Analysis of Two-Level Overlapping Additive Schwarz Preconditioners for a Discontinuous Galerkin Method

The Schwarz method refers to a general methodology, based on the idea of divide-andconquer, for solving the systems of linear algebraic equations resulting from numerical discretizations of partial differential equations. In the past fifteen years extensive research has been done on the method to solve different types of algebraic systems which arise from various discretizations of partial differential equations such as finite difference/element/volume methods, spectral methods and mortar finite element methods (cf. [SBG96, Xu92] and references therein). On the other hand, very few results on the Schwarz method have been known in the literature for discontinuous Galerkin methods (cf. [FK01a, LT00, RVW96]). Discontinuous Galerkin methods use piecewise, totally discontinuous polynomial trial and test function spaces, that is, no continuity constraints are explicitly imposed on the functions across the element interfaces. As a consequence, weak formulations must include jump terms across interfaces and typically penalty terms are (artificially) added to control the jump terms (cf. [Arn82, DD76, Whe78]). Discontinuous Galerkin methods have several advantages over other types of finite element methods. For example, the trial and test spaces are very easy to construct; they can naturally handle inhomogeneous boundary conditions and curved boundaries; they also allow the use of highly nonuniform and unstructured meshes. In addition, the fact that the mass matrices are block diagonal is an attractive feature in the context of time-dependent problems, especially if explicit time discretizations are used. On the other hand, discontinuous Galerkin methods would seem to be at a disadvantage in view of a relatively larger number of degrees of freedom per element. Therefore, to offset this disadvantage, effective remedies must be found at the level of solution of the systems of algebraic equations. The objective of this paper is to develop some two-level overlapping additive Schwarz preconditioners for a discontinuous Galerkin method for solving second order elliptic problems. In Section 2, the discontinuous Galerkin and some known facts about the method, as well as a trace inequality and a generalized Poincaré inequality for discontinuous, piecewise functions are recalled. In Section 3, some two-level overlapping additive Schwarz preconditioners are proposed and analyzed for the discontinuous Galerkin method. The main result is to show that the condition numbers of the preconditioned systems are of the order , where and stand for the coarse mesh size and the size of overlaps between subdomains. This paper is the second in a sequel devoted to developing Schwarz methods for discontinuous Galerkin methods. [FK01a] contains non-overlapping Schwarz methods for discontinuous Galerkin methods. The condition number estimates of the order are established and numerical experiments are presented. In [FK01b], Schwarz methods are developed for the discontinuous Galerkin method of Baker [Bak77] for the biharmonic problems.