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 $L^2$-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 $k+1$.