Well-Posedness and Approximation of Weak Solutions to Time Dependent Maxwell's Equations with $L^2$-Data

TL;DR

This paper proves well-posedness of weak solutions to time-dependent Maxwell's equations with rough data and material coefficients, and introduces a structure-preserving finite element method with proven convergence.

Contribution

It provides a direct proof of well-posedness for Maxwell's equations with $L^2$-data and develops a finite element scheme that preserves key physical properties and converges to the weak solution.

Findings

Well-posedness of weak solutions established.

Finite element scheme preserves Gauss law and energy identity.

Convergence of the scheme to the weak solution proven.

Abstract

We study Maxwell's equations in conducting media with perfectly conducting boundary conditions on Lipschitz domains, allowing rough material coefficients and $L^2$-data. Our first contribution is a direct proof of well-posedness of the first-order weak formulation, including solution existence and uniqueness, an energy identity, and continuous dependence on the data. The argument uses interior-in-time mollification to show uniqueness while avoiding reflection techniques. Existence is via the well-known Galerkin method (cf.~Duvaut and Lions \cite[Eqns.~(4.31)--(4.32), p.~346; Thm.~4.1]{GDuvaut_JLLions_1976a}). For completeness, and to make the paper self-contained, a complete proof has been provided. Our second contribution is a structure-preserving semi-discrete finite element method based on the N\'ed\'elec/Raviart--Thomas de Rham complex. The scheme preserves a discrete Gauss law for all times and satisfies a continuous-in-time energy identity with stability for nonnegative conductivity. With a divergence-free initialization of the magnetic field (via potential reconstruction or constrained $L^2$ projection), we prove convergence of the semi-discrete solutions to the unique weak solution as the mesh is refined. The analysis mostly relies on projector consistency, weak-* compactness in time-bounded $L^2$ spaces, and identification of time derivatives in dual spaces.