no. 2
A hybrid-mixed method for elasticity
Harder, Christopher1 ; Madureira, Alexandre L.2 ; Valentin, Frédéric3
1 Mathematical and Computer Sciences Department, Metropolitan State University of Denver, P.O. Box 173362, Campus Box 38, Denver, CO 80217-3362, USA
2 National Laboratory for Scientific Computing − LNCC, and Fundação Getúlio Vargas − FGV, Brazil
3 National Laboratory for Scientific Computing − LNCC, Av. Getúlio Vargas, 333, 25651-070 Petrópolis − RJ, Brazil
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 311-336.

This work presents a family of stable finite element methods for two- and three-dimensional linear elasticity models. The weak form posed on the skeleton of the partition is a byproduct of the primal hybridization of the elasticity problem. The unknowns are the piecewise rigid body modes and the Lagrange multipliers used to relax the continuity of displacements. They characterize the exact displacement through a direct sum of rigid body modes and solutions to local elasticity problems with Neumann boundary conditions driven by the multipliers. The local problems define basis functions which are in a one-to-one correspondence with the basis of the subspace of Lagrange multipliers used to discretize the problem. Under the assumption that such a basis is available exactly, we prove that the underlying method is well posed, and the stress and the displacement are super-convergent in natural norms driven by (high-order) interpolating multipliers. Also, a local post-processing computation yields strongly symmetric stress which is in local equilibrium and possesses continuous traction on faces. A face-based a posteriori estimator is shown to be locally efficient and reliable with respect to the natural norms of the error. Next, we propose a second level of discretization to approximate the basis functions. A two-level numerical analysis establishes sufficient conditions under which the well-posedness and super-convergent properties of the one-level method is preserved.

Reçu le :
DOI : 10.1051/m2an/2015046
Classification : 65N30, 65N55, 65Y05, 65N12
Mots clés : Elasticity equation, mixed method, hybrid method, finite element, multiscale, Elasticity equation, mixed method, hybrid method, finite element, multiscale
Harder, Christopher 1 ; Madureira, Alexandre L. 2 ; Valentin, Frédéric 3

1 Mathematical and Computer Sciences Department, Metropolitan State University of Denver, P.O. Box 173362, Campus Box 38, Denver, CO 80217-3362, USA
2 National Laboratory for Scientific Computing − LNCC, and Fundação Getúlio Vargas − FGV, Brazil
3 National Laboratory for Scientific Computing − LNCC, Av. Getúlio Vargas, 333, 25651-070 Petrópolis − RJ, Brazil 
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Harder, Christopher; Madureira, Alexandre L.; Valentin, Frédéric. A hybrid-mixed method for elasticity. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 2, pp. 311-336. doi : 10.1051/m2an/2015046. http://archive.numdam.org/articles/10.1051/m2an/2015046/

S. Adams and B. Cockburn, A mixed finite element method for elasticity in three dimensions. J. Sci. Comput. 25 (2005) 515–521. | DOI | MR | Zbl

M. Amara and J.M. Thomas, Equilibrium finite elements for the linear elastic problem. Numer. Math. 33 (1979) 367–383. | DOI | MR | Zbl

R. Araya, C. Harder, D. Paredes and F. Valentin, Multiscale hybrid-mixed method. SIAM J. Numer. Anal. 51 (2013) 3505–3531. | DOI | MR | Zbl

T. Arbogast and K. Boyd, Subgrid upscaling and mixed multiscale finite elements. SIAM J. Numer. Anal. 44 (2006) 1150–1171. | DOI | MR | Zbl

D.N. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions. Math. Comput. 77 (2008) 1229–1251. | DOI | MR | Zbl

D.N. Arnold and R. Winther, Mixed finite elements for elasticity. Numer. Math. 92 (2002) 401–419. | DOI | MR | Zbl

D.N. Arnold, F. Brezzi and J. Douglas, Peers: a new mixed finite element for plane elasticity. Japan J. Appl. Math. 1 (1984) 347–367. | DOI | MR | Zbl

D.N. Arnold, J.J. Douglas and C.P. Gupta, A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45 (1984) 1–22. | DOI | MR | Zbl

