Domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry

TL;DR

This paper introduces domain decomposition and multiscale mortar mixed finite element methods for linear elasticity with weak stress symmetry, providing efficient solutions and error analysis for complex grid configurations.

Contribution

It develops new domain decomposition and mortar finite element methods for linear elasticity with weak stress symmetry, including analysis and computational validation.

Findings

Condition number analysis of interface problems

Optimal convergence for stress, displacement, and rotation

Superconvergence for displacement

Abstract

Two non-overlapping domain decomposition methods are presented for the mixed finite element formulation of linear elasticity with weakly enforced stress symmetry. The methods utilize either displacement or normal stress Lagrange multiplier to impose interface continuity of normal stress or displacement, respectively. By eliminating the interior subdomain variables, the global problem is reduced to an interface problem, which is then solved by an iterative procedure. The condition number of the resulting algebraic interface problem is analyzed for both methods. A multiscale mortar mixed finite element method for the problem of interest on non-matching multiblock grids is also studied. It uses a coarse scale mortar finite element space on the non-matching interfaces to approximate the trace of the displacement and impose weakly the continuity of normal stress. A priori error analysis is performed. It is shown that, with appropriate choice of the mortar space, optimal convergence on the fine scale is obtained for the stress, displacement, and rotation, as well as some superconvergence for the displacement. Computational results are presented in confirmation of the theory of all proposed methods.