Optimal a priori error estimates for the $hp$-version of the local discontinuous Galerkin method for convection--diffusion problems

Abstract

We study the convergence properties of the $hp$-version of the local discontinuous Galerkin finite element method for convection-diffusion problems; we consider a model problem in a one-dimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh-width $h$, in the polynomial degree $p$, and in the regularity of the exact solution. We identify a special numerical flux for which the estimates are optimal in both $h$ and $p$. The theoretical results are confirmed in a series of numerical examples.