Superconvergent discontinuous Galerkin methods for second-order elliptic problems

Abstract

We identify discontinuous Galerkin methods for second-order elliptic problems in several space dimensions having superconvergence properties similar to those of the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods. These methods use polynomials of degree k ≥ 0 for both the potential as well as the flux. We show that the approximate flux converges in L² with the optimal order of k + 1, and that the approximate potential and its numerical trace superconverge, in L²-like norms, to suitably chosen projections of the potential, with order k + 2. We also apply element-by-element postprocessing of the approximate solution to obtain new approximations of the flux and the potential. The new approximate flux is proven to have normal components continuous across inter-element boundaries, to converge in L² with order k + 1, and to have a divergence converging in L² also with order k + 1. The new approximate potential is proven to converge with order k + 2 in L² . Numerical experiments validating these theoretical results are presented.