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 $k\ge 0$ for both the potential as well as the flux. We show that the approximate flux converges in $L^2$ with the optimal order of $k+1$, and that the approximate potential and its numerical trace superconverge, in $L^2$-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 $L^2$ with order $k+1$, and to have a divergence converging in $L^2$ also with order $k+1$. The new approximate potential is proven to converge with order $k+2$ in $L^2$. Numerical experiments validating these theoretical results are presented.