Variational Convergence of IP−DGFEM

Abstract

In this paper, we develop the theory required to perform a variational convergence analysis for discontinuous Galerkin nite element methods when applied to minimization problems. For Sobolev indices in \left[1;∈fty\right), we prove generalizations of many techniques of classical analysis in Sobolev spaces and apply them to a typical energy minimization problem for which we prove convergence of a variational interior penalty discontinuous Galerkin nite element method (VIPDGFEM). Our main tool in this analysis is a theorem which allows the extraction of a \weakly" converging subsequence of a family of discrete solutions and which shows that any \weak limit" is a Sobolev function.