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arXiv · 1905.07383

Superconvergent HDG methods for Maxwell's equations via the $M$-decomposition

Gang Chen·Peter Monk·Yangwen Zhang
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Abstract

The concept of the $M$-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 $M$-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 $M$-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.

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Gang Chen, Peter Monk, Yangwen Zhang. 2019-05-17. Superconvergent HDG methods for Maxwell's equations via the $M$-decomposition. https://arxiv.org/abs/1905.07383

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