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：处理复杂的地貌和进入更高顺序的能力是空间-时间限制元素方法的主要优势。 因此, 我们想要开发一个坚实的背景, 我们可以从中构建适当的空间- 时间方法。 在本文中, 我们将将时间作为另一个空间方向, 这是空间- 时间方法的主要理念。 首先, 我们将简要讨论矢量波方程是如何在空间- 时间结构中从 Maxwell 的方程中产生出来的, 同时考虑到 Ohm 的法律 。 然后我们将利用不同的试验和测试空间为矢量波方程得出一个空间- 时间变异的配方。 本文有两个主要目标。 首先, 我们证明由此产生的Galerkin- Petrov 变异性配方是独一无二的。 其次, 我们分析一个高压产品中的方程的离散等值, 并显示有条件的稳定性, 即 CFL 条件 。 了解矢量波方程和相应的空间- 时限元素方法对于改进Maxwell 方程的现有理论至关重要, 并且为计算更复杂的电磁问题铺路 。