Frequency-Explicit A Posteriori Error Estimates for Finite Element Discretizations of Maxwell's Equations
Théophile Chaumont-Frelet, Patrick Vega
Abstract
We consider residual-based a posteriori error estimators for Galerkin 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 three-dimensional case and covers conforming Nédélec discretizations of the first and second families. We also present numerical experiments in the two-dimensional case, where Maxwell's equations are discretized with Nédélec elements of the first family. These illustrating examples perfectly fit our key theoretical findings and suggest that our estimates are sharp.