A Finite Element Method to a Periodic Steady-State Problem for an Electromagnetic Field System Using the Space-Time Finite-Element Exterior Calculus

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 \(\mathbb {R}^4\) ${\mathbb{R}}^4$ using the standard de Rham complex. Consequently, we can directly apply the finite-element exterior calculus (FEEC) theory in \(\mathbb {R}^4\) ${\mathbb{R}}^4$ to deduce the well-posedness, stability, and convergence. Numerical examples using the cubical element are reported to validate the theoretical results.