A finite element method to a periodic steady-state problem for an electromagnetic field system using the space-time finite element exterior calculus

TL;DR

This paper develops a finite element method based on exterior calculus to efficiently solve periodic steady-state electromagnetic problems modeled by Poisson equations, ensuring well-posedness and convergence.

Contribution

It introduces a novel reformulation of Poisson problems into a single Hodge-Laplace problem in four dimensions using FEEC, enabling direct application of exterior calculus techniques.

Findings

The method guarantees well-posedness and stability.

Numerical examples validate theoretical convergence.

The approach effectively solves periodic electromagnetic problems.

Abstract

This paper proposes a finite element method for solving the periodic steady-state problem for the scalar-valued and vector-valued Poisson equations, a simple reduction model of the Maxwell equations under the Coulomb gauge. Introducing a new potential variable, we reformulate two systems composed of the scalar-valued and vector-valued Poisson problems to a single Hodge-Laplace problem for the 1-form in $\mathbb{R}^4$ using the standard de Rham complex. Consequently, we can directly apply the Finite Element Exterior Calculus (FEEC) theory in $\mathbb{R}^4$ to deduce the well-posedness, stability, and convergence. Numerical examples using the cubical element are reported to validate the theoretical results.