We introduced a new augmented variational formulation for the elasticity problem in the plane that involves four unknowns, namely, the displacement, the stress tensor, the strain tensor of small deformations and the pressure. We proved that this problem is well posed for appropriate values of a stabilization parameter. We also gave sufficient conditions for the well posedness of the corresponding Galerkin scheme, and detailed concrete examples of discrete spaces satisfying these conditions. We provided error
estimates for these cases

Barrios Faúndez, Tomás Patricio

Gatica Pérez, Gabriel Nibaldo

Gatica Simpertigue, Luis Fernando

González Taboada, María

idUS. Depósito de Investigación Universidad de Sevilla

XX Congreso de Ecuaciones Diferenciales y AplicacionesX Congreso de Matema´tica AplicadaSevilla, 24-28 septiembre 2007(pp. 1–8)A new primal-mixed finite elementmethod for the linear elasticity problemT.P. Barrios1, G.N. Gatica2, L.F. Gatica1, M. Gonza´lez31 Fac. de Ingenier´ıa, Universidad Cato´lica de la Sant´ısima Concepcio´n, Casilla 297, Concepcio´n, Chile.E-mail: tomas@ucsc.cl, lgatica@ucsc.cl.2 Dpto. de Ingenier´ıa Matema´tica, Universidad de Concepcio´n, Casilla 160-C, Concepcio´n, Chile.E-mail: ggatica@ing-mat.udec.cl.3 Dpto. de Matema´ticas, Universidad de La Corun˜a, Campus de Elvin˜a s/n, 15071, La Corun˜a, Spain.E-mail: mgtaboad@udc.es.Palabras clave: linear elasticity, primal-mixed finite element methodAbstractWe introduced a new augmented variational formulation for the elasticity problemin the plane that involves four unknowns, namely, the displacement, the stress tensor,the strain tensor of small deformations and the pressure. We proved that this problemis well posed for appropriate values of a stabilization parameter. We also gave sufficientconditions for the well posedness of the corresponding Galerkin scheme, and detailedconcrete examples of discrete spaces satisfying these conditions. We provided errorestimates for these cases.1 IntroductionRecently, J.K. Djoko and B.D. Reddy [8] proposed a new class of mixed formulations forthe linear elasticity problem in the plane. The new formulations are based on the discreteEVSS (Elastic-Viscous-Split-Stress) method (see [9, 10]), and involve the displacement, thestrain and the stress as unknowns. A stabilization term of the form 2αdiv(t−e(u)), wheret denotes the infinitesimal strain, is added, and two different variational formulations arederived. The first one fits the abstract framework of [15, 3], and the second one has thesame structure as the problem considered in [7].Here, we introduce the infinitesimal strain and the pressure as further unknowns,and obtain a four-field variational formulation with a saddle point structure. Using theabstract theory from [5], we show that this problem is well posed. However, the bilinearform a(·, ·) is not coercive on the whole space and thus, it would be difficult to define finite1T.P. Barrios, G.N. Gatica, L.F. Gatica, M. Gonza´lezelement