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 first discuss superconvergence results for nonselfadjoint linear elliptic problems using discontinuous 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 flux, it is shown, in this article, that the error estimate for the discrete flux in L2-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 L2-norm. These results confirm superconvergent results for linear elliptic problems.