=1 T i(x), 6
were T i(x) are te barycentric coordinates of te element T associated wit vertices x T i of T, i = 1,..., d + 1. Te constant c b is computed in suc a way tat te value of te bubble function at te barycenter of T is one. Let N be te number of nodes in te finite element mes, and {ϕ 1,..., ϕ N } be te finite element basis of S. Starting wit te basis of S, we construct a dual space M spanned by te basis {ξ 1,..., ξ N } so tat te basis functions of S and M satisfy a condition of biortogonality relation ξ i ϕ j dx = c j δ ij, c j 0, 1 i, j N, (14) were δ ij is te Kronecker symbol, and c j a scaling factor, wic is cosen so tat T ξ i dx = T ϕ i dx. Hence, te sets of basis functions of S and M form a biortogonal system. Te basis functions of M are constructed locally on a reference element ˆT so tat te basis functions of S and M ave te same port [23], and in eac element te sum of all te basis functions of M is one. In te following, we give te local basis functions of M on te reference triangle ˆT := {(x, y) : 0 x, 0 y, x + y 1} and on te reference tetraedron ˆT := {(x, y, z) : 0 x, 0 y, 0 z, x + y + z 1}. For te te reference triangle, we ave ˆξ 1 := 3 4x 4y, ˆξ 2 := 4x 1, and ˆξ 3 := 4y 1, associated wit its tree vertices (0, 0), (1, 0) and (0, 1), respectively, and for te reference tetraedron ˆT := {(x, y, z) : 0 x, 0 y, 0 z, x + y + z 1}, we ave ˆξ 1 := 4 5x 5y 5z, ˆξ 2 := 5x 1, and ˆξ 3 := 5y 1, ˆξ 4 := 5z 1, associated wit its four vertices (0, 0, 0), (1, 0, 0), (0, 1, 0) and (0, 0, 1), respectively. Moreover, d+1 ˆξ i=1 i = 1. Te global basis functions for te space M are constructed by glueing te local basis functions togeter. Tis procedure of constructing global basis functions for M from te local ones is te same as of constructing global basis functions for te standard finite element space S from te local ones. Tese global basis functions ten satisfy te condition of biortogonality (14) wit global finite element basis functions, and pϕ i = pξ i, 1 i N. We can see tat a local basis function of M does not assume value one at one vertex and zero at oter vertices. Since te global basis functions are extended by zero beyond teir port, tey are not continuous. Remark 3 Suc a biortogonal system as been very popular in te context of mortar finite elements [35, 22]. Construction of basis functions of M satisfying te biortogonality and an optimal approximation property for a iger order finite element space is considered in [22]. However, construction of suc a basis for iger order finite element in a simplicial mes is not so easy [27]. 7
We also need a subspace of S and a subspace of M aving zero average on defined as { } { } S 0 := v S : v dx = 0, M 0 := ξ M : ξ dx = 0. In [23, 25], te first equation of (7) is discretized by using a Galerkin formulation, and te second equation of (7) is discretized by using a Petrov Galerkin formulation. Te Petrov Galerkin formulation is cosen so tat te pressure solution is taken from S, wereas te test functions are taken from M. Hence te discrete formulation of variational equation (7) is a non-symmetric saddle point problem. Altoug te existence, uniqueness and stability of te solution are establised in [23] using te teory of non-symmetric saddle point problems [28, 4], many well-known linear solvers perform better for symmetric linear systems. Terefore, we consider te discretization of (10) using our biortogonal system. Te discrete formulation is to find (u, φ, ξ ) V S 0 M 0 suc tat a((u, φ ), (v, ψ )) + b((v, ψ ), ξ ) = l(v ), (v, ψ ) V S 0, b((u, φ ), η ) = 0, η M 0. (15) Te goal of coosing te different finite element bases to discretize φ and ξ is to be able to statically condense out tese variables from te system leading to a displacement-based formulation. In order to obtain an algebraic formulation of te