Frequency-explicit approximability estimates for time-harmonic Maxwell's equations

TL;DR

This paper derives frequency-explicit approximability estimates for time-harmonic Maxwell's equations in heterogeneous media, facilitating improved error analysis for high-order finite element methods.

Contribution

It introduces a novel regularity splitting approach that generalizes scalar Helmholtz results to Maxwell's equations, enabling sharp approximability estimates.

Findings

Provides frequency-explicit error bounds for Maxwell's equations

Demonstrates the effectiveness of high-order Nédélec finite elements

Extends scalar Helmholtz approximation results to vector Maxwell problems

Abstract

We consider time-harmonic Maxwell's equations set in an heterogeneous medium with perfectly conducting boundary conditions. Given a divergence-free right-hand side lying in $L^2$, we provide a frequency-explicit approximability estimate measuring the difference between the corresponding solution and its best approximation by high-order N\'ed\'elec finite elements. Such an approximability estimate is crucial in both the a priori and a posteriori error analysis of finite element discretizations of Maxwell's equations, but the derivation is not trivial. Indeed, it is hard to take advantage of high-order polynomials given that the right-hand side only exhibits $L^2$ regularity. We proceed in line with previously obtained results for the simpler setting of the scalar Helmholtz equation, and propose a regularity splitting of the solution. In turn, this splitting yields sharp approximability estimates generalizing known results for the scalar Helmoltz equation and showing the interest of high-order methods.