Implementing Overlapping Domain Decomposition Methods on a Virtual Parallel Machine

Abstract

To solve many partial differential equations of different types domain decomposition techniques were developed. Such algorithms are generally very well suited for implementation on a virtual parallel machine, simulated on a distributed system. While such algorithms are readily available and well established in the literature, authors do usually not concern themselves with questions of the practical implementability of their algorithms. In particular issues such as finding the optimal size of overlap in domain decompositions, finding the most effective number of subdomains or deciding whether to use exact or inexact subdomain solvers are beyond the scope of these results. In this paper we will address these questions. We first develop suitable domain decomposition algorithms for our virtual parallel machine. Through numerical experiments using our algorithms we then show that smaller linear systems work well even without any overlap while larger systems require that at least 10% of the subdomain size overlap to have convergency. The data also indicates that between 20% to 35% of the subdomain is the optimal overlap size. We next increase the number of subdomains and analyze its effect on the parallel solver. Our data shows that for a sufficiently large linear system computational speed of convergence improves significantly as the number of subdomains increases. We finally compare the effectiveness of exact and inexact domain solvers and show that the appropriate choice of the number of iterations in the worker algorithm, is much more efficient in the inexact solver than in the exact solver.