Superconvergent discontinuous Galerkin methods for second-order elliptic problems

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.