discrete saddle point problem (15), we use te same notation for te vector representation of te solution and te solution as elements in V, S and M. Using ψ = 0 and v = 0 subsequently in (15), we ave 2µ ε(u ) : ε(v ) dx + div v ξ dx = l(v ), v V φ ψ ψ ξ dx = 0, ψ S 0. (16) Let A, M, B and D be te matrices associated wit te bilinear forms 2µ ε(u ) : ε(v ) dx, φ ψ dx, div u : η dx, and φ η dx, respectively. Ten te algebraic formulation of te saddle point problem (15) can be written as A 0 B T 0 M 1 D T B 1 D 0 u φ ξ = f 0 0, (17) were f is te vector of discretization of te linear form l( ). Note tat te first two equations in (17) correspond to te two equations in (16), and te last equation in (17) corresponds to te last equation in (15). If we look closely at te linear system (17), we find tat if te matrix D is diagonal, we can easily eliminate te degrees of freedom corresponding to φ and ξ arriving at te formulation involving only one unknown u. After statically condensing out variables φ and ξ, we arrive at te reduced system ( A + B T D 1 MD 1 B ) u = f. As te matrix D 1 is also diagonal, te system matrix is sparse. Te variational formulation of tis reduced system is given in (24). 8
4 A priori error estimate Now we turn our attention to sowing te well-posedness and error estimate for our discrete formulation. In order to sow te existence, uniqueness and stability of te solution of (15), we ave to prove tree conditions of well-posedness stated in Section 2 before Lemma 1. As te discretization is conforming te linear form l( ), and te bilinear forms a(, ) and b(, ) are continuous on te discrete spaces as well. An application of Korn s inequality again guarantees te coercivity of te bilinear form a(, ): a((u, φ ), (u, φ )) = 2µ ε(u ) 2 0, + φ 2 0, C( u 2 1, + φ 2 0,). (18) Hence te bilinear form a(, ) is coercive on te wole space V S 0. It remains to sow te tird condition. Tat is, te bilinear form b(, ) satisfies te discrete inf- condition: (v,ψ ) V S 0 b((v, ψ ), η ) v 1, + ψ 0, β η 0,, η M 0 (19) for a constant β > 0 independent of mes-size and. To sow tat te bilinear form b(, ) satisfies te inf- condition (19), we need te following lemma. A proof of tis lemma can be found in [23]. Lemma 4 Tere exists a constant ˆβ independent of te mes-size suc tat div v η dx v V v ˆβ η 0,, η M. 0 (20) 1, Wit te elp of tis lemma, we can sow tat te bilinear form b(, ) satisfies te inf- condition (19). Te proof of tis lemma is similar to te one in te continuous case but we give te proof for completeness. Lemma 5 Tere exists β > 0 independent of mes-size and suc tat (v,ψ ) V S 0 Proof Let η M 0. = 1 2 (v,ψ ) V S 0 ( 1 2 v V (v,ψ ) V S 0 b((v, ψ ), η ) v 1, + ψ 0, β η 0,, η M 0. (21) b((v, ψ ), η ) v 1, + ψ 0, b((v, ψ ), η ) + v 1, + ψ 0, div v η dx v 1, + 1 2 ψ S 0 (v,ψ ) V S 0 ψ η ψ 0,, ) b((v, ψ ), η ) v 1, + ψ 0, 9
were te last step is obtained by using ψ = 0 in te remum for te first term, and v = 0 in te remum for te second term from te previous step. Te result follows (wit β = ˆβ 2 ) by using Lemma 4. As all conditions of well-posedness are satisfied, te standard teory of saddle point problem yields te following two teorems [11]: Teorem 6 Te discrete problem (15) as exactly one solution (u, φ, ξ ) V S 0 M 0, wic is uniformly stable wit respect to te data f, and tere exists a constant C independent of mes-size and Lamé parameter suc tat u 1, + φ 0, + ξ 0, C f 0,. Teorem 7 Assume tat (u, φ, ξ) V R R and (u, φ, ξ ) V S 0 M 0 are te solutions of problems (10) and (15), respectively. Ten, te following error estimate olds uniformly in : u u 1, + φ φ 0, + ξ ξ 0, ( C inf u v 1, + inf φ ψ 0, + v V ψ S 0 inf η M 0 ξ η 0, ). It is well-known tat te space V and S 0 ave optimal approximation property. We now establis an approximation property of te space M. For tat purpose, we