A new error analysis for discontinuous finite element methods for linear elliptic problems

Abstract

The standard a priori error analysis of discontinuous Galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the Galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm. In this paper, a new error analysis of discontinuous Galerkin methods is developed using only the H^k weak formulation of a boundary value problem of order 2^k. This is accomplished by replacing the Galerkin orthogonality with estimates borrowed from a posteriori error analysis and by using a discrete energy norm that is well defined for functions in H^k.