  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.
