A mixed finite element method for nearly incompressible elasticity and Stokes equations using primal and dual meshes with quadrilateral and hexahedral grids

TL;DR

This paper introduces a mixed finite element method utilizing primal and dual meshes with quadrilateral and hexahedral grids to effectively approximate solutions for nearly incompressible elasticity and Stokes equations.

Contribution

It presents a novel finite element approach combining primal and dual meshes with bubble functions for improved accuracy in nearly incompressible problems.

Findings

Achieves stable and accurate approximations for elasticity and Stokes equations.

Effectively handles nearly incompressible materials with mixed finite element spaces.

Demonstrates convergence properties through numerical experiments.

Abstract

We consider a mixed finite element method for approximating the solution of nearly incompressible elasticity and Stokes equations. The finite element method is based on quadrilateral and hexahedral triangulation using primal and dual meshes. We use the standard bilinear and trilinear finite element space enriched with element-wise defined bubble functions with respect to the primal mesh for the displacement or velocity, whereas the pressure space is discretised by using a piecewise constant finite element space with respect to the dual mesh.