Superconvergent HDG methods for Maxwell's equations via the $M$-decomposition
It provides a systematic construction of superconvergent HDG methods for Maxwell's equations, addressing a non-coercive problem, which is significant for computational electromagnetics.
This paper extends the M-decomposition framework to construct superconvergent HDG methods for Maxwell's equations on unstructured 2D meshes, achieving optimal error estimates and superconvergence for the curl of the solution, as confirmed by numerical experiments.
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.