Preconditioning of discrete Helmholtz operators perturbed by a diagonal complex matrix. (English) Zbl 0972.65031

Magolu monga Made, Mardochée; Beauwens, Robert; Warzée, Guy

Commun. Numer. Methods Eng. 16, No. 11, 801-817 (2000).

The authors study a new class of preconditioning methods to solve large and sparse linear systems with highly indefinite complex-symmetric systems. The starting observation is that the Krylov subspace method sometimes degrades when incomplete Cholesky factorizations are used as a preconditioner. The main underlying reason for this is that the eigenvalues with negative real part, on their move towards 1 have to cross the origin. This can lead to a clustering of eigenvalues near zero. The proposed remedy is to slightly move the spectrum of the original system matrix along the imaginary axis.
The paper contains a theoretical analysis and extensive numerical tests in the case of the Helmholtz equation in several domains. The superiority of their method is clearly visible in these cases.

Reviewer: Willy Govaerts (Gent)

MSC:

65F35	Numerical computation of matrix norms, conditioning, scaling
65F10	Iterative numerical methods for linear systems
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

preconditioning; large and sparse linear systems; Krylov subspace method; incomplete Cholesky factorizations; clustering of eigenvalues; numerical tests; Helmholtz equation

References:

[1]	Harari, Archives of Computational Methods in Engineering 3 pp 131– (1996)
[2]	Saad, SIAM Journal on Scientific and Statistical Computing 7 pp 856– (1986) · Zbl 0599.65018
[3]	Freund, SIAM Journal on Scientific and Statistical Computing 13 pp 425– (1992) · Zbl 0761.65018
[4]	Freund, SIAM Journal on Scientific Computing 15 pp 313– (1994) · Zbl 0803.65036
[5]	van der Vorst, SIAM Journal on Scientific and Statistical Computing 13 pp 631– (1992) · Zbl 0761.65023
[6]	Sleijpen, Electronic Transactions on Numerical Analysis 1 pp 11– (1993)
[7]	Iterative Solution Methods. Cambridge University Press: Cambridge, 1994.
[8]	Numerical Linear Algebra for High-Performance Computers. SIAM, Philadelphia, 1998. · Zbl 0914.65014
[9]	Matrix Computations (3rd edn). The John Hopkins University Press: Baltimore, MD, 1996. · Zbl 0865.65009
[10]	Iterative Methods for Sparse Linear Systems. PWS Publishing, Co.: Boston, MA, 1996.
[11]	Axelsson, Numerical Mathematics 48 pp 479– (1986) · Zbl 0564.65016
[12]	Modified incomplete factorization strategies. In Preconditioned Conjugate Gradient Methods, (eds). Lectures Notes in Mathematics, vol. 1457. Springer: Berlin, 1990; 1-16.
[13]	Meijerink, Mathematics of Computation 31 pp 148– (1977)
[14]	Elman, Journal of Computational Physics 142 pp 163– (1998) · Zbl 0929.65089
[15]	A domain decomposition approach to solving the Helmholtz equation with a radiation boundary condition. In Domain Decomposition Methods in Science and Engineering, (eds). American Mathematical Society: Providence, RI, 1994; 177-192. · Zbl 0811.65109
[16]	Incomplete factorization based preconditionings for solving the Helmholtz equation. Technical Report, Service des Milieux Continus, Université Libre de Bruxelles, May 1999.
[17]	Matrix Analysis. Cambridge University Press: Cambridge, 1985. · Zbl 0576.15001
[18]	Magolu monga Made, SIAM Journal on Scientific Computing 19 pp 1083– (1998) · Zbl 0914.65029
[19]	A set of gmres routines for real and complex arithmetics. Technical Report TR-PA-97-49, CERFACS, Toulouse, France, 1997.
[20]	de La Bourdonnaye, Contemporary Mathematics 218 pp 42– (1998)
[21]	Freund, ACM Transactions on Mathematical Software 22 pp 46– (1996) · Zbl 0884.65027
[22]	Malhotra, Computer Methods in Applied Mechanics and Engineering 146 pp 173– (1997) · Zbl 0901.76059