Mixed finite elements for elasticity on quadrilateral meshes

TL;DR

This paper introduces stable mixed finite element methods for planar linear elasticity on quadrilateral meshes, achieving high convergence rates and including a simple first-order element for practical use.

Contribution

The paper develops a family of stable mixed finite element methods with high convergence rates for elasticity on quadrilateral meshes, and presents a simple first-order element.

Findings

Methods achieve convergence rates of order r in L2 norm for all variables.

A simple first-order element is introduced with first-order convergence.

The methods are stable and applicable to general quadrilateral meshes.

Abstract

We present stable mixed finite elements for planar linear elasticity on general quadrilateral meshes. The symmetry of the stress tensor is imposed weakly and so there are three primary variables, the stress tensor, the displacement vector field, and the scalar rotation. We develop and analyze a stable family of methods, indexed by an integer $r \geq 2$ and with rate of convergence in the $L^2$ norm of order $r$ for all the variables. The methods use Raviart-Thomas elements for the stress, piecewise tensor product polynomials for the displacement, and piecewise polynomials for the rotation. We also present a simple first order element, not belonging to this family. It uses the lowest order BDM elements for the stress, and piecewise constants for the displacement and rotation, and achieves first order convergence for all three variables.