subspaces such that the corresponding Galerkin scheme is well posed. For thisreason, we add a Galerkin least-squares term that renders coercive the bilinear form onthe whole space. The new augmented variational formulation is showed to be well posedfor appropriate values of the stabilization parameter. We also give sufficient conditions forthe well posedness of the corresponding Galerkin scheme. We showed that to approximatethe displacement-pressure pair, we can choose any pair of stable spaces for the Stokesproblem. We consider in particular, the approximation of the pair displacement-pressureusing the mixed finite element introduced in [4] and the mini-element, and provide errorestimates in both cases.2 The augmented variational formulationLet Ω ⊂ R2 be a bounded and simply connected domain with Lipschitz-continuous boun-dary Γ. We consider the following problem: Given a volume force f , determine the dis-placement vector field u and the symmetric stress tensor field σ of a linear elastic materialoccupying the region Ω, and thus satisfying the equationsσ = C e(u) in Ω−div(σ) = f in Ωu = 0 on Γ(1)We denote by e(u) := 12 (∇u+ (∇u)t) the strain tensor of small deformations, and by Cthe elasticity tensor determined by Hooke’s law, that is,C ζ := λ tr(ζ) I + 2µ ζ ∀ ζ ∈ [L2(Ω)]2×2where I is the identity matrix of R2×2 and λ, µ > 0 are the Lame´ parameters.In order to obtain a new variational formulation of problem (1), we introduce twoauxiliary unknowns in Ω, the strain tensor of small deformations t := e(u) and the pressurep := λ tr(t). Then, the constitutive law can be written as followsσ = C t = p I + 2µ t in Ω (2)Testing equation (2) with a function s ∈ [L2(Ω)]2×2sym , we obtain that−∫Ωσ : s+∫Ωp tr(s) + 2µ∫Ωt : s = 0 ∀ s ∈ [L2(Ω)]2×2sym (3)On the other hand, testing the equilibrium equation with a function v ∈ [H10 (Ω)]2, inte-grating by parts in Ω, and using the symmetry of σ and the boundary condition, we havethat ∫Ωσ : e(v) =∫Ωf · v ∀v ∈ [H10 (Ω)]2 (4)From now on, we assume that f ∈ [L2(Ω)]2.Now we remark that in case of pure homogeneous Dirichlet boundary conditions,∫Ωp = λ∫Ωdiv(u) = λ∫Γu · n = 02New primal-mixed FEM for the linear elasticity problemwhere n is the unit outward normal to Γ. Therefore, we look for p ∈ L20(Ω) := {q ∈L2(Ω) :∫Ω q = 0}. Testing appropriately the relations defining the auxiliary unknowns tand p, we obtain that∫Ωe(u) : τ −∫Ωt : τ = 0 ∀ τ ∈ [L2(Ω)]2×2sym (5)− 1λ∫Ωp q +∫Ωtr(t) q = 0 ∀ q ∈ L20(Ω) (6)Let us denote byX = [L2(Ω)]2×2sym ×[H10 (Ω)]2 and byM = [L2(Ω)]2×2sym ×L20(Ω). Summingup equations (3) and (4), and equations (5) and (6), we arrive at the following variationalformulation of problem (1): Find ((t,u), (σ, p)) ∈ X ×M such thata((t,u), (s,v)) + b((s,v), (σ, p)) =∫Ωf · v ∀ (s,v) ∈ Xb((t,u), (τ , q)) − c((σ, p), (τ , q)) = 0 ∀ (τ , q) ∈M(7)where the bilinear forms a : X ×X → R, b : X ×M → R and c :M ×M → R are definedbya((t,u), (s,v)) = 2µ∫Ωt : s c((τ , q), (ϕ, r)) =1λ∫Ωq rb((s,v), (τ , q)) = −∫Ωτ : s+∫Ωq tr(s) +∫Ωτ : e(v)In the following Theorem, we prove that problem (7) is well posed.Theorem 2.1 Problem (7) has a unique solution ((t,u), (σ, p)) ∈ X ×M . Moreover,there exists a constant C > 0, independent of λ, such that||((t,u), (σ, p))||X×M ≤ C ||f ||[L2(Ω)]2Proof. Problem (7) is a saddle point problem that fits the abstract framework of [5](see Section II.1.2). Therefore, in order to prove that this problem is well posed, witha continuous dependence constant independent of λ, we first remark that all the bilinearforms, a(·, ·), b(·, ·) and c(·, ·), are continuous. Now, let B : X →M ′ be the linear operatorassociated with the bilinear form b(·, ·). It is easy to see thatKer(B) ={(s,v) ∈ X : s = e(v) in Ω; div(v) = 0 in Ω}Then, by virtue of Korn’s inequality, the bilinear form a(·, ·) is coercive on Ker(B). Indeed,given (s,v) ∈ Ker(B), we have thata((s,v), (s,v)) = µ(||s||2[L2(Ω)]2×2 + ||e(v)||2[L2(Ω)]2×2)≥ µmin(1, c2K)||(s,v)||2XOn the other hand, it is clear that the bilinear form c(·, ·) is positive semi-definite andsymmetric.Let us prove that b(·, ·) satisfies an inf-sup condition in X×M . Let (σ, p) ∈M . Then,from Corollary 2.4 in [13], there exists a unique w ∈ [H10 (Ω)]2 and a positive constant c13T.P. Barrios, G.N. Gatica, L.F. Gatica, M. Gonza´lezsuch that div(w) = p in Ω and ||w||[H1(Ω)]2 ≤ c1||p||L2(Ω). Taking (s,v) = (e(w),w) ∈ X,we have thatsup(s,v)∈X(s,v) 6=0b((s,v), (σ, p))||(s,v)||X≥ b((e(w),w), (σ, p))||(e(w),w)||X≥||p||2L2(Ω)√2 ||w||[H1(Ω)]2≥ 1c1√2||p||L2(Ω) (8)On the other hand, taking (s,v) = (−σ,0) ∈ X and applying the Cauchy-Schwarz in-equality, we obtain thatsup(s,v)∈X(s,v) 6=0b((s,v), (σ, p))||(s,v)||X≥||σ||2[L2(Ω)]2×2 −∫Ωp tr(σ)||σ||[L2(Ω)]2×2≥ ||σ||[L2(Ω)]2×2 −√2 ||p||L2(Ω) (9)Finally, we deduce from (8) and (9) thatsup(s,v)∈X(s,v) 6=0b((s,v), (σ, p))||(s,v)||X≥ β ||(σ, p)||Mand therefore, the result follows. In the previous Theorem, we proved that problem (7) is well posed. However, sincethe bilinear form a(·, ·) is not coercive on the whole space X, it will be difficult to findfinite-dimensional subspaces such that the corresponding Galerkin scheme is well posed.For this reason, we enrich the variational formulation (7) with the least-squares termα∫Ω(e(u)− t) : (e(v) + s) = 0 ∀ (s,v) ∈ Xwhere α is a positive parameter. This term seems superfluous at the continuous level, butit will play an important role in the discrete setting. Adding this term to the bilinear forma(·, ·), we obtain the following augmented variational formulation of problem (1): Find((t,u), (σ, p)) ∈ X ×M such thata˜((t,u), (s,v)) + b((s,v), (σ, p)) =∫Ωf · v ∀ (s,v) ∈ Xb((t,u), (τ , q)) − c((σ, p), (τ , q)) = 0 ∀ (τ , q) ∈M(10)where the bilinear form a˜ : X ×X → R is given bya˜((t,u), (s,v)) = a((t,u), (s,v)) + α∫Ω(e(u)− t) : (e(v) + s)and b(·, ·) and c(·, ·) are the bilinear forms defined above.We remark that, by virtue of Korn’s inequality, if 0 < α < 2µ, then the bilinear