The MHM Method for Elasticity on Polytopal Meshes

Antônio Tadeu A. Gomes (1) , Weslley S. Pereira (1, 2) , Frédéric Valentin (1, 3)

1 LNCC / MCT - Laboratorio Nacional de Computação Cientifica [Rio de Janeiro]
2 UFJF - Universidade Federal de Juiz de Fora
3 CRISAM - Inria Sophia Antipolis - Méditerranée

The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed. Also, discrete traction stays in local equilibrium with external forces. We show by means of a multi-level analysis that the MHM method achieves optimal convergence under local regularity conditions without refining the coarse partition. The upshot is that the Poincar\'e and Korn's inequalities do not degenerate, and then convergence arises on general meshes. We employ two- and three-dimensional numerical tests to assess theoretical results and to verify the robustness of the method through a multi-layer media case. Also, we address computational aspects of the underlying parallel algorithm associated with different configurations of the MHM method; our aim is to find the best compromise between execution time and memory allocation to achieve a given error threshold.

Domaines: Analyse numérique [math.NA]; Analyse numérique [cs.NA]

Antônio Tadeu A. Gomes, Weslley S. Pereira, Frédéric Valentin. The MHM Method for Elasticity on Polytopal Meshes. 2020. ⟨hal-02931170v2⟩