A Variation Of The Schwarz Alternating Method: The Domain Reduction Method

. Domain decomposition methods are highly parallel methods for solving elliptic partial differential equations. In many domain decomposition variants, the domain is partitioned into a number of (possibly overlapping) subdomains before computation begins. In contrast, the domain reduction method folds the domain into a number of smaller domains covering only a small portion of the entire domain. The solution over the entire domain is recovered by unfolding the solutions on the subdomains and summing them. The cost of the folding and unfolding is negligible. All steps of the algorithm are embarrassingly parallel. 1. Introduction. In this paper, the solution to an elliptic boundary value problem is approximated using a combination of domain decomposition, multigrid, and projection method techniques (see [1], [2], [4], [7], [13], [14], [16], and [17]). Consider problems of the following form. 8 ? ! ? : Lu = f in\Omega ae IR d ; d ? 0; u = 0 on @\Omega : (1.1) More complicated boun..