forma˜(·, ·) is coercive on X. Indeed, given any (s,v) ∈ X, we have thata˜((s,v), (s,v)) = (2µ− α) ||s||2[L2(Ω)]2×2 + α ||e(v)||2[L2(Ω)]2×2≥ min(2µ− α, α c2K) ||(s,v)||2X(11)Now, we are able to prove the following result.4New primal-mixed FEM for the linear elasticity problemTheorem 2.2 For α ∈ (0, 2µ), problem (10) has a unique solution ((t,u), (σ, p)) ∈ X ×M . Moreover, there exists a constant C > 0, independent of λ, such that||((t,u), (σ, p))||X×M ≤ C ||f ||[L2(Ω)]2Proof. Clearly, the bilinear form a˜(·, ·) is continuous. Moreover, from (11), a˜(·, ·) iscoercive on X and, in particular, on Ker(B). Therefore, the result follows from the proofof Theorem 2.1 and the abstract theory in [5]. 3 The augmented primal-mixed finite element methodIn this section, we consider the Galerkin scheme associated to (10) and provide sufficientconditions on the finite-dimensional subspaces that allow us to guarantee that the discreteproblem is well posed. We also define explicit finite element subspaces satisfying theseconditions and give the corresponding error estimates.Let h be a positive parameter and let us consider finite-dimensional subspaces Xth ⊂[L2(Ω)]2×2sym , Xuh ⊂ [H10 (Ω)]2, Mσh ⊂ [L2(Ω)]2×2sym and Mph ⊂ L20(Ω). We define Xh :=Xth × Xuh and Mh := Mσh ×Mph . The Galerkin scheme associated with the augmentedvariational formulation (10) reads: Find ((th,uh), (σh, ph)) ∈ Xh ×Mh such thata˜((th,uh), (sh,vh)) + b((sh,vh), (σh, ph)) =∫Ωf · vh ∀ (sh,vh) ∈ Xhb((th,uh), (τ h, qh)) − c((σh, ph), (τ h, qh)) = 0 ∀ (τ h, qh) ∈Mh(12)We can prove the following result.Theorem 3.1 Assume that α ∈ (0, 2µ) and that the finite element subspaces Xth, Xuh ,Mσh and Mph are such that1. e(Xuh ) ⊂ Xth2. (Xuh ,Mph) is a stable pair for the Stokes problem3. Mσh ⊂ XthThen the discrete problem (12) has a unique solution ((th,uh), (σh, ph)) ∈ Xh ×Mh andthere exists a positive constant C, independent of h and λ, such that||((t,u), (σ, p))− ((th,uh), (σh, ph))||X×M ≤≤ C inf((sh,vh),(τ h,qh))∈Xh×Mh||((t,u), (σ, p))− ((sh,vh), (τ h, qh))||X×M(13)Proof. The properties of the bilinear forms a˜(·, ·) and c(·, ·) are satisfied in any subspacesof X and M , respectively. Therefore, it only remains to prove that the bilinear form b(·, ·)satisfies a discrete inf-sup condition in Xh ×Mh, that is, there exists a positive constantβ˜, independent of h, such thatsup(sh,vh)∈Xh(sh,vh) 6=0b((sh,vh), (σh, ph))||(sh,vh)||X≥ β˜ ||(σh, ph)||M ∀ (σh, ph) ∈Mh (14)5T.P. Barrios, G.N. Gatica, L.F. Gatica, M. Gonza´lezLet (σh, ph) ∈Mh. Since we assume that e(Xuh ) ⊂ Xth, we can take (sh,vh) = (e(vh),vh) ∈Xh. Then, we have thatsup(sh,vh)∈Xh(sh,vh) 6=0b((sh,vh), (σh, ph))||(sh,vh)||X≥ supvh∈Xuhvh 6=0b((e(vh),vh), (σh, ph))||(e(vh),vh)||X≥ 1√2supvh∈Xuhvh 6=0∫Ωphdiv(vh)||vh||[H1(Ω)]2Now, using that (Xuh ,Mph) is a stable pair for the Stokes problem, there exists a positiveconstant γ, independent of h, such thatsupwh∈Xuhwh 6=0∫Ωqhdiv(wh)||wh||[H1(Ω)]2≥ γ||qh||L2(Ω) ∀ qh ∈MphTherefore, we