Domain Decomposition Preconditioners for Discontinuous Galerkin Finite Element Methods

In recent years, much attention has been given to domain decomposition methods for linear elliptic problems that are based on a partitioning of the domain of the physical problem. Since the subdomains can be handled independently, such methods are very attractive for coarse-grain parallel computers.
In this talk we shall present in a unified framework a class of non-overlapping Schwarz domain decomposition methods for the solution of the algebraic linear systems of equations arising from discontinuous Galerkin (DG) approximations of self-adjoint elliptic problems. We shall present the convergence theory, discuss some interesting features which have no analogue in the conforming case, and show that the proposed Schwarz methods can be successfully accelerated by suitable Krylov space based iterative solvers. The issue of preconditioning DG approximations of non-self-adjoint elliptic problems will be also addressed.
Numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods will be shown.