Wavenumber-Explicit hp-FEM Analysis for Maxwell’s Equations with Impedance Boundary Conditions

Abstract

The time-harmonic Maxwell equations at high wavenumber k in domains with an analytic boundary and impedance boundary conditions are considered. A wavenumber-explicit stability and regularity theory is developed that decomposes the solution into a part with finite Sobolev regularity that is controlled uniformly in k and an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nédélec elements of order p on a mesh with mesh size h is shown under the k-explicit scale resolution condition that (a) kh/p is sufficient small and (b) \(p/\ln k\) is bounded from below.