Superconvergence of the Local Discontinuousgalerkin Method for Elliptic Problems on Cartesiangridsbernardo

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 ux for which the L 2-norm of the gradient and the L 2-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.