Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation. (English) Zbl 1094.65041

The authors present some analysis of Krylov subspace methods for solving the Helmholtz equation for an inhomogeneous medium wave problem with first order radiation boundary condition. For approximating the inverse of the shifted-Laplace operator a multigrid algorithm is designed.


65F35 Numerical computation of matrix norms, conditioning, scaling
65F10 Iterative numerical methods for linear systems
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs


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[1] Bayliss, A.; Goldstein, C.I.; Turkel, E., An iterative method for Helmholtz equation, J. comput. phys., 49, 443-457, (1983) · Zbl 0524.65068
[2] Bourgeois, A.; Bourget, M.; Lailly, P.; Poulet, M.; Ricarte, P.; Versteeg, R., Marmousi, model and data, (), 5-16
[3] Erlangga, Y.A.; Vuik, C.; Oosterlee, C.W., On a class of preconditioners for the Helmholtz equation, Appl. numer. math., 50, 409-425, (2004) · Zbl 1051.65101
[4] Gander, M.J.; Nataf, F., AILU for Helmholtz problems: a new preconditioner based on the analytical parabolic factorization, J. comput. acoustics, 9, 4, 1499-1506, (2001) · Zbl 1360.76181
[5] Hackbusch, W., Multi-grid methods and applications, (2003), Springer Berlin
[6] Kechroud, R.; Soulaimani, A.; Saad, Y., Preconditioning techniques for the solution of the Helmholtz equation by the finite element method, () · Zbl 1059.65105
[7] A.L. Laird, M.B. Giles, Preconditioned iterative solution of the 2D Helmholtz equation, Report NA-02/12, Oxford University Computing Laboratory, 2002
[8] Lahaye, D.; De Gersem, H.; Vandewalle, S.; Hameyer, K., Algebraic multigrid for complex symmetric systems, IEEE trans. magn., 36, 4, 1535-1538, (2000)
[9] Monga Made, M.M., Incomplete factorization-based preconditionings for solving the Helmholtz equation, Internat. J. numer. methods engrg., 50, 1077-1101, (2001) · Zbl 0977.65102
[10] Manteuffel, T.A.; Parter, S.V., Preconditioning and boundary conditions, SIAM J. numer. anal., 27, 3, 656-694, (1990) · Zbl 0713.65064
[11] Plessix, R.E.; Mulder, W.A., Separation-of-variables as a preconditioner for an iterative Helmholtz solver, Appl. numer. math., 44, 385-400, (2003) · Zbl 1013.65117
[12] Saad, Y.; Schultz, M.H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear system, SIAM J. sci. comput., 7, 3, 856-869, (1986) · Zbl 0599.65018
[13] Saad, Y., Iterative methods for sparse linear systems, (2003), SIAM Philadelphia, PA · Zbl 1002.65042
[14] Trottenberg, U.; Oosterlee, C.W.; Schüller, A., Multigrid, (2001), Academic Press London
[15] Varga, R.S., Matrix iterative analysis, (1962), Prentice-Hall Englewood Cliffs, NJ · Zbl 0133.08602
[16] van der Vorst, H.A., Bi-CGSTAB: A fast and smoothly converging variant of bicg for the solution of nonsymmetric linear systems, SIAM J. sci. comput., 13, 2, 631-644, (1992) · Zbl 0761.65023
[17] van der Vorst, H.A.; Vuik, C., The superlinear convergence behaviour of GMRES, J. comput. appl. math., 48, 327-341, (1993) · Zbl 0797.65026
[18] de Zeeuw, P.M., Matrix-dependent prolongations and restrictions in a blackbox multigrid solver, J. comput. appl. math., 33, 1-27, (1990) · Zbl 0717.65099
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