An Introduction to Domain Decomposition Methods for PDEs

The numerical solution of differential problems of practical interest can be a difficult task to solve. Problems issued from computational mechanics or life science modeling are usually set on complex geometries and discretized on very fine grid, leading to very large-scale algebraic system, for which parallel approaches are mandatory. Domain Decomposition Methods are powerful tool to properly devising parallel algorithms that can benefit from multiprocessors architectures. The basic idea is the following. The given computational domain is partitioned into subdomains that may or may not overlap. The original problem is the reformulated upon each subdomain, yielding a family of subproblems of reduced size that are coupled through suitable conditions at the subdomains interfaces. The interface coupling is removed at the expense of introducing an iterative procedure among subdomains, which intrinsically embody a preconditioner for the system induced on the interface unknowns.

In this course we will present the mathematical foundations of classical Domain Decomposition Methods and their discrete formulation. We will describe classical overlapping Schwarz algorithms and Optimized Schwarz Methods, which guarantee convergence also in the absence of overlap between the subdomains. We will describe the use of domain decomposition methods in the heterogeneous coupling of multiphysics problems, and discuss the numerical aspects of the method.

BCAMCOURSE2012GERARDO-GIORDA