Class of Preconditioners for Discontinuous Galerkin Approximations of Elliptic Problems

Abstract

Domain decomposition (DD) methods provide powerful preconditioners for the iterative solution of the large algebraic linear systems of equations that arise in finite element approximations of partial differential equations. Many DD algorithms can conveniently be described and analyzed as Schwarz methods, and, if on the one hand a general theoretical framework has been previously developed for classical conforming discretizations (see, e.g., [7]), on the other hand, only few results can be found for discontinuous Galerkin (DG) approximations (see, e.g., [6, 4, 2, 1]). Based on discontinuous finite element spaces, DG methods have become increasing popular thanks to their great flexibility for providing discretizations on matching and non-matching grids and their high degree of locality. In this paper we present and analyze, in the unified framework based on the flux formulation proposed in [3], a class of Schwarz preconditioners for DG approximations of second order elliptic problems. Schwarz methods for a wider class of DG discretizations are studied in [2, 1]. The issue of preconditioning non-symmetric DG approximations is also discussed. Numerical experiments to asses the performance of the proposed preconditioners and validate our convergence results are presented.