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The ability to deal with complex geometries and to go to higher orders is themain advantage of space-time finite element methods. Therefore, we want todevelop a solid background from which we can construct appropriate space-timemethods. In this paper, we will treat time as another space direction, which isthe main idea of space-time methods. First, we will briefly discuss how exactlythe vectorial wave equation is derived from Maxwell's equations in a space-timestructure, taking into account Ohm's law. Then we will derive a space-timevariational formulation for the vectorial wave equation using different trialand test spaces. This paper has two main goals. First, we prove uniquesolvability for the resulting Galerkin--Petrov variational formulation. Second,we analyze the discrete equivalent of the equation in a tensor product and showconditional stability, i.e. a CFL condition. Understanding the vectorial waveequation and the corresponding space-time finite element methods is crucial forimproving the existing theory of Maxwell's equations and paves the way tocomputations of more complicated electromagnetic problems.

Space-Time FEM for the Vectorial Wave Equation under Consideration of Ohm's Law