Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

Abstract

We consider the hp-version interior penalty discontinuous Galerkin finite-element method (hp-DGFEM) for second-order linear reaction-diffusion equations. To the best of our knowledge, the sharpest known error bounds for the hp-DGFEM. are due to Rivière et al. (1999, Comput. Geosci., 3, 337-360) and Houston et al. (2002, SIAM J. Numer. Anal., 99, 2133-2163). These are optimal with respect to the meshsize h but suboptimal with respect to the polynomial degree p by half an order of p. We present improved error bounds in the energy norm, by introducing a new function space framework. More specifically, assuming that the solutions belong element-wise to an augmented Sobolev space, we deduce fully hp-optimal error bounds.