Convergence and comparison analysis of some numerical Schwarz methods

SummaryDomain decomposition methods are a natural means for solving partial differential equations on multi-processors. The spatial domain of the equation is expressed as a collection of overlapping subdomains and the solution of an associated equation is solved on each of these subdomains. The global solution is then obtained by piecing together the subsolutions in some manner. For elliptic equations, the global solution is obtained by iterating on the subdomains in a fashion that resembles the classical Schwarz alternating method. In this paper, we examine the convergence behavior of different subdomain iteration procedures as well as different subdomain approximations. For elliptic equations, it is shown that certain iterative procedures are equivalent to block Gauss-Siedel and Jacobi methods. Using different subdomain approximations, an inner-outer iterative procedure is defined.M-matrix analysis yields a comparison of different inner-outer iterations.