Numerical study of conforming space-time methods for Maxwell's equations

TL;DR

This paper explores conforming space-time finite element methods for Maxwell's equations, addressing stability issues and proposing a new unconditionally stable Galerkin--Bubnov method with numerical validation.

Contribution

It introduces a novel conforming space-time Galerkin--Bubnov finite element method that is unconditionally stable for Maxwell's equations, overcoming CFL restrictions.

Findings

The new method is unconditionally stable in numerical experiments.

CFL condition's sharpness is analyzed and addressed.

Different right-hand side projections affect convergence rates.

Abstract

Time-dependent Maxwell's equations govern electromagnetics. Under certain conditions, we can rewrite these equations into a partial differential equation of second order, which in this case is the vectorial wave equation. For the vectorial wave, we investigate the numerical application and the challenges in the implementation. For this purpose, we consider a space-time variational setting, i.e. time is just another spatial dimension. More specifically, we apply integration by parts in time as well as in space, leading to a space-time variational formulation with different trial and test spaces. Conforming discretizations of tensor-product type result in a Galerkin--Petrov finite element method that requires a CFL condition for stability. For this Galerkin--Petrov variational formulation, we study the CFL condition and its sharpness. To overcome the CFL condition, we use a Hilbert-type transformation that leads to a variational formulation with equal trial and test spaces. Conforming space-time discretizations result in a new Galerkin--Bubnov finite element method that is unconditionally stable. In numerical examples, we demonstrate the effectiveness of this Galerkin--Bubnov finite element method. Furthermore, we investigate different projections of the right-hand side and their influence on the convergence rates. This paper is the first step towards a more stable computation and a better understanding of vectorial wave equations in a conforming space-time approach.