Wavenumber-explicit $hp$-FEM analysis of Maxwell's equations with impedance boundary conditions in piecewise smooth media

TL;DR

This paper analyzes the $hp$-finite element method for Maxwell's equations with impedance boundary conditions in piecewise smooth media, establishing conditions for quasi-optimal discretization based on wavenumber and mesh parameters.

Contribution

It provides a wavenumber-explicit analysis of the $hp$-FEM for Maxwell's equations, including scale resolution conditions ensuring quasi-optimality.

Findings

Quasi-optimality holds under specific wavenumber-dependent mesh and polynomial degree conditions.

The analysis applies to media with analytic properties and jumps across interfaces.

Provides guidelines for mesh refinement and polynomial degree choice in high-frequency regimes.

Abstract

We consider the time-harmonic Maxwell equations with impedance boundary conditions on a bounded Lipschitz domain $\Omega$ with analytic boundary $\Gamma$. We suppose that $\Omega$ consists of multiple subdomains, and that the permeability and permittivity tensors are analytic on every subdomain, but may jump across subdomain interfaces. Under these conditions we show that for any wavenumber $k\in\mathbb{C}$ with $|k|\geq 1$ for which Maxwell's equations are polynomially well-posed, a Galerkin discretization based on N\'{e}d\'{e}lec elements of order $p$ on a mesh with mesh width $h$ is quasi-optimal, provided that there holds the wavenumber-explicit scale resolution condition a) that $|k|h/p$ is sufficiently small and b) that $p/\log |k|$ is bounded from below.