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

¹ Faculdade de Engenharia Civil, Arquitetura e Urbanismo, Universidade Estadual de Campinas, Rua Josiah Willard Gibbs, 85, Campinas, São Paulo 13083-839, Brazil
² Instituto Federal do Norte de Minas Gerais, Fazenda Varginha, km 02 Rodovia MG-404, Salinas, Minas Gerais 39560-000, Brazil
³ Instituto de Matemática, Estatística e Computação Científica, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, Campinas, São Paulo 13083-859, Brazil
⁴ Laboratório Nacional de Computação Científica, Petrópolis, Rio de Janeiro 25651-075, Brazil
⁵ Centro de Informática, Universidade Federal da Paraíba, R. dos Escoteiros, s/n – Mangabeira, João Pessoa – PB, João Pessoa, Paraíba 58055-000, Brazil

^* Corresponding author: soniag@unicamp.br

Received: 27 August 2020
Accepted: 11 March 2021

Abstract

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 using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontinuous polynomial spaces, and of computing elementwise rigid body modes. In the second level, the methods are made effective by solving completely independent local boundary Neumann elasticity problems written in a mixed form with weak symmetry enforced via the rotation multiplier. Since the finite-dimensional space for the traction variable constraints the local stress approximations, 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 pairs of finite element spaces 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 verifications assess theoretical results and highlight the high precision of the new methods on coarse meshes for multilayered heterogeneous material problems.