Frequency-explicit a posteriori error estimates for finite element discretizations of Maxwell's equations

TL;DR

This paper develops frequency-explicit a posteriori error estimates for finite element discretizations of Maxwell's equations, showing that these estimates become independent of frequency on sufficiently refined meshes, extending previous results from scalar wave problems.

Contribution

It introduces frequency-explicit reliability and efficiency estimates for Maxwell's equations, demonstrating their independence from frequency on refined meshes, and extends known results from scalar wave equations.

Findings

Estimates are frequency-independent on refined meshes.

Numerical experiments confirm the sharpness of the estimates.

Results apply to various Nédélec and discontinuous Galerkin discretizations.

Abstract

We consider residual-based a posteriori error estimators for Galerkin-type discretizations of time-harmonic Maxwell's equations. We focus on configurations where the frequency is high, or close to a resonance frequency, and derive reliability and efficiency estimates. In contrast to previous related works, our estimates are frequency-explicit. In particular, our key contribution is to show that even if the constants appearing in the reliability and efficiency estimates may blow up on coarse meshes, they become independent of the frequency for sufficiently refined meshes. Such results were previously known for the Helmholtz equation describing scalar wave propagation problems and we show that they naturally extend, at the price of many technicalities in the proofs, to Maxwell's equations. Our mathematical analysis is performed in the 3D case, and covers conforming N\'ed\'elec discretizations of the first and second family, as well as first-order (and hybridizable) discontinuous Galerkin schemes. We also present numerical experiments in the 2D case, where Maxwell's equations are discretized with N\'ed\'elec elements of the first family. These illustrating examples perfectly fit our key theoretical findings, and suggest that our estimates are sharp.