Stability properties of Discontinuous Galerkin method in mixed form

Abstract

In this article, the problem of finding the necessary stabilization for a class of Discontinuous Galerkin methods in mixed form for elliptic problems is considered. In particular, a new stabilized formulation of the (unstable) Bassi-Rebay method and a new formulation of the Local Discontinuous Galerkin (LDG) method are presented. The stability properties of the new formulations are studied and error estimates are derived. The theoretical results are validated in a series of numerical tests. This article summarizes and comments the main ideas presented in my PhD Thesis, defended on the 19th of January, 2007, at the Mathematics Department of the University of Pavia.