Convergence Analysis of a Discontinuous Galerkin Method for Wave Equations in Second-Order Form.

In this paper we study the convergence property of a spatial discontinuous Galerkin method for wave equations. We prove an optimal convergence rate in the energy norm, which improves an existing suboptimal a priori error estimate. In addition, by adding a penalty term to the variational form, we obtain a supercloseness result, based on which we prove superconvergence of a postprocessed gradient, where the postprocessing operator is the polynomial preserving recovery. All theoretical findings are verified by numerical tests.