Efficient Iterative Solution of the Three-Dimensional Helmholtz Equation

TitleEfficient Iterative Solution of the Three-Dimensional Helmholtz Equation
Publication TypeJournal Articles
Year of Publication1998
AuthorsElman H, O'Leary DP
JournalJournal of Computational Physics
Pagination163 - 181
Date Published1998/05/01/
ISBN Number0021-9991

We examine preconditioners for the discrete indefinite Helmholtz equation on a three-dimensional box-shaped domain with Sommerfeld-like boundary conditions. The preconditioners are of two types. The first is derived by discretization of a related continuous operator that differs from the original only in its boundary conditions. The second is derived by a block Toeplitz approximation to the descretized problem. The resulting preconditioning matrices allow the use of fast transform methods and differ from the discrete Helmholtz operator by an operator of low rank. We present experimental results demonstrating that when these methods are combined with Krylov subspace iteration, convergence rates depend only mildly on both the wave number and discretization mesh size. In addition, the methods display high efficiencies in an implementation on an IBM SP-2 parallel computer.