A priori and computable a posteriori error estimates for an HDG method for the coercive Maxwell equations

In this paper we present and analyze a hybridizable discontinuous Galerkin (HDG) method for a mixed curl-curl formulation of the steady state coercive Maxwell equations. With a discrete Sobolev embedding type estimates for the discontinuous polynomials, we provide a priori error estimates for the electric field and the Lagrange multiplier in the energy norm. With the smooth or minimal regularity assumption on the exact solution, we have optimal convergence rate for the electric field and the Lagrange multiplier in the energy norm. The a priori error estimate for the electric field in the L2-norm is also obtained by the duality argument, and the approximation is also optimal for the electric field in the L2-norm. Moreover, by employing suitable Helmholtz decompositions of the error, together with the upper bound estimate for the Lagrange multiplier, we provide a computable residual-based a posteriori error estimator which is derived based on the error measured in terms of a mesh-dependent energy norm. The efficiency of the a posteriori error estimator is also established. Three dimensional numerical results testing the performance of the a priori and a posteriori error estimates for the Maxwell equations are given.