Mixed Discontinuous Galerkin Methods with Minimal Stabilization

Abstract

We will address the problem of finding the minimal necessary stabilization for a class of Discontinuous Galerkin (DG) methods in mixed form. In particular, we will present a new stabilized formulation of the Bassi-Rebay method (see Ref. 1 for the original unstable method) and a new formulation of the Local Discontinuous Galerkin method (LDG), introduced in 1998 by Cockburn and Shu. 2 It will be shown that, in order to reach stability, it is enough to add jump terms only over a part of the boundary of the domain, instead of over all the skeleton of the mesh, as it is usually done (see Ref. 3, for instance).