Optimal Error Estimates for the hp–Version Interior Penalty Discontinuous Galerkin Finite Element Method

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 Riviére, Wheeler and Girault [8] and due to Houston, Schwab and Süli [5] which 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 hp-optimal error bounds.