Wavenumber-explicit hp-FEM analysis for Maxwell's equations with impedance boundary conditions

TL;DR

This paper develops a wavenumber-explicit analysis for the hp-FEM applied to Maxwell's equations with impedance boundary conditions, providing stability, regularity, and quasi-optimality results that depend explicitly on the wavenumber k.

Contribution

It introduces a novel wavenumber-explicit stability and regularity framework for Maxwell's equations with impedance boundary conditions, enabling precise finite element error analysis.

Findings

Decomposition of solutions into finite regularity and analytic parts.

Establishment of k-explicit stability and regularity estimates.

Proof of quasi-optimality of Nedelec element discretization under specific scale resolution 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 Nedelec 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.