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

TL;DR

The paper develops a frequency-explicit residual-based a posteriori error estimator for discontinuous Galerkin discretizations of time-harmonic Maxwell equations in first-order form. It proves reliability and efficiency with constants that remain controlled as frequency varies on refined meshes, and shows asymptotic constant-free behavior for smooth solutions, enabling robust h- and hp-adaptivity. The estimator accommodates a range of DG fluxes via the flux form and TF-SF interface, and is supported by a rigorous Aubin–Nitsche–type analysis and regular decomposition, along with comprehensive 2D numerical experiments. These results advance adaptive Maxwell simulations by providing a theoretically sound, practical estimator that drives efficient refinement across frequencies and complex geometries. The work has direct implications for high-frequency electromagnetics, providing a principled tool to balance accuracy and computational cost in DG methods.

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.

Figures (16)

• Figure 1: Planewave example: error and estimator
• Figure 2: Planewave example: efficiencies
• Figure 3: Nearly resonant example: error and estimator
• Figure 4: Nearly resonant example: efficiencies with upwind fluxes
• Figure 5: Nearly resonant experiment: errors in $h$-adaptive refinements

Theorems & Definitions (20)

• Remark 2.1: Hanging nodes
• Lemma 4.1: Jump form with gradient arguments
• proof
• Remark 4.2: General jump conditions
• Lemma 4.3: Control of the residual
• proof
• Lemma 4.4: General control of the residual
• proof
• Lemma 4.5: Approximation factor
• proof