Preconditioning discontinuous Galerkin saddle point systems

Abstract

The efficient application of discontinuous Galerkin methods to elliptic problems requires multilevel preconditioning. This chapter introduces a suitable approach for the interior penalty method. This method has been extended to the local discontinuous Galerkin method (LDG) and to other discontinuous Galerkin methods. This method is used as a building block for preconditioners for the full saddle point system arising from LDG discretizations of Poisson's and Stokes' equations. These preconditioners are based on multilevel methods for the interior penalty method. The chapter proposes a general framework for the construction of saddle point preconditioners. The essence of this approach is that an efficient preconditioner can be constructed using preconditioners for the upper left block and the Schur complement of a saddle point system.