Optimal convergence of the original DG method for the transport-reaction equation on special meshes

We show that the approximation given by the original discontinuous Galerkin method for the transport-reaction equation in d space dimensions is optimal provided the meshes are suitably chosen: the L²-norm of the error is of order k + 1 when the method uses polynomials of degree k. These meshes are not necessarily conforming and do not satisfy any uniformity condition; they are required only to be made of Simplexes, each of which has a unique outflow face. We also find a new, element-by-element postprocessing of the derivative in the direction of the flow which superconverges with order k + 1.