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Incomplete block factorization preconditioning for linear systems arising in the numerical solution of the Helmholtz equation. (English) Zbl 0854.65086

The paper is concerned with the numerical solution of the Dirichlet boundary value problem in a plane domain for the inhomogeneous Helmholtz equation with a complex coefficient. A five-point finite difference discretization yields a linear system with a complex symmetric coefficient matrix, being block-tridiagonal and also being a complex perturbation of an \(M\)-matrix. It is to be solved by conjugate gradient type methods, here by the biconjugate gradient method, see e.g. D. A. H. Jacobs [IMA J. Numer. Anal. 6, 447-452 (1986; Zbl 0614.65028)].
Hereby the problem of choosing an effective preconditioner remains. The suggestion to use preconditioners found to be good for the unperturbed block-tridiagonal matrix also for the perturbed matrices tends to be unsatisfactory when the perturbation is relatively large. For this case, two incomplete block factorizations for the complex system matrix and for its real part are established (in the case of small mesh size). Numerical test results (Dirichlet problem in the unit square) show that the use of these factorizations as preconditioners gives considerably better convergence results than the use of preconditioners used earlier.

MSC:

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

Citations:

Zbl 0614.65028
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References:

[1] Axelsson, O., A general incomplete block-matrix factorization method, Linear Algebra Appl., 74, 179-190 (1986) · Zbl 0622.65023
[2] Axelsson, O.; Polman, B., On approximate factorization methods for block matrices suitable for vector and parallel processors, Linear Algebra Appl., 77, 3-26 (1986) · Zbl 0587.65013
[3] Bayliss, A.; Goldstein, C. I.; Turkel, E., An iterative method for the Helmholtz equation, J. Comput. Phys., 49, 443-457 (1983) · Zbl 0524.65068
[4] Beauwens, R.; Ben Bouzid, M., On sparse block factorization iterative methods, SIAM J. Numer. Anal., 24, 1066-1076 (1987) · Zbl 0633.65035
[5] Concus, P.; Golub, G. H.; Meurańt, G., Block preconditioning for the conjugate gradient method, SIAM J. Sci. Statist. Comput., 6, 220-252 (1985) · Zbl 0556.65022
[6] Fan, K., Note on M-matrices, Quart. J. Math. Oxford (2), 11, 43-49 (1960) · Zbl 0104.01203
[7] Freund, R. W., Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices, SIAM J. Sci. Statist. Comput., 13, 425-448 (1992) · Zbl 0761.65018
[8] Guo, C.-H., Some results on sparse block factorization iterative methods, Linear Algebra Appl., 145, 187-199 (1991) · Zbl 0723.65019
[9] Jacobs, D. A.H., A generalization of the conjugate-gradient method to solve complex systems, IMA J. Numer. Anal., 6, 447-452 (1986) · Zbl 0614.65028
[10] Magolu, M.-M., Modified block-approximate factorization strategies, Numer. Math., 61, 91-110 (1992) · Zbl 0724.65029
[11] Meijerink, J. A.; van der Vorst, H. A., An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix, Math. Comp., 31, 148-162 (1977) · Zbl 0349.65020
[12] Stoer, J.; Freund, R., On the solution of large indefinite systems of linear equations by conjugate gradient algorithms, (Glowinski, R.; Lions, J. L., Computing Methods in Applied Sciences and Engineering, V (1982), North-Holland: North-Holland Amsterdam), 35-53 · Zbl 0499.65018
[13] 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, 631-644 (1992) · Zbl 0761.65023
[14] Vassilevski, P. S., Indefinite elliptic problem preconditioning, Comm. Appl. Numer. Methods, 8, 257-264 (1992) · Zbl 0765.65106
[15] Yserentant, H., Preconditioning indefinite discretization matrices, Numer. Math., 54, 719-734 (1989) · Zbl 0663.65042
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