On the necessity of the inf-sup condition for a mixed finite element formulation

TL;DR

This paper investigates a dual mixed finite element formulation for the Poisson problem, highlighting the importance of the inf-sup condition for stability and error estimates.

Contribution

It demonstrates that even without uniform inf-sup stability, existence, uniqueness, and optimal error estimates can be achieved under certain conditions.

Findings

Finite element scheme is not uniformly inf-sup stable.

Existence and uniqueness of solutions are established.

Optimal error estimates are derived for the gradient variable.

Abstract

We study a non standard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in H(div) for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some light on the sources of instability for mixed formulations.