define a quasi-projection operator Q : L 2 () M by Q v ϕ dx = vϕ dx, v L 2 (), ϕ S. Since Q v M, we ave Q v = N j=1 d jξ j. We multiply bot sides of Q v = N j=1 d jξ j by ϕ i, integrate in and use te biortogonality relation between te basis functions of S and M to get d i = ϕ i v dx. c i Tis allows us to write te action of te operator Q on a function v L 2 () as N Q v = ϕ i v dx ξ i, (22) c i i=1 wic tells tat te operator Q is local in te sense tat for any v L 2 (), te value of Q v at any point in T T only depends on te value of v in S(T ), were S(T ) is te patc of an element T T. Precisely, S(T ) is defined as te interior of te closed set S(T ) = { T T : T T }. (23) 10
Te definition of Q allows us to eliminate te auxiliary variables φ and ξ from te discrete saddle point problem (15) arriving at a symmetric and positivedefinite problem of finding u V suc tat 2µ ε(u ) : ε(v ) dx + Q div u Q div v dx = l(v ), v V. (24) Moreover, Q is stable in te L 2 -norm [22]. Lemma 8 For eac element T T, if v L 2 (S(T )), Q v 0,T C v 0,S(T ). (25) Lemma 9 Let v H 1 (). Ten tere exists a constant C > 0 suc tat v Q v 0, C v 1,. (26) Proof We first sow tat if w L 2 () and w = 1 on S(T ), ten Q w T = w on T. (27) Let l 1,, l ns(t ) be te indices of vertices in S(T ) ordered in suc a way tat l 1,, l nt are vertices of element T, and l nt +1,, l ns(t ) are indices of remaining vertices in S(T ). Denoting te port of ϕ i by S i and using te expression of Q w from (22), we ave n T S li ϕ li w dx Q w T = ξ li. (28) c li i=1 Since w = 1 on S(T ), and te sum of all basis functions of M at eac element is one, w = n S(T ) i=1 ξ li on S(T ). We substitute tis expression of w in (28) and obtain Q w = wic concludes tat Q w = w on T. Let S(T ) be te area or te volume of te set S(T ), v H 1 (), and c v := 1 S(T ) v dx. Ten using a triangle S(T ) inequality and Lemma 8, we obtain n T i=1 v Q v 0,T v c v 0,T + Q v c v 0,T = v c v 0,T + Q (v c v ) 0,T C v c v 0,S(T ). An application of te Bramble Hilbert lemma [15] ten yields a constant C > 0 suc tat v Q v 0,T C v 1,S(T ). (29) Te estimate (26) is ten obtained by summing (29) over all elements of T and noting tat eac element is contained in only a fixed number of S(T ) for T T. Ten Teorem 7 yields te linear convergence of te energy norm of te error to zero wit respect to te mes-size if u [H 2 ()] d, φ H 1 () and ξ H 1 (). ξ li, 11
5 Extension to finite elasticity In tis section, we briefly outline te extension of tis symmetric formulation to te nonlinear elasticity. Te non-symmetric formulation proposed in [23] is extended to nonlinear yperelastic material law in [24]. Here we propose a symmetric mixed formulation for finite elasticity, were we can apply a biortogonal system for te discretization so tat oter auxiliary variables can be statically condensed out from te system as in te linear case. Te variational approac to te nonlinear elasticity is based on te minimization of te energy functional of te form (W (F (u)) l(u)) dx (30) over a suitable class W of te displacements [16, 13, 33], were F (u) is te deformation gradient defined as F (u) = u + 1. Let J = det(f ). We assume tat te polyconvex energy function W (F ) can be written as W (F ) = H(, J) 2 + G 2 (F ), were G 2 is independent of, and H(, J) is a function of and J. For te isotropic material te energy function W depends only on te tree principal invariants I C, II C and III C of C = F T F, were I C = tr(c), II C = 1 2 (tr2 (C) tr(c 2 )), and III C = det(c) = J 2. If te material law satisfies te two-term Mooney Rivlin law [33], we ave W (F ) = U (J) + µ 2 [(1 c m) (I C 3 2 ln(j)) + c m (II C 3 2 ln(j))] were and µ are Lamé parameters and 0 c m 1 is a material constant. Te real-valued function U is given by U(J) = 1 2 (J 1)2 or U(J) = 1 2 (ln J)2 or U(J) = 1 4 (J 2 1 2 ln J). For tis case H(, J) = U(J), and G 2 (F ) = µ 2 [(1 c m) (I C 3 2 ln(j)) + c m (II C 3 2 ln(j))]. We recall tat te standard neo-hookean law is recovered wen c m = 0. Introducing a pressure-like variable φ = H(, J), te mixed formulation for te finite elasticity can be obtained by minimizing te functional (φ 2 + G 2 (F ) l(u)) dx (31) over te function space L 2 () W under te weak constraint (φ H(, J))η dx = 0, η L 2 (). 12
