Superconvergence of a class of expanded discontinuous Galerkin methods for fully nonlinear elliptic problems in divergence form

Abstract

For fully nonlinear elliptic boundary value problems in divergence form, improved error estimates are derived in the frame work of a class of expanded discontinuous Galerkin methods. It is shown that the error estimate for the discrete flux in L 2 -norm is of order k + 1 , when piecewise polynomials of degree k ≥ 1 are used to approximate both potential as well as flux variables. Then, solving a discrete linear elliptic problem in each element locally, a suitable post-processing of the discrete potential is proposed and it is proved that the resulting post-processed potential converges with order of convergence k + 2 in L 2 -norm. By choosing stabilizing parameters appropriately, similar results are derived for the expanded HDG methods for nonlinear elliptic problems.