Optimal L2 L² ‐Norm Error Estimates for Steady and Unsteady Maxwell Equations Using Skeletal Discontinuous Galerkin Methods

ABSTRACT This article proposes an optimal convergence analysis of two model problems: the stationary Maxwell equations, which represent the ‐elliptic problem, and the time‐dependent Maxwell equations in cold plasma. We employ skeletal discontinuous Galerkin (DG) methods for spatial discretization. First, we introduce a skeletal DG method for the ‐elliptic problem with variable coefficients and discuss the optimal convergence analysis in the energy and norms. Next, we propose a continuous in time skeletal DG method for the Maxwell problem in cold plasma. The proof of error converging at an optimal rate for the cold plasma equations in and discrete energy norms hinges on a suitably defined Ritz projection derived from the previously discussed stationary Maxwell problem. We also present numerical computations in two and three dimensions for the stationary and time‐dependent Maxwell equations, including implicit and explicit time integration techniques for the time‐dependent case. These computations verify the theoretical rates we have presented.