Optimal error estimate and superconvergence of the DG method for first-order hyperbolic problems

We consider the original discontinuous Galerkin method for the first-order hyperbolic problems in d -dimensional space. We show that, when the method uses polynomials of degree k , the L 2 -error estimate is of order k + 1 provided the triangulation is made of rectangular elements satisfying certain conditions. Further, we show the O ( h 2 k + 1 ) -order superconvergence for the error on average on some suitably chosen subdomains (including the whole domain) and their outflow faces. Moreover, we also establish a derivative recovery formula for the approximation of the convection directional derivative which is superconvergent with order k + 1 .