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/

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