A New Minimisation Principle for Poisson Equation Leading to a Flexible Finite Element Approach

TL;DR

This paper presents a novel minimisation principle for the Poisson equation that enables flexible finite element discretizations without inf-sup conditions, demonstrated through numerical examples.

Contribution

It introduces a new variational principle involving both the solution and its gradient, allowing for more flexible finite element choices.

Findings

Finite element spaces can be chosen freely without inf-sup conditions.

Numerical examples show the approach's effectiveness.

The method improves flexibility in finite element discretization.

Abstract

We introduce a new minimisation principle for Poisson equation using two variables: the solution and the gradient of the solution. This principle allows us to use any conforming finite element spaces for both variables, where the finite element spaces do not need to satisfy a so-called inf-sup condition. A numerical example demonstrates the superiority of the approach.