New H(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

Philippe R B Devloo (1) , Agnaldo M Farias (2) , Sonia M. Gomes (3) , Weslley S Pereira (4) , Antonio J B dos Santos (5) , Frédéric Valentin (6)

1 UNICAMP - Universidade Estadual de Campinas = University of Campinas
2 IFNMG - Instituto Federal do Norte de Minas Gerais
3 IMECC - Instituto de Matemática, Estatística e Computação Científica [Brésil]
4 UFJF - Universidade Federal de Juiz de Fora
5 UFPB - Universidade Federal da Paraiba / Federal University of Paraiba
6 LNCC / MCT - Laboratorio Nacional de Computação Cientifica [Rio de Janeiro]

This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new methods approximate displacement, stress, and rotation variables using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontinuous polynomial spaces, and of computing rigid-body modes element wisely. In the second scale level, the methods are made effective by solving completely independent local Newmann elasticity problems written in a mixed form with weak symmetry enforced via a rotation multiplier. Since the finite-dimensional space for the traction variable constrains the discrete space for the local stress tensor, the discrete stress field lies in the H(div) space globally and stays in local equilibrium with external forces. We propose different choices to approximate local problems based on locally stable pairs of finite elements defined on affine second-level meshes. Those choices generate the family of multiscale finite element methods for which stability and convergence are proved in a unified framework. Notably, we prove that the methods are optimal and high-order convergent in the natural norms. Also, it emerges that the approximate displacement and stress divergence are super-convergent in the L2-norm. Numerical verification assess theoretical results and highlight the high precision of the new methods on coarse meshes for multilayered heterogeneous material problems.

Mots clés: Multiscale; Mixed finite elements; Linear elasticity; Hybridization

Domaines: Analyse numérique [math.NA]

Philippe R B Devloo, Agnaldo M Farias, Sonia M. Gomes, Weslley S Pereira, Antonio J B dos Santos, et al.. New H(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes. 2019. ⟨hal-02415020v1⟩