Space Decompositions and Solvers for Discontinuous Galerkin Methods

We present a brief overview of the different domain and space decomposition techniques that enter in developing and analyzing solvers for discontinuous Galerkin methods. Emphasis is given to the novel and distinct features that arise when considering DG discretizations over conforming methods. Connections and differences with the conforming approaches are emphasized.Comment: 2 pages 2 figures no table

Two-level additive schwarz preconditioners for a weakly over-penalized symmetric interior penalty method

We propose and analyze several two-level additive Schwarz preconditioners for a weakly over-penalized symmetric interior penalty method for second order elliptic boundary value problems. We also report numerical results that illustrate the parallel performance of these preconditioners. © 2010 Springer Science+Business Media, LLC

Schwarz methods for a preconditioned WOPSIP method for elliptic problems

We construct and analyze non-overlapping Schwarz methods for a preconditioned weakly over-penalized symmetric interior penalty (WOPSIP) method for elliptic problems

Domain Decomposition Methods for Discontinuous Galerkin Approximations of Elliptic Problems

The application of the techniques of domain decomposition to construct effective preconditioners for systems generated by standard methods such as finite difference or finite element methods has been well-researched in the past few decades. However, results concerning the application of these techniques to systems created by the discontinuous Galerkin method (DG) are much more rare. This dissertation represents the effort to extend the study of two-level nonoverlapping and overlapping additive Schwarz methods for DG discretizations of second- and fourth-order elliptic partial differential equations. In particular, the general Schwarz framework is used to find theoretical bounds for the condition numbers of the preconditioned systems corresponding to both the nonoverlapping and overlapping additive Schwarz methods. In addition, the impact on the performance of the preconditioners caused by varying the penalty parameters inherent to DG methods is investigated. Another topic of investigation is the choice of course subspace made for the two-level Schwarz methods. The results of in-depth computational experiments performed to validate and study various aspects of the theory are presented. In addition, the design and implementation of the methods are discussed