On a class of preconditioners for solving the Helmholtz equation.(English)Zbl 1051.65101

Summary: In 1983, a preconditioner was proposed by A. Bayliss, C. I. Goldstein, and E. Turkel [J. Comput. Phys. 49, 443–457 (1983; Zbl 0524.65068)] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. A. L. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year’s Report, St. Hugh’s College, Oxford (2001)] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign.
Both approaches are further generalized to a new class of preconditioners, the so-called “shifted Laplace” preconditioners of the form $$\Delta \varphi - \alpha k^2 \varphi$$ with $$\alpha \in \mathbb C$$. Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.

MSC:

 65N06 Finite difference methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65F35 Numerical computation of matrix norms, conditioning, scaling 65F10 Iterative numerical methods for linear systems

Zbl 0524.65068

CGS; AILU
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References:

 [1] Bamberger, A; Joly, P; Roberts, J.E, Second-order absorbing boundary conditions for the wave equation: A solution for corner problem, SIAM J. numer. anal., 27, 2, 323-352, (1990) · Zbl 0716.35036 [2] Bayliss, A; Goldstein, C.I; Turkel, E, An iterative method for Helmholtz equation, J. comput. phys., 49, 443-457, (1983) · Zbl 0524.65068 [3] Berenger, J.-P, A perfectly matched layer for the absorption of electromagnetic waves, J. comput. phys., 114, 185-200, (1994) · Zbl 0814.65129 [4] Clayton, R.W; Engquist, B, Absorbing boundary conditions for wave-equation migration, Geophysics, 45, 5, 895-904, (1980) [5] Engquist, B; Majda, A, Absorbing boundary conditions for the numerical simulation of waves, Math. comp., 31, 629-651, (1977) · Zbl 0367.65051 [6] Fletcher, R, Conjugate gradient methods for indefinite systems, (), 73-89 · Zbl 0326.65033 [7] R.W. Ereund, Preconditioning of symmetric, but highly indefinite linear systems, Numer. Anal. Manus. 97-3-03, Bell Labs., Murray Hill, NJ, 1997 [8] Freund, R.W; Nachtigal, N.M, QMR: A quasi minimum residual method for non-Hermitian linear systems, Numer. math., 60, 315-339, (1991) · Zbl 0754.65034 [9] Gander, M.J; Nataf, F, AILU for Helmholtz problems: A new preconditioner based on the analytic parabolic factorization, J. comput. acoustics, 9, 4, 1499-1509, (2001) · Zbl 1360.76181 [10] Lahaye, D; De Gersem, H; Vandewalle, S; Hameyer, K, Algebraic multigrid for complex symmetric systems, IEEE trans. magn., 36, 4, 1535-1538, (2000) [11] Gozani, J; Nachshon, A; Turkel, E, Conjugate gradient coupled with multi-grid for an indefinite problem, (), 425-427 [12] Higham, N.J, Factorizing complex symmetric matrices with positive definite real and imaginary parts, Math. comp., 67, 3, 1591-1599, (1998) · Zbl 0909.65016 [13] Kechroud, R; Soulaimani, A; Saad, Y, Preconditioning techniques for the solution of the Helmholtz equation by the finite element method, () · Zbl 1059.65105 [14] A.L. Laird, Preconditioned iterative solution of the 2D Helmholtz equation, First Year’s Report, St. Hugh’s College, Oxford, 2001 [15] Made, M.M.M, Incomplete factorization-based preconditionings for solving the Helmholtz equation, Internat. J. numer. methods engng., 50, 1077-1101, (2001) · Zbl 0977.65102 [16] Paige, C.C; Saunders, M.A, Solution of sparse indefinite systems of linear equations, SIAM J. numer. anal., 12, 4, 617-629, (1975) · Zbl 0319.65025 [17] 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 [18] Saad, Y, A flexible inner-outer preconditioned GMRES algorithm, SIAM J. sci. statist. comput., 14, 461-469, (1993) · Zbl 0780.65022 [19] Saad, Y, Iterative methods for sparse linear systems, (1996), PWS Boston · Zbl 1002.65042 [20] Saad, Y; Schultz, M.H, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. statist. comput., 7, 12, 856-869, (1986) · Zbl 0599.65018 [21] Sonneveld, P, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. sci. statist. comput., 10, 1, 36-52, (1986) · Zbl 0666.65029 [22] van der Vorst, H.A, Bi-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems, SIAM J. sci. statist. comput., 13, 2, 631-644, (1992) · Zbl 0761.65023 [23] van der Vorst, H.A; Melissen, J.B.M, A petrov – galerkin type method for solving ax=b, where A is symmetric complex, IEEE trans. magnetics, 26, 2, 706-708, (1990) [24] van der Vorst, H.A; Vuik, C, GMRESR: A family of nested GMRES methods, Numer. linear algebra appl., 1, 4, 369-386, (1994) · Zbl 0839.65040
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