deduce thatsup(sh,vh)∈Xh(sh,vh) 6=0b((sh,vh), (σh, ph))||(sh,vh)||X≥ γ√2||ph||L2(Ω) (15)On the other hand, since Mσh ⊂ Xth, we can take (sh,vh) = (−σh,0) ∈ Xh. Then,applying the Cauchy-Schwarz inequality, we obtain thatsup(sh,vh)∈Xh(sh,vh) 6=0b((sh,vh), (σh, ph))||(sh,vh)||X≥ ||σh||[L2(Ω)]2×2 −√2 ||ph||L2(Ω) (16)Thus, inequality (14) follows from inequalities (15) and (16). Next, we define explicit finite element subspaces satisfying the assumptions of Theo-rem 3.1. In what follows, we let {Th}h>0 be a regular family of triangulations of Ω¯ andassume that Ω¯ = ∪T∈Th T . We denote by hT the diameter of a triangle T ∈ Th and definethe mesh size h := maxT∈ThhT . Given a triangle T ∈ Th, we denote by a1, a2 and a3 itsvertices, and by si the side opposite ai, for i = 1, 2, 3. We also let ni be the unit outwardnormal to si. Finally, λ1, λ2 and λ3 denote the barycentric coordinates of triangle T . Inaddition, given an integer k ≥ 0, we denote by Pk(T ) the space of polynomials in twovariables defined in T of total degree at most k.3.1 An approximation using discontinuous pressuresWe first consider the pair (Xuh ,Mph) introduced by Bernardi and Raugel (see [4] or [13])to solve the Stokes problem. Then, we defineXuh = {v ∈ [H10 (Ω) ∩ C0(Ω¯)]2 : v|T ∈ [P1(T )]2 ⊕ 〈n1λ2λ3,n2λ3λ1,n3λ1λ2〉, ∀T ∈ Th }andMph = {q ∈ L20(Ω) : q|T ∈ P0(T ), ∀T ∈ Th}6New primal-mixed FEM for the linear elasticity problemIn order to approximate the symmetric stress tensor field σ, we define Mσh as thespace of piecewise-constant symmetric tensors:Mσh :={τ ∈ [L2(Ω)]2×2 : τ |T ∈ S0 ∀T ∈ Th}whereS0 :=〈(1 00 0),(0 11 0),(0 00 1)〉Finally, to satisfy the assumptions of Theorem 3.1, we define the finite element subspaceXth as follows:Xth :={s ∈ [L2(Ω)]2×2 : s|T ∈ S0 ⊕〈e(n1λ2λ3), e(n2λ3λ1), e(n3λ1λ2)〉, ∀T ∈ Th}Using the approximation properties of the subspaces involved (see [6, 13]), we deducefrom (13) the following result.Theorem 3.2 Assume that u ∈ [H10 (Ω)]2 ∩ [H2(Ω)]2, σ ∈ [L2(Ω)]2×2sym ∩ [H1(Ω)]2×2, t ∈[L2(Ω)]2×2sym ∩ [H1(Ω)]2×2 and that p ∈ L20(Ω)∩H1(Ω). Then there exists a constant C > 0,independent of h and λ, such that||((t,u), (σ, p))− ((th,uh), (σh, ph))||X×M ≤≤ C h(||u||[H2(Ω)]2 + ||σ||[H1(Ω)]2×2 + ||t||[H1(Ω)]2×2 + ||p||H1(Ω))3.2 An approximation using continuous pressuresNow we consider the so-called mini-element, introduced by Arnold et al. (see [2, 13]). Inthis case,Xuh = {v ∈ [H10 (Ω) ∩ C0(Ω¯)]2 : v|T ∈ [P1(T )⊕ 〈λ1λ2λ3〉]2, ∀T ∈ Th }andMph = {q ∈ L20(Ω) ∩ C0(Ω¯) : q|T ∈ P1(T ), ∀T ∈ Th}To approximate the symmetric stress tensor field σ, we consider the finite elementspace Mσh defined in the previous subsection. Then, in order to satisfy the assumptionsof Theorem 3.1, we define the finite element subspace Xth as follows:Xth :={s ∈ [L2(Ω)]2×2 : s|T ∈ S0 ⊕〈e(λ1λ2λ30), e(0λ1λ2λ3)〉, ∀T ∈ Th}Using the approximation properties of the subspaces involved (see [6, 13]), we deducefrom (13) the following result, analogous to Theorem 3.2.Theorem 3.3 Assume