Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations

TL;DR

This paper establishes sharp, wavenumber-explicit error bounds for edge-element discretizations of high-frequency Maxwell equations, including scattering problems, providing the first preasymptotic error bounds for these methods.

Contribution

It provides the first preasymptotic error bounds for Nédélec edge-element discretizations of Maxwell equations, with explicit dependence on the wavenumber and applicability to complex scattering scenarios.

Findings

First preasymptotic error bounds for Nédélec elements in Maxwell problems.

Wavenumber-explicit error estimates for high-frequency Maxwell discretizations.

Results applicable to scattering problems with PEC obstacles and variable coefficients.

Abstract

We prove sharp wavenumber-explicit error bounds for first- or second-family-N\'ed\'elec-element (a.k.a. edge-element) conforming discretisations, of arbitrary (fixed) order, of the variable-coefficient time-harmonic Maxwell equations posed in a bounded domain with perfect electric conductor (PEC) boundary conditions. The PDE coefficients are allowed to be piecewise regular and complex-valued; this set-up therefore includes scattering from a PEC obstacle and/or variable real-valued coefficients, with the radiation condition approximated by a perfectly matched layer (PML). In the analysis of the $h$-version of the finite-element method, with fixed polynomial degree $p$, applied to the time-harmonic Maxwell equations, the $\textit{asymptotic regime}$ is when the meshwidth, $h$, is small enough (in a wavenumber-dependent way) that the Galerkin solution is quasioptimal independently of the wavenumber, while the $\textit{preasymptotic regime}$ is the complement of the asymptotic regime. The results of this paper are the first preasymptotic error bounds for the time-harmonic Maxwell equations using first-family N\'ed\'elec elements or higher-than-lowest-order second-family N\'ed\'elec elements. Furthermore, they are the first wavenumber-explicit results, even in the asymptotic regime, for Maxwell scattering problems with a non-empty scatterer.