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:取决于时间的 Maxwell 方程式管理电磁。 在某些条件下, 我们可以重写这些方程式, 将其转换成一个部分差异的第二顺序方程式, 在本案中, 这是矢量波方程式 。 对于矢量波, 我们调查数字应用和执行中的挑战 。 为此, 我们考虑一个时- 时间变异设置, 即时间只是另一个空间层面 。 更具体地说, 我们应用时- 时间和空间的整合, 导致与不同的试验和测试空间形成时- 时间变异配制 。 将慢产品类型分解成一个Galerkin- Petrov 限值元素方法, 从而形成一个需要 CFL 条件来稳定。 对于这个 Galerkin- Petrov 的变异配方程式, 我们研究CFL 条件及其敏锐度。 为了克服 CFLL 条件, 我们使用希尔伯特型变换式转换, 导致与同等试验和测试空间空间的变换配制配制。 将时- Butnov 的分解导致新的 Galerkin- Bruft 元素的新的 Gal 方法, 我们用一个更稳定、 的平平平平平平时- 的平平平平平平平平平的平的平平平平的平的平平的平的平比的平比的平平平平平的平的平平的平的平的平平的平的平的平的平的平的平比的平的平的平的平的平的平的平的平的平比的平的平比。</s>