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.

翻译：时间依赖的Maxwell方程组是电磁学的基础控制方程。在特定条件下，可将该方程组改写为二阶偏微分方程，即矢量波动方程。本文针对矢量波动方程，研究其数值实现方法及应用挑战。为此，我们采用时空变分框架——即将时间视为另一空间维度。具体而言，通过在时间和空间方向分别进行分部积分，构建了具有不同试探空间与检验空间的时空变分形式。采用张量积型协调离散化后，得到需要满足CFL条件以保证稳定性的Galerkin-Petrov有限元方法。针对该Galerkin-Petrov变分形式，我们研究了CFL条件及其最优性。为突破CFL条件限制，引入Hilbert型变换，构建了具有相同试探与检验空间的变分形式。采用协调时空离散化后，得到无条件稳定的新型Galerkin-Bubnov有限元方法。数值算例验证了该Galerkin-Bubnov有限元方法的有效性。此外，我们进一步研究了右端项不同投影方式对收敛阶的影响。本文是建立协调时空框架下矢量波动方程稳定计算方法与深化理论认知的初始探索。