A mixed finite element for weakly-symmetric elasticity

TL;DR

This paper introduces a novel finite element discretization for weakly symmetric linear elasticity equations on tetrahedral meshes, ensuring stability and optimal approximation, and enabling a compact finite volume scheme for displacement.

Contribution

It develops a stable finite element method combining specific polynomial spaces for displacement, stress, and multipliers, with an innovative approach to eliminate variables for efficiency.

Findings

Proves stability and optimal approximation properties of the new element.

Enables variable elimination to produce a cell-centered finite volume scheme.

Applicable to tetrahedral meshes with polynomial degree r .

Abstract

We develop a finite element discretization for the weakly symmetric equations of linear elasticity on tetrahedral meshes. The finite element combines, for $r \geq 0$, discontinuous polynomials of $r$ for the displacement, $H(\mathrm{div})$-conforming polynomials of order $r+1$ for the stress, and $H(\mathrm{curl})$-conforming polynomials of order $r+1$ for the vector representation of the multiplier. We prove that this triplet is stable and has optimal approximation properties. The lowest order case can be combined with inexact quadrature to eliminate the stress and multiplier variables, leaving a compact cell-centered finite volume scheme for the displacement.