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

翻译：时空有限元方法的主要优势在于能够处理复杂几何形状并实现高阶精度。为此，我们旨在建立坚实的理论基础，以构建合适的时空方法。本文遵循时空方法的核心思想，将时间视为另一空间维度。首先，我们简要讨论如何在时空结构下从麦克斯韦方程组中推导出含欧姆定律的矢量波动方程。随后，通过采用不同的试验空间与检验空间，推导出矢量波动方程的时空变分形式。本文有两个主要目标：第一，证明所得Galerkin-Petrov变分形式解的唯一存在性；第二，分析该方程在张量积结构下的离散等价形式，并证明其条件稳定性（即CFL条件）。深入理解矢量波动方程及其对应的时空有限元方法，对于完善麦克斯韦方程组的现有理论至关重要，并为解决更复杂的电磁问题计算铺平道路。