Spectral Correctness of the Simplicial Discontinuous Galerkin Approximation of the First-Order Form of Maxwell’s Equations with Discontinuous Coefficients

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 661-684, April 2025.
Abstract. The paper analyzes the discontinuous Galerkin approximation of Maxwell’s equations written in first-order form and with nonhomogeneous magnetic permeability and electric permittivity. Although the Sobolev smoothness index of the solution may be smaller than [math], it is shown that the approximation converges strongly and is therefore spectrally correct. The convergence proof uses the notion of involution and is based on a deflated inf-sup condition and a duality argument. One essential idea is that the smoothness index of the dual solution is always larger than [math] irrespective of the regularity of the material properties.