Superconvergence of the Local Discontinuous Galerkin Method for Elliptic Problems on Cartesian Grids

Bernardo Cockburn, Guido Kanschat, Ilaria Perugia, Dominik Schötzau

Abstract

In this paper, we present a super-convergence result for the Local Discontinuous Galerkin method for a model elliptic problem on Cartesian grids. We identify a special numerical flux for which the L2-norm of the gradient and the L2-norm of the potential are of order k+1/2 and k+1, respectively, when tensor product polynomials of degree at most k are used; for arbitrary meshes, this special LDG method gives only the orders of convergence of k and k+1/2, respectively. We present a series of numerical examples which establish the sharpness of our theoretical results.