D.N. Arnold, G. Awanou and R. Winther, Nonconforming tetrahedral mixed finite elements for elasticity. Math. Models Methods Appl. Sci. 23 (2014) 783–796. | DOI | MR | Zbl

I. Babuska and E. Osborn, Generalized finite element methods: Their performance and their relation to mixed methods. SIAM J. Numer. Anal. 20 (1983) 510–536. | DOI | MR | Zbl

L. Beirão Da Veiga, F. Brezzi and L.D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51 (2013) 794–812. | DOI | MR | Zbl

J. Bramwell, L. Demkowicz, J. Gopalakrishnan and W. Qiu, A locking-free hp dpg method for linear elasticity with symmetric stresses. Numer. Math. 122 (2012) 671–707. | DOI | MR | Zbl

S.C. Brenner, Korn’s inequalities for piecewise H 1 vector fields. Math. Comput. 73 (2004) 1067–1087. | DOI | MR | Zbl

S.C. Brenner and L.R. Scott, The Mathematical Foundations of the Finite Element Methods. Springer (2002). | MR

F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Vol. 15 of Springer Ser. Comput. Math. Springer-Verlag, Berlin, New-York (1991). | MR | Zbl

Z. Chen and T. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comput. 72 (2002) 541–576. | DOI | MR | Zbl

B. Cockburn and K. Chi, Superconvergent hdg methods for linear elasticity with weakly symmetric stresses. IMA J. Numer. Anal. (2012) 1–24. | MR | Zbl

L. Demkowicz and J. Gopalakrishnan, A primal dpg method without a first order reformulation. Comput. Math. Appl. 66 (2013) 1058–1064. | DOI | MR

A. Ern and J.-L. Guermond, Theory and practice of finite elements. Springer-Verlag, Berlin, New-York (2004). | MR | Zbl

J. Gopalakrishnan and J. Guzmán, Symmetric nonconforming mixed finite elements for linear elasticity. SIAM J. Numer. Anal. 49 (2011) 1504–1520. | DOI | MR | Zbl

J. Gopalakrishnan and W. Qiu, An analysis of the practical dpg method. Math. Comput. 83 (2014) 537–552. | DOI | MR | Zbl

J. Guzmán and M. Neilan, Symmetric and conforming mixed finite elements for plane elasticity using rational bubble functions. Numer. Math. 126 (2014) 153–171. | DOI | MR | Zbl

C. Harder, D. Paredes and F. Valentin, A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coefficients. J. Comput. Phys. 245 (2013) 107–130. | DOI | MR | Zbl

C. Harder, D. Paredes and F. Valentin, On a multiscale hybrid-mixed method for advective-reactive dominated problems with heterogenous coefficients. SIAM Multiscale Model. Simul. 13 (2015) 491–518. | DOI | MR | Zbl

T.Y. Hou and X. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. | DOI | MR | Zbl

J. Hu, A new family of efficient conforming mixed finite elements on both rectangular and cuboid meshes for linear elasticity in the symmetric formulation. Preprint [math.NA] (2015). | arXiv | MR

L.E. Payne and H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5 (1960) 286–292. | DOI | MR | Zbl

T. Pian and P. Tong, Basis of finite element methods for solid continua, Int. J. Numer. Methods Engrg. 1 (1969) 3–28. | DOI | Zbl

W. Qiu and K. Shi, An hdg method for linear elasticity with strong symmetric stresses. Preprint [math.NA] (2014). | arXiv

P. Raviart and J. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspect of finite element methods, No. 606 in Lect. Notes Math. Springer-Verlag, New York (1977) 292–315. | MR | Zbl

P. Raviart and J. Thomas, Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comput. 31 (1977) 391–413. | DOI | MR | Zbl

S. Soon, B. Cockburn and H. Stolarski, A hybridizable discontinuous galerkin method for linear elasticity. Int. J. Numer. Methods Engrg. 80 (2009) 1058–1092. | DOI | MR | Zbl

R. Stenberg, On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math. 48 (1986) 447–462. | DOI | MR | Zbl

a. Toselli and O. Widlund, Domain decomposition methods-algorithms and theory. Vol. 34 of Springer Ser. Comput. Math. Springer-Verlag, Berlin (2005). | MR | Zbl

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