Mixed finite element approximation of the vector Laplacian with Dirichlet boundary conditions

TL;DR

This paper investigates the performance of mixed finite element methods for the vector Laplace equation with Dirichlet boundary conditions, revealing suboptimal convergence and implications for related PDEs.

Contribution

It demonstrates that mixed finite element methods are not optimal for Dirichlet boundary conditions and analyzes the resulting suboptimal convergence, extending understanding of their limitations.

Findings

Mixed finite elements do not perform optimally with Dirichlet boundary conditions.

Suboptimal convergence occurs in the finite element solutions.

Results impact the use of mixed formulations for biharmonic and Stokes equations.

Abstract

We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite element method, in which the rotation of the solution is introduced as a second unknown, is advantageous, and appropriate choices of mixed finite element spaces lead to a stable, optimally convergent discretization. However, the theory that leads to these conclusions does not apply to the case of Dirichlet boundary conditions, in which both components of the solution vanish on the boundary. We show, by computational example, that indeed such mixed finite elements do not perform optimally in this case, and we analyze the suboptimal convergence that does occur. As we indicate, these results have implications for the solution of the biharmonic equation and of the Stokes equations using a mixed formulation involving the vorticity.