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

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 equation, we examine numerical schemes and their challenges. For this purpose, we consider a space‐time variational setting, that is, 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 which we study. 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 toward a more stable computation and a better understanding of vectorial wave equations in a conforming space‐time approach.