Space-time finite element methods for nonlinear wave equations via elliptic regularisation

TL;DR

This paper introduces a new conforming space-time Galerkin method for semi-linear wave equations, based on elliptic regularisation, with proven stability, error bounds, and validated by numerical experiments.

Contribution

It develops a novel variational formulation and discretisation approach for nonlinear wave equations using elliptic regularisation, ensuring stability and accuracy.

Findings

Method is well-posed and unconditionally stable.

A priori error bounds are established.

Numerical experiments confirm theoretical results.

Abstract

We present and analyse a new conforming space-time Galerkin discretisation of a semi-linear wave equation, based on a variational formulation derived from De Giorgi's elliptic regularisation viewpoint of the wave equation in second-order formulation. The method is shown to be well-posed through a minimisation approach, and also unconditionally stable for all choices of conforming discretisation spaces. Further, a priori error bounds are proven for sufficiently smooth solutions. Special attention is given to the conditioning of the method and its stable implementation. Numerical experiments are provided to validate the theoretical findings.