Positive stable preconditioners for symmetric indefinite linear systems arising from Helmholtz equations. (English) Zbl 1231.78010

Summary: Using the finite difference method to discretize Helmholtz equations usually leads to a large spare linear system of equations \(Ax=b\). Since the coefficient matrix \(A\) is frequently indefinite, it is difficult to solve iteratively. The approach taken in this Letter is to precondition this linear system with positive stable preconditioners and then to solve it iteratively using Krylov subspace methods. Numerical experiments are given in order to demonstrate the efficiency of the presented preconditioners.


78A25 Electromagnetic theory (general)
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
65F10 Iterative numerical methods for linear systems
Full Text: DOI


[1] Guo, C. H., Appl. Numer. Math., 19, 495 (1996) · Zbl 0854.65086
[2] Erlangga, Y. A.; Vuik, C.; Oosterlee, C. W., Appl. Numer. Math., 50, 409 (2004) · Zbl 1051.65101
[3] Bayliss, A.; Goldstein, C. I.; Turkel, E., J. Comput. Phys., 49, 443 (1983) · Zbl 0524.65068
[4] Gozani, J.; Nachshon, A.; Turkel, E., Conjugate gradient compiled with multi-grid for an indefinite problem, (Vichnevestsky, R.; Tepelman, R. S., Advances in Computer Methods for Partial Differential Equations, vol. V (1984), IMACS: IMACS New Brunswick, NJ), 425-427
[5] A.L. Laird, Preconditioned iterative solution of the 2D Helmholtz equation, First Year’s Report, No. 02/12, St. Hugh’s College, Oxford, 2002; A.L. Laird, Preconditioned iterative solution of the 2D Helmholtz equation, First Year’s Report, No. 02/12, St. Hugh’s College, Oxford, 2002
[6] Benzi, M., J. Comput. Phys., 182, 418 (2002) · Zbl 1015.65018
[7] Saad, Y., Iterative Methods for Sparse Linear Systems (1996), PWS: PWS Boston · Zbl 1002.65042
[8] Paige, C. C.; Saunders, M. A., SIAM J. Numer. Anal., 12, 617 (1975) · Zbl 0319.65025
[9] Hestenes, M. R.; Stiefel, E., J. Res. Nat. Bur. Standards, 49, 409 (1952) · Zbl 0048.09901
[10] van der Vost, H. A.; Melissen, J. B.M., IEEE Trans. Mag., 26, 706 (1990)
[11] Freund, R. W., SIAM J. Sci. Stat. Comput., 13, 425 (1992) · Zbl 0761.65018
[12] Saad, Y.; Schultz, M. H., SIAM J. Sci. Stat. Comput., 7, 856 (1986) · Zbl 0599.65018
[13] Erlangga, Y. A.; Oosterlee, C. W.; Vuik, C., SIAM J. Sci. Comput., 27, 1471 (2006) · Zbl 1095.65109
[14] Laird, A. L.; Giles, M. B., AAIA J., 44, 2654 (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.