An ultra-weak three-field finite element formulation for the biharmonic and extended Fisher--Kolmogorov equations

TL;DR

This paper introduces an ultra-weak three-field finite element method for solving biharmonic and extended Fisher--Kolmogorov equations, establishing well-posedness and demonstrating numerical effectiveness.

Contribution

It develops a novel three-field formulation with Raviart--Thomas discretisations, providing new theoretical analysis and error estimates for these complex PDEs.

Findings

The formulation is well-posed and stable.

Error estimates are rigorously established.

Numerical examples confirm the method's effectiveness.

Abstract

This paper discusses a so-called ultra-weak three-field formulation of the biharmonic problem where the solution, its gradient, and an additional Lagrange multiplier are the three unknowns. We establish the well-posedness of the problem using the abstract theory for saddle-point problems, and develop a conforming finite element scheme based on Raviart--Thomas discretisations of the two auxiliary variables. The well-posedness of the discrete formulation and the corresponding a priori error estimate are proved using a discrete inf-sup condition. We further extend the analysis to the time-dependent semilinear equation, namely extended Fisher--Kolmogorov equation. We present a few numerical examples to demonstrate the performance of our approach.