Well-posedness and semi-discretization of Maxwell’s equations with L 2 -data

Abstract

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 (existence, uniqueness and continuous dependence on data) of the first-order weak formulation. An energy identity is also derived. 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 [G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics , volume 219 of Grundlehren der Mathematischen Wissenschaften , Berlin-New York, Springer-Verlag, 1976. Translated from the French by C. W. John, Eqns. (4.31)–(4.32), p. 346; Thm. 4.1]). 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 Nédélec/Raviart–Thomas de Rham complex. The semi-discrete problem is shown to be well-posed. 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. The results at the semi-discrete level are novel.