A priori and a posteriori analysis of the discontinuous Galerkin approximation of the time-harmonic Maxwell's equations under minimal regularity assumptions

TL;DR

This paper provides minimal-regularity error estimates for a discontinuous Galerkin method solving time-harmonic Maxwell's equations, demonstrating asymptotic optimality and stability independent of frequency for fine meshes.

Contribution

It introduces new a priori and a posteriori error estimates for the interior penalty dG method under minimal regularity assumptions, with frequency-independent constants.

Findings

dG solution is asymptotically optimal in an augmented energy norm

Error estimates are valid under minimal regularity assumptions

Stability constants are comparable to the continuous problem for fine meshes

Abstract

We derive a priori and a posteriori error estimates for the discontinuous Galerkin (dG) approximation of the time-harmonic Maxwell's equations. Specifically, we consider an interior penalty dG method, and establish error estimates that are valid under minimal regularity assumptions and involving constants that do not depend on the frequency for sufficiently fine meshes. The key result of our a priori error analysis is that the dG solution is asymptotically optimal in an augmented energy norm that contains the dG stabilization. Specifically, up to a constant that tends to one as the mesh is refined, the dG solution is as accurate as the best approximation in the same norm. The main insight is that the quantities controlling the smallness of the mesh size are essentially those already appearing in the conforming setting. We also show that for fine meshes, the inf-sup stability constant is as good as the continuous one up to a factor two. Concerning the a posteriori analysis, we consider a residual-based error estimator under the assumption of piecewise constant material properties. We derive a global upper bound and local lower bounds on the error with constants that (i) only depend on the shape-regularity of the mesh if it is sufficiently refined and (ii) are independent of the stabilization bilinear form.