Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids

In this paper, we present a superconvergence result for the local discontinuous Galerkin (LDG) 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 orders 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.