A Superconvergent HDG Method for the Maxwell Equations

Abstract

We present and analyze a new hybridizable discontinuous Galerkin (HDG) method for the steady state Maxwell equations. In order to make the problem well-posed, a condition of divergence is imposed on the electric field. Then a Lagrange multiplier p is introduced, and the problem becomes the solution of a mixed curl–curl formulation of the Maxwell’s problem. We use polynomials of degree $k+1$, k, k to approximate ${{\varvec{u}}},\nabla \times {{\varvec{u}}}$ and p respectively. In contrast, we only use a non-trivial subspace of polynomials of degree $k+1$ to approximate the numerical tangential trace of the electric field and polynomials of degree $k+1$ to approximate the numerical trace of the Lagrange multiplier on the faces. On the simplicial meshes, we show that the convergence rates for $\varvec{u}$ and $\nabla \times \varvec{u}$ are independent of the Lagrange multiplier p. If we assume the dual operator of the Maxwell equation on the domain has adequate regularity, we show that the convergence rate for $\varvec{u}$ is $O(h^{k+2})$. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this HDG method achieves superconvergence for the electric field without postprocessing. Finally, we show that the superconvergence of the HDG method is also derived on general polyhedral elements. Numerical results are given to verify the theoretical analysis.