that u ∈ [H10 (Ω)]2 ∩ [H2(Ω)]2, σ ∈ [L2(Ω)]2×2sym ∩ [H1(Ω)]2×2, t ∈[L2(Ω)]2×2sym ∩ [H1(Ω)]2×2 and that p ∈ L20(Ω)∩H1(Ω). Then there exists a constant C > 0,independent of h and λ, such that||((t,u), (σ, p))− ((th,uh), (σh, ph))||X×M ≤≤ C h(||u||[H2(Ω)]2 + ||σ||[H1(Ω)]2×2 + ||t||[H1(Ω)]2×2 + ||p||H1(Ω))7T.P. Barrios, G.N. Gatica, L.F. Gatica, M. Gonza´lezAcknowledgmentsThis research was partially supported by MEC (project MTM2004-05796-C02-01), Xuntade Galicia (project PGIDIT05PXIC30302PN and Bolsas para realizar estad´ıas no estran-xeiro) and Universidad de Concepcio´n.References[1] D.N. Arnold, F. Brezzi and J. Douglas, PEERS: A new mixed finite element for plane elasticity,Japan J. Appl. Math., 1, 347-367, (1984).[2] D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for Stokes equations, Calcolo, vol.XXI, 337-344, (1984).[3] C. Bernardi, C. Canuto and Y. Maday, Generalized inf-sup conditions for Chebyshev spectral approx-imation of the Stokes problem, SIAM J. Numer. Anal. vol 25, 6, pp. 1237-1271, (1988).[4] C. Bernardi and G. Raugel, Analysis of some finite elements for the Stokes problem, Math. Comp.44, (1985), pp. 71-79.[5] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991.[6] P.G. Ciarlet, The Finite Element Methods for Elliptic Problems, North-Holland, Amsterdam, 1978.[7] P. Ciarlet, J. Huang and J. Zou, Some observations on generalized saddle-point problems, SIAM J.Matrix Anal. Appl. vol. 25, 1, pp. 224-236, (2003).[8] J.K. Djoko and B.D. Reddy, An extended Hu-Washizu formulation for elasticity. Comput. MethodsApply. Mech. Engrg. vol. 195, 39&40, pp. 6330-6346, (2006).[9] M. Fortin, R. Guenette and R. Pierre, Numerical analysis of the modified EVSS method, Comput.Methods Appl. Mech. Engrg. 143 (1997) 79-95.[10] A. Fortin, R. Guenette and R. Pierre, On the discrete EVSS method, Comput. Methods Appl. Mech.Engrg. 189 (2000) 121-139.[11] G.N. Gatica, Analysis of a new augmented mixed finite element method for linear elasticity allowingRT0-P1-P0 approximations, ESAIM: M2AN vol. 40, 1, pp. 1-28 (2006).[12] T.P. Barrios, G.N. Gatica, M. Gonza´lez and N. Heuer, A residual based a posteriori error estimatorfor an augmented mixed finite element method in linear elasticity, ESAIM: M2AN vol. 40, 5, pp.843-869 (2006).[13] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory andAlgorithms, Springer Series in Computational Mathematics 5, Springer-Verlag, Berlin, 1986.[14] R.B. Kellogg and B. Liu, A finite element method for the compressible Stokes equations, SIAM J.Numer. Anal. vol. 33, 2, pp. 780-788, (1996).[15] R.A. Nicolaides, Existence uniqueness and approximation for generalized saddle point problems,SIAM J. Numer. Anal. vol 19, 2, pp. 349-357, (1982).8

A new primal-mixed finite element method for the linear elasticity problem

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A new primal-mixed finite element method for the linear elasticity problem

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