Wavenumber-Explicit <i>hp</i>-FEM Analysis for Maxwell's Equations with Impedance Boundary Conditions.

Abstract

The time-harmonic Maxwell equations at high wavenumber <i>k</i> 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 <i>k</i> and an analytic part. Using this regularity, quasi-optimality of the Galerkin discretization based on Nédélec elements of order <i>p</i> on a mesh with mesh size <i>h</i> is shown under the <i>k</i>-explicit scale resolution condition that (a) <i>kh</i>/<i>p</i> is sufficient small and (b) p / ln k is bounded from below.