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Zbl 0929.65089
Elman, Howard C.; O'Leary, Dianne P.
Efficient iterative solution of the three-dimensional Helmholtz equation. (English)
[J] J. Comput. Phys. 142, No.1, 163-181 (1998). ISSN 0021-9991

The authors examine two types of preconditioners for the discrete indefinite Helmholtz equation with Sommerfeld-like boundary conditions. The first is derived by discretization of a related continuous operator, the second uses the block Toeplitz approximation to the desired problem. The resulting preconditioning matrices allow the use of fast transform methods (e.g. fast Fourier transform) and differ from the discrete Helmholtz operator by an operator of low rank. Some numerical experiments presented in the paper demonstrate the efficiency of the method when combined with Krylov subspace iteration. The authors show that the performance of restared GMRES with the proposed preconditioners is relatevely insensitive to the discretization mesh size and the wave number, and the algorithms are highly parallelizable. The presented technique is potentially applicable to inhomogeneous media, exterior domain problems and non-Cartesian grids.
[Mária Lukáčová (Brno)]

MSC 2000:: *65N06 Finite difference methods (BVP of PDE)
35J05 Laplace equation, etc.
65F10 Iterative methods for linear systems
65F35 Matrix norms, etc. (numerical linear algebra)

Keywords: Helmholtz equation; Sommerfeld-like boundary conditions; preconditioners; Krylov subspace iterations; parallel computation; numerical experiments

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