Domain Decomposition Using a Schwarz Method

Abstract

Realistic modeling of physical problems often involves systems of partial differential equations (PDE), usually nonlinear, and defined on domains that can have both large size and complex shape. In most cases, the selected numerical method requires that one discretize the domain, and the number of degrees of freedom can easily be more than what the available computer will handle. Modeling of the air flow around an aircraft with 3D finite elements requires, for instance, the discretization of the surrounding domain with a few million points, with several unknowns to determine at each point. The numerical scheme can furthermore be implicit and hence involve a linearsystems solution with this impressive number of unknowns. If we do not have a supercomputer at hand, which only very specialized research centers do, the matrix of such a system cannot even fit within the memory of the computer.