hp-optimal convergence of the original DG method for linear hyperbolic problems on special simplicial meshes

Abstract

Abstract We prove $hp$-optimal error estimates for the original discontinuous Galerkin (DG) method when approximating solutions to first-order hyperbolic problems with constant convection fields in the $L^{2}$ and DG norms. The main theoretical tools used in the analysis are novel $hp$-optimal approximation properties of the special projector introduced in Cockburn et al. (2008, Optimal convergence of the original DG method for the transportreaction equation on special meshes. SIAM J. Numer. Anal., 46:1250-1265). We assess the theoretical findings on some test cases.