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 L₂-error estimate is of order k+1 provided the triangulation is made of rectangular elements satisfying certain conditions. Further, we show the O(h^2k+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.