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

Julia I. M. Hauser

TL;DR

This work develops a space-time finite element framework for the vectorial wave equation derived from Maxwell's equations with Ohm's law, treating time as an additional spatial dimension to handle complex geometries and higher-order accuracy. It establishes a space-time variational formulation and proves existence and uniqueness of solutions, along with energy-based norm estimates. On the discretization side, a tensor-product FEM using second-order in time and Nédélec/RT spatial elements is proposed, yielding a Kronecker-structured linear system and a CFL-type stability condition, with numerical tests confirming convergence behavior and conditional stability. The approach lays groundwork for more advanced electromagnetic simulations, including eddy-current problems, on unstructured space-time meshes and paves the way for tackling complex materials and geometries in practical applications.

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.

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

TL;DR

This work develops a space-time finite element framework for the vectorial wave equation derived from Maxwell's equations with Ohm's law, treating time as an additional spatial dimension to handle complex geometries and higher-order accuracy. It establishes a space-time variational formulation and proves existence and uniqueness of solutions, along with energy-based norm estimates. On the discretization side, a tensor-product FEM using second-order in time and Nédélec/RT spatial elements is proposed, yielding a Kronecker-structured linear system and a CFL-type stability condition, with numerical tests confirming convergence behavior and conditional stability. The approach lays groundwork for more advanced electromagnetic simulations, including eddy-current problems, on unstructured space-time meshes and paves the way for tackling complex materials and geometries in practical applications.

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.
Paper Structure (20 sections, 12 theorems, 197 equations, 1 figure, 9 tables)

This paper contains 20 sections, 12 theorems, 197 equations, 1 figure, 9 tables.

Table of Contents

  1. Introduction
  2. Derivation of the vectorial wave equation
  3. The vectorial wave equation under consideration of Ohm's law
  4. The space-time Sobolev spaces
  5. Basis representations
  6. The variational formulation
  7. Existence and uniqueness
  8. Existence
  9. Uniqueness
  10. Norm estimate
  11. A space-time finite element method in a tensor product
  12. Space-time finite element spaces
  13. The finite element discretization
  14. Remark
  15. The CFL condition
  16. ...and 5 more sections

Key Result

Lemma 1.1

Let $d=2,3$. The space $H^1(0,T;L^2(\Omega;\mathbb R^d))$ is isometric to the Hilbert tensor product $H^1(0,T)\hat{ {\tensop} {\otimes}}L^2(\Omega;\mathbb R^d)$ and the space $L^2(0,T;H(\operatorname{curl};\Omega))$ is isometric to the Hilbert tensor product $L^2(0,T)\hat{ {\tensop} {\otimes}}H( and $C^\infty([0,T]){ {\tensop} {\otimes}} [C_0^\infty(\Omega)]^d$ is dense in $L^2(0,T)\hat{ {\

Figures (1)

  • Figure 1: The domain $\Omega$ and the support $\Omega_\sigma =\text{supp}(\sigma)$ of the conductivity $\sigma.$

Theorems & Definitions (23)

  • Lemma 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • proof
  • Theorem 2.1
  • Lemma 2.2
  • ...and 13 more