A multipoint stress mixed finite element method for elasticity on simplicial grids

TL;DR

This paper introduces a novel multipoint stress mixed finite element method for linear elasticity on simplicial grids, enabling local stress elimination and achieving optimal convergence rates.

Contribution

It develops two variants of the method with local stress and rotation elimination, extending multipoint finite element techniques to elasticity problems.

Findings

First-order convergence for stress, displacement, and rotation.

Second-order superconvergence of displacement at cell centers.

Numerical results confirm theoretical error estimates.

Abstract

We develop a new multipoint stress mixed finite element method for linear elasticity with weakly enforced stress symmetry on simplicial grids. Motivated by the multipoint flux mixed finite element method for Darcy flow, the method utilizes the lowest order Brezzi-Douglas-Marini finite element spaces for the stress and the vertex quadrature rule in order to localize the interaction of degrees of freedom. This allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation and results in a cell-centered system for displacement and rotation. The second uses a piecewise linear rotation and a quadrature rule for the asymmetry bilinear form. This allows for further elimination of the rotation, resulting in a cell-centered system for the displacement only. Stability and error analysis is performed for both variants. First-order convergence is established for all variables in their natural norms. A duality argument is further employed to prove second order superconvergence of the displacement at the cell centers. Numerical results are presented in confirmation of the theory.