An a priori error analysis of the local discontinuous Galerkin method for elliptic problems

In this paper, we present the first a priori error analysis for the local discontinuous Galerkin (LDG) method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L²-norm of the gradient and the L²-norm of the potential are of order k and k + 1/2, respectively, when polynomials of total degree at least k are used; if stabilization parameters of order h^-1 are taken, the order of convergence of the potential increases to k+1. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains.