An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems

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Abstract
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.