Domain Decomposition and Incomplete Factorisation Methods for Partial Differential Equations

In this paper we develop and study a method which tries to combine the merits of Domain Decompoxition (DD) and Incomplete Cholesky preconditioned Con,iugate Gradient method (ICCG) for the parallel solution of linear elliptic Partial Differential Equations (PDEs) on rectangular domains. We frst discretise the PDE problem, using Spline Collocation, a method of Finite Element type based on smooth splines. This gives rise to a sparse linear system of equations. The ICCG method provides us with a very effient, but not straightfarward parallelisable linear solver for such systems. On the (other hand, DD methods are very effective for elliptic PD.Es. A combination of DD and ICCG methods, in which the subdomain solves are carried out with ICCG, leads to eflcient and highly parallelisable solvers. We implement this hybrid DD-ICCG method on a hypercube, discuss its parallel eflciency, and show results from expieriments on configurations with up to 32 processors. We apply a totally local communication scheme and discuss its performance on the iPSCI2 hypercube. A similsrr approach can be used with other PDE discretisation methods.