Superconvergent discontinuous Galerkin methods for nonlinear elliptic equations

Abstract

Based on the analysis of Cockburn et. al. (Math. Comp. 78 (2009), pp. 1-24) for a selfadjoint linear elliptic equation, we rst discuss su- perconvergence results for nonselfadjoint linear elliptic problems using discon- tinuous Galerkin methods. Further, we have extended our analysis to derive superconvergence results for quasilinear elliptic problems. When piecewise polynomials of degree k 1 are used to approximate both the potential as well as the ux, it is shown, in this article, that the error estimate for the discrete ux in L 2 -norm is of order k + 1: Further, based on solving a discrete linear elliptic problem at each element, a suitable postprocessing of the discrete potential is developed and then, it is proved that the resulting post-processed potential converges with order of convergence k + 2 in L 2 -norm. These results conrm superconvergent results for linear elliptic problems.