Huangxin Chen, Weifeng Qiu, Ke Shi, Manuel Solano:
A superconvergent HDG method for the Maxwell equations
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 problem. We use polynomials of degree k+1, k and k to approximate u, curl 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, a special choice of the stabilization parameters is applied, and the HDG system is shown to be well-posed. Moreover, we show that the convergence rates for u and curl 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 u is order 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 on general polyhedral elements, by a particular choice of the stabilization parameters again, the HDG system is also well-posed and the superconvergence of the HDG method is derived.
This preprint gave rise to the following definitive publication(s):
Huangxin CHEN, Weifeng QIU, Ke SHI, Manuel SOLANO: A superconvergent HDG method for the Maxwell equations. Journal of Scientific Computing, vol. 70, 3, pp. 1010-1029, (2017).