Ten te symmetric non-linear saddle point problem is given by te Euler- Lagrange equations of te constrained minimization problem: min (φ 2 + G 2 (F ) l(u)) dx (u,φ) W L 2 () subject to (φ H(, J))η dx = 0, η L 2 (). 6 A posteriori error estimator In tis section, using a similar approac as in [3], we derive an a posteriori error estimator of residual type. Let T be te diameter of element T, and E te size of edge E. We assume sape regularity of te mes, wic, in particular, means T T for all T, T T wit T T =. Here and in te following R S means tat te quantity R is bounded from above by C S, were C is a constant not depending on te mes size. Teorem 10 Let (u, φ, ξ) V R R and (u, φ, ξ ) V S 0 M 0 denote te solution of problems (10) and (15), respectively. Ten tere olds te a posteriori estimate + u u 1, + φ φ 0, + ξ ξ 0, [ T f + div 2µε(u ) 0,T + 1/2 E [2µε(u )n T ] 0,E T T E T φ ξ + ξ 0,T + 0,T div u φ ]. 0,T Proof As in [12], we apply Brezzi s teory [10] to obtain te following inf- condition wit a constant C > 0: Let u 1, + φ 0, + ξ 0, C (v,ψ,η) V R R {a((u, φ), (v, ψ)) + b((u, φ), η) + b((v, ψ), ξ)}. v 1,+ ψ 0,+ η 0,=1 A := a((u u, φ φ ), (v, ψ)) + b((v, ψ), ξ ξ ) I := u u 1, + φ φ 0, + ξ ξ 0, and B := div u φ. 0, Using Caucy-Scwarz inequality, we ave ( b((u u, φ φ ), η) = div (u u ) φ φ ) η dx = b((u, φ ), η) η 0, div u φ 0,, 13
and ence Since I C we ave (v,ψ,η) V R R v 1,+ ψ 0,+ η 0,=1 I C b((u u, φ φ ), η) η 0, B. (v,ψ,η) V R R v 1,+ ψ 0,+ η 0,=1 { a((u u, φ φ ), (v, ψ))+ b((u u, φ φ ), η) + b((v, ψ), ξ ξ ) A + b((u u, φ φ ), η). Ten wit te Galerkin ortogonality for arbitrary v A = 2µ ε(u u ) : ε(v) dx + (φ φ )ψ dx + (div v ψ ) (ξ ξ ) dx = 2µ ε(u u ) : ε(v v ) dx + (φ φ )(ψ ψ ) dx ( + div (v v ) ψ ψ ) (ξ ξ ) dx = f (v v ) dx 2µ ε(u ) : ε(v v ) dx φ (ψ ψ ) dx ( div (v v ) ψ ψ ) ξ dx = [ (f + div 2µε(u )) (v v ) dx 2µ(ε(u )n T ) (v v ) dσ T T T T ( φ ξ ) ] (ψ ψ ) dx ξ div (v v ) dx, T T were n T is te unit outward normal vector on T, and ε(u )n T is interpreted as matrix-vector product, wic produces a vector, and ence (ε(u )n T ) (v v ) is a scalar function. Tus using Caucy-Scwarz inequality and trace teorem and coosing v an interpolant to v, we obtain A C [ T f + div 2µε(u ) 0,T v 1,T + 1/2 E [2µε(u )n T ] 0,E v 1, T + T T E T φ ξ ] ψ ψ 0,T + ξ 0,T div (v v ) 0,T 0,T C [ T f + div 2µε(u ) 0,T + 1/2 E [2µε(u )n T ] 0,E T T + ξ 0,T + φ ξ ], 0,T E T }, 14
were [u] represents te jump of te function u across te interelement boundaries, and C now depends on v and ψ. We ave also used te fact tat v v 0,T C v 1,T. Let [ D = T f + div 2µε(u ) 0,T + 1/2 E [2µε(u )n T ] 0,E T T E T + ξ 0,T + φ ξ ]. 0,T Tis sows tat I B + D. Tus te total error I is bounded by computable error B + D. In te following, we sow tat te part B of te computable error B + D converges asymptotically to zero wen 0. To sow tis, we introduce anoter quasi-projection operator Q : L2 () S, Q v η dx = v η dx, η M. Tis operator is uniquely defined due to te biortogality relation between S and M. Since Q v S and Q w M, for v, w L 2 (), we ave Q v Q w dx = w Q v dx = v Q w dx. Moreover, we ave Q v = v for v S, and ence Q is an projection onto S. Terefore, Q is stable in te L2 -norm and as te following approximation property in te L 2 -norm: v Q v 0, v 1,, v H 1 (). Tat means Lemmas 8 and 9 can be reproduced for Q. Ten using φ = Q div u, we ave for 0 < r 1, B = div u φ = div u div u + div u Q 0, div u 0, C r div u r,, were we ave assumed tat te exact solution u H 1+r () wit r > 0. Tis sows tat B converges to zero asymptotically wen 0. Te displacement solution is computed by inverting te reduced system after statically condensing out φ and ξ. Ten te solutions φ and ξ can be efficiently computed just by inverting a diagonal matrix. Tis gives an efficient implementation of te error estimator. 15
