A Superconvergent HDG Method for the Maxwell Equations

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. of problems posed in two dimensions. To see how this is possible, consider the usual time harmonic Maxwell system for the electric field E (a complex valued vector function):

A Three-Dimensional Hybrid High-Order Method for Magnetostatics

“…It indeed hinges, as in [3], on face unknowns for the magnetic field belonging to a subtle subspace of P k+1 (F; R 2 ). However, there are two main differences between our method and the one in [3]. First, taking advantage of the fact that Problem (1) is actually first-order, we do not (locally) reconstruct a discrete curl operator.…”

“…Their connections with HDG methods have been later discussed in [4] in the context of scalar variable diffusion problems. The method we introduce here shares some similarities with the HDG method of [3]. It indeed hinges, as in [3], on face unknowns for the magnetic field belonging to a subtle subspace of P k+1 (F; R 2 ).…”

“…The method we introduce here shares some similarities with the HDG method of [3]. It indeed hinges, as in [3], on face unknowns for the magnetic field belonging to a subtle subspace of P k+1 (F; R 2 ). However, there are two main differences between our method and the one in [3].…”

“…Let Ω ⊂ R 3 denote an open, bounded and connected polyhedral domain, with boundary ∂ Ω and unit outward normal n. We assume that Ω is topologically trivial, and that ∂ Ω is connected. For any X ⊂ Ω , we denote by (•, •) X and || • || X the usual inner product and norm on L 2 (X, R l ), l ∈ {1, 2, 3}.…”

“…We can, in particular, cite the seminal work of [9] on simplicial elements. On more general element shapes, one can mention the Discontinuous Galerkin method of [11], the Hybridizable Discontinuous Galerkin (HDG) methods of [10] and [3], or the Virtual Element method of [12].…”