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

TL;DR

This paper analyzes the high-wavenumber Maxwell equations discretized with edge elements, establishing conditions for quasi-optimality of the Galerkin method with transparent boundary conditions.

Contribution

It provides a $k$-explicit analysis of the $hp$-FEM for Maxwell's equations, detailing conditions for quasi-optimality with transparent boundary conditions.

Findings

Quasi-optimality shown under specific $kh/p$ and $p/ ext{log}k$ conditions.

Analysis applies to Maxwell's equations with transparent boundary conditions.

Provides guidelines for mesh and polynomial degree choices at high frequencies.

Abstract

The time-harmonic Maxwell equations at high wavenumber $k$ are discretized by edge elements of degree $p$ on a mesh of width $h$. For the case of a ball and exact, transparent boundary conditions, we show quasi-optimality of the Galerkin method under the $k$-explicit scale resolution condition that a) $kh/p$ is sufficient small and b) $p/\log k$ is bounded from below.