Numerical results for an unconditionally stable space-time finite element method for the wave equation

TL;DR

This paper presents a new unconditionally stable space-time finite element method for the wave equation, using a novel variational formulation and tensor-product discretization, with demonstrated stability and optimal convergence in numerical examples.

Contribution

Introduces a new variational formulation and finite element discretization for the wave equation that achieves unconditional stability without CFL constraints.

Findings

Method is unconditionally stable in numerical tests.

Achieves optimal convergence rates in space-time norms.

Applicable to 1D and 2D spatial domains.

Abstract

In this work, we introduce a new space-time variational formulation of the second-order wave equation, where integration by parts is also applied with respect to the time variable, and a modified Hilbert transformation is used. For this resulting variational setting, ansatz and test spaces are equal. Thus, conforming finite element discretizations lead to Galerkin--Bubnov schemes. We consider a conforming tensor-product approach with piecewise polynomial, continuous basis functions, which results in an unconditionally stable method, i.e., no CFL condition is required. We give numerical examples for a one- and a two-dimensional spatial domain, where the unconditional stability and optimal convergence rates in space-time norms are illustrated.