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.