On the suboptimality of the p-version discontinuous Galerkin methods for first order hyperbolic problems

We address the issue of the suboptimality in the p-version discontinuous Galerkin (dG) methods for first order hyperbolic problems. The convergence rate is derived for the upwind dG scheme on tensor product meshes in any dimension. The standard proof in seminal work [14] leads to suboptimal convergence in terms of the polynomial degree by 3/2 order for general convection fields, with the exception of piecewise multi-linear convection fields, which rather yield optimal convergence. Such suboptimality is not observed numerically. Thus, it might be caused by a limitation of the analysis, which we partially overcome: for a special class of convection fields, we shall show that the dG method has a p-convergence rate suboptimal by 1/2 order only.