An optimal-order error estimate for the discontinuous Galerkin method

In this paper a new approach is developed for analyzing the discontinuous Galerkin method for hyperbolic equations. For a model problem in ${R^2}$, the method is shown to converge at a rate $O({h^{n + 1}})$ when applied with nth degree polynomial approximations over a semiuniform triangulation, assuming sufficient regularity in the solution.