Frequency-explicit a posteriori error estimates for discontinuous Galerkin discretizations of Maxwell's equations

TL;DR

This paper introduces a new residual-based a posteriori error estimator for discontinuous Galerkin methods applied to Maxwell's equations, analyzing its reliability, efficiency, and frequency dependence, supported by numerical examples.

Contribution

It generalizes existing error estimates to Maxwell's equations with discontinuous Galerkin discretizations, including frequency dependence and asymptotic behavior for smooth solutions.

Findings

Estimator is reliable and efficient across frequencies.

Estimator is asymptotically constant-free for smooth solutions.

Numerical examples confirm theoretical properties and adaptive refinement suitability.

Abstract

We propose a new residual-based a posteriori error estimator for discontinuous Galerkin discretizations of time-harmonic Maxwell's equations in first-order form. We establish that the estimator is reliable and efficient, and the dependency of the reliability and efficiency constants on the frequency is analyzed and discussed. The proposed estimates generalize similar results previously obtained for the Helmholtz equation and conforming finite element discretization of Maxwell's equations. In addition, for the discontinuous Galerkin scheme considered here, we also show that the proposed estimator is asymptotically constant-free for smooth solutions. We also present two-dimensional numerical examples that highlight our key theoretical findings and suggest that the proposed estimator is suited to drive $h$- and $hp$-adaptive iterative refinements.