Superconvergent HDG methods for Maxwell's equations via the $M$-decomposition
Abstract
The concept of the -decomposition was introduced by Cockburn et al.\ in Math. Comp.\ vol.\ 86 (2017), pp.\ 1609-1641 {to provide criteria to guarantee optimal convergence rates for the Hybridizable Discontinuous Galerkin (HDG) method for coercive elliptic problems}. In that paper they systematically constructed superconvergent hybridizable discontinuous Galerkin (HDG) methods to approximate the solutions of elliptic PDEs on unstructured meshes. In this paper, we use the -decomposition to construct HDG methods for the Maxwell's equations on unstructured meshes in two dimension. In particular, we show the any choice of spaces having an -decomposition, together with sufficiently rich auxiliary spaces, has an optimal error estimate and superconvergence even though the problem is not in general coercive. Unlike the elliptic case, we obtain a superconvergent rate for the curl of the solution, not the solution, and this is confirmed by our numerical experiments.
Cite
@article{arxiv.1905.07383,
title = {Superconvergent HDG methods for Maxwell's equations via the $M$-decomposition},
author = {Gang Chen and Peter Monk and Yangwen Zhang},
journal= {arXiv preprint arXiv:1905.07383},
year = {2019}
}