×

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.

MSC:

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Guo, C.H., Appl. numer. math., 19, 495, (1996)
[2] Erlangga, Y.A.; Vuik, C.; Oosterlee, C.W., Appl. numer. math., 50, 409, (2004)
[3] Bayliss, A.; Goldstein, C.I.; Turkel, E., J. comput. phys., 49, 443, (1983)
[4] Gozani, J.; Nachshon, A.; Turkel, E., Conjugate gradient compiled with multi-grid for an indefinite problem, (), 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
[6] Benzi, M., J. comput. phys., 182, 418, (2002)
[7] Saad, Y., Iterative methods for sparse linear systems, (1996), PWS Boston · Zbl 1002.65042
[8] Paige, C.C.; Saunders, M.A., SIAM J. numer. anal., 12, 617, (1975)
[9] Hestenes, M.R.; Stiefel, E., J. res. nat. bur. standards, 49, 409, (1952)
[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)
[12] Saad, Y.; Schultz, M.H., SIAM J. sci. stat. comput., 7, 856, (1986)
[13] Erlangga, Y.A.; Oosterlee, C.W.; Vuik, C., SIAM J. sci. comput., 27, 1471, (2006)
[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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.