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

TL;DR

This paper develops a space-time finite element method for the vectorial wave equation derived from Maxwell's equations, incorporating Ohm's law, and analyzes its solvability and stability for advanced electromagnetic simulations.

Contribution

It introduces a novel space-time variational formulation for the vectorial wave equation with proof of unique solvability and conditional stability analysis.

Findings

Proves unique solvability of the Galerkin--Petrov formulation

Establishes a CFL condition for stability

Provides a foundation for complex electromagnetic computations

Abstract

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.