Higher-Order Space-Time Continuous Galerkin Methods for the Wave Equation

TL;DR

This paper extends a stabilization technique for space-time finite element methods solving the wave equation from low-order to higher-order polynomial degrees, achieving unconditional stability and optimal convergence.

Contribution

It generalizes the stabilization approach to higher-order space-time finite element methods for the wave equation, removing the CFL restriction.

Findings

Unconditional stability demonstrated numerically.

Optimal convergence rates achieved in space-time norms.

Extension from lowest-order to arbitrary polynomial degree.

Abstract

We consider a space-time variational formulation of the second-order wave equation, where integration by parts is also applied with respect to the time variable. Conforming tensor-product finite element discretisations with piecewise polynomials of this space-time variational formulation require a CFL condition to ensure stability. To overcome this restriction in the case of piecewise multilinear, continuous ansatz and test functions, a stabilisation is well-known, which leads to an unconditionally stable space-time finite element method. In this work, we generalise this stabilisation idea from the lowest-order case to the higher-order case, i.e. to an arbitrary polynomial degree. We give numerical examples for a one-dimensional spatial domain, where the unconditional stability and optimal convergence rates in space-time norms are illustrated.