EFFICIENT ITERATIVE SOLUTION OF THE 3-DIMENSIONAL HELMHOLTZ-EQUATION

We examine preconditioners for the discrete indefinite Helmholtz equat ion on a three-dimensional box-shaped domain with Sommerfeld-like boun dary conditions. The preconditioners are of two types. The first is de rived by discretization of a related continuous operator that differs from the original only in its boundary conditions. The second is deriv ed by a block Toeplitz approximation to the descretized problem. The r esulting preconditioning matrices allow the use of fast transform meth ods and differ from the discrete Helmholtz operator by an operator of low rank. We present experimental results demonstrating that when thes e methods are combined with Krylov subspace iteration, convergence rat es depend only mildly on both the wave number and discretization mesh size. In addition, the methods display high efficiencies in an impleme ntation on an IBM SP-2 parallel computer. (C) 1998 Academic Press.