Introduction aux méthodes de décomposition de domaine
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Jean Roberts
INRIA Rocquencourt



Abstract
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Among the fastest methods for solving sparse linear systems are block  cyclic reduction, FFT, and multigrid methods.  All of these work very well for very regular  problems but less well for problems that are not particularly regular.  The idea of domain decomposition is to solve sub-problems that are smaller and more regular and to obtain a solution to the  global problem by using iterative techniques for combining solutions of the sub-problems.

Over the past twenty years domain decomposition has become very  important for high performance computing as techniques that decompose large complicated problems into smaller simpler problems that can be solved simultaneously on different processors are naturally well suited to parallelization. Domain decomposition is thus a widely employed algebraic tool for the solution of large linear systems. The techniques of domain decomposition algorithms are equally important for complex physical problems for  which it is desirable to solve different problems or to use different numerical methods or simply  to use different computation grids in different parts of the domain.

Domain decomposition methods divide into two categories, those in  which the subdomains overlap and those in which they do not.  In this presentation we will give a  brief introduction to domain decomposition methods with more emphasis  on non overlapping methods. We will be, in particular, concerned with domain decomposition  methods for mixed finite element models.  Several domain  decomposition methods, Dirichlet-Neumann, Neumann-Neumann, and Robin-Robin will be introduced.  The Neumann-Neumann method will be  developed in somewhat more detail and a balancing domain preconditioner will be briefly  outlined.