Convergence Analysis of an Adaptive Interior Penalty Discontinuous Galerkin Method

Abstract

We study the convergence of an adaptive interior penalty discontinuous Galerkin (IPDG) method for a two-dimensional model second order elliptic boundary value problem. Based on a residual-type a posteriori error estimator, we prove that after each refinement step of the adaptive scheme we achieve a guaranteed reduction of the global discretization error in the mesh-dependent energy norm associated with the IPDG method. In contrast to recent work on adaptive IPDG methods [O. Karakashian and F. Pascal, Convergence of Adaptive Discontinuous Galerkin Approximations of Second-order Elliptic Problems, preprint, University of Tennessee, Knoxville, TN, 2007], the convergence analysis does not require multiple interior nodes for refined elements of the triangulation. In fact, it will be shown that bisection of the elements is sufficient. The main ingredients of the proof of the error reduction property are the reliability and a perturbed discrete local efficiency of the estimator, a bulk criterion that takes care of a proper selection of edges and elements for refinement, and a perturbed Galerkin orthogonality property with respect to the energy inner product. The results of numerical experiments are given to illustrate the performance of the adaptive method.