7 Numerical examples In tis section, we demonstrate te performance of our new formulation for linear elasticity wit some numerical examples. Te material parameters are given in compatible units. Bot following examples do not ave pure Diriclet boundary condition, and terefore, we do not need to satisfy te zero mean condition for functions in M and S. However, in case of pure Diriclet boundary condition, bot trial and test functions in M and S sould satisfy te zero mean condition. Tat means te solutions ξ M and φ S are determined up to an additive constant. We ten searc for solutions ξ M and φ S wose values at one arbitrary vertex are prescribed a priori. Example 1: Cook s membrane Our first problem is te popular bencmark problem called Cook s membrane problem. Te Cook s membrane is a twodimensional tapered panel := conv{(0, 0), (48, 44), (48, 60), (0, 44)}, were conv{ξ} represents te convex ull of te set ξ. Te left boundary of te panel is clamped in bot directions and te rigt boundary is subjected to an inplane sear load in te positive y- direction as sown in te left picture of Figure 1. We consider te material parameters E = 250 and ν = 0.49999. We compute te vertical tip displacement at te top-rigt corner of te membrane using uniform refinement of te initial mes given in te rigt picture of Figure 1. In te rigt picture of Figure 1, we sow te convergence of te numerical results wit respect to te number of elements using tree finite element formulations for linear elasticity. We can see te good convergence beaviour of te two mixed formulations, wereas te standard displacement formulation sows te locking effect. Te new mixed formulation sows even better accuracy tan te mini finite element in te coarse mes. Te mini finite element proposed in [2] is applied to te linear elasticity. Example 2: Rectangular beam In tis second example, we consider a linear elastic beam of rectangular size subjected to a couple at one end, as sown in Figure 2. Along te edge x = 0, te orizontal displacement and vertical surface traction are zero. At te point (0, 0), te vertical displacement is also zero. Te exact solution is given by u(x, y) = 2f(1 ( ) ν2 ) l x El 2 y, and v(x, y) = f(1 [ ν2 ) x 2 + ν ] y(y l). El 1 ν We set L = 10, l = 2, E = 1500, ν = 0.4999, and f = 3000. We ave sown te setting of te problem in Figure 2, and te discretization errors wit respect to te number of elements are presented in Figure 3. As can be seen from Figure 3, te standard approac locks completely, wereas we get very good numerical approximations wit te new approac and mini-element. 16
16 44 48 T 100N vertical tip displacement at T 7 6 5 4 3 standard mini new mixed 10 1 10 2 number of elements per edge Figure 1: Problem setting and initial mes (left) and vertical tip displacement at T versus number of elements using different formulations (rigt) l f L Figure 2: Te rectangular beam wit initial mes and problem setting 10 1 L 2 error 10 2 H 1 error 10 1 10 3 standard mini new mixed O( 2 ) 10 2 standard mini new mixed O() 10 2 10 3 10 4 number of elements 10 2 10 3 10 4 number of elements Figure 3: Error plot versus number of elements, L 2 -norm (left) and H 1 -norm (rigt), rectangular beam Acknowledgement We are grateful to te anonymous referees for teir valuable suggestions to improve te quality of te earlier version of tis work. 17
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