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A PRECONDITIONED METHOD FOR THE SOLUTION OF THE ROBBINS PROBLEM FOR THE HELMHOLTZ EQUATION

Part of: Numerical linear algebra

Published online by Cambridge University Press:  31 March 2011

JIANG LE ,
HUANG JIN ,
XIAO-GUANG LV  and
QING-SONG CHENG
JIANG LE*
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, PR China (email: [email protected], [email protected]) School of Science, HuaiHai Institute of Technology, Lianyungang, Jiangsu, PR China (email: [email protected], [email protected])
HUANG JIN
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, PR China (email: [email protected], [email protected])
XIAO-GUANG LV
Affiliation:
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, PR China (email: [email protected], [email protected]) School of Science, HuaiHai Institute of Technology, Lianyungang, Jiangsu, PR China (email: [email protected], [email protected])
QING-SONG CHENG
Affiliation:
School of Science, HuaiHai Institute of Technology, Lianyungang, Jiangsu, PR China (email: [email protected], [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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A preconditioned iterative method for the two-dimensional Helmholtz equation with Robbins boundary conditions is discussed. Using a finite-difference method to discretize the Helmholtz equation leads to a sparse system of equations which is too large to solve directly. The approach taken in this paper is to precondition this linear system with a sine transform based preconditioner and then solve it using the generalized minimum residual method (GMRES). An analytical formula for the eigenvalues of the preconditioned matrix is derived and it is shown that the eigenvalues are clustered around 1 except for some outliers. Numerical results are reported to demonstrate the effectiveness of the proposed method.

Keywords

Helmholtz equationRobbins boundary conditionsGMRES methodsine transformpreconditioner

MSC classification

Secondary: 65F10: Iterative methods for linear systems
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 1 , July 2010 , pp. 87 - 100
Copyright
Copyright © Australian Mathematical Society 2011

Footnotes

This research is supported by NSFC (10871034) and Research Grant Z2008038 from HuaiHai Institute of Technology.

References

[1]Arnold, A. and Ehrhardt, M., “Discrete transparent boundary conditions for wide angle parabolic equations in underwater acoustics”, J. Comput. Phys. 145 (1998) 611–638.CrossRefGoogle Scholar
[2]Axelsson, O. and Barker, V., Finite element solution of boundary value problems: theory and computation (Academic Press, Orlando, FL, 1984).Google Scholar
[3]Bayliss, A., Goldstein, C. I. and Turkel, E., “An iterative method for the Helmholtz equation”, J. Comput. Phys. 49 (1983) 443–457.CrossRefGoogle Scholar
[4]Chan, R. and Ng, M., “Conjugate gradient methods for Toeplitz systems”, SIAM Rev. 38 (1996) 427–482.CrossRefGoogle Scholar
[5]Erlangga, Y. A., “A robust and efficient iterative method for the numerical solution of the Helmholtz equation”, Ph. D. Thesis, Delft University of Technology, 2005.Google Scholar
[6]Erlangga, Y. A., Vuik, C. and Oosterlee, C. W., “On a class of preconditioners for solving the Helmholtz equation”, Appl. Numer. Math. 50 (2004) 409–425.CrossRefGoogle Scholar
[7]Gander, M. J. and Nataf, F., “AILU for Helmholtz problems: a new preconditioner based on the analytical parabolic factorization”, J. Comput. Acoust. 9 (2001) 1499–1506.CrossRefGoogle Scholar
[8]Greenbaum, A., Iterative methods for solving linear systems, Volume 17 of Frontiers in Applied Mathematics (SIAM, Philadelphia, PA, 1997).Google Scholar
[9]Guo, C. H., “Incomplete block factorization preconditioning for linear systems arising in the numerical solution of the Helmholtz equation”, Appl. Numer. Math. 19 (1996) 495–508.Google Scholar
[10]Haldenwang, P., Labrosse, G., Abboudi, S. and Deville, M., “Chebyshev 3-D spectral and 2-D pseudospectral solvers for the Helmholtz equation”, J. Comput. Phys. 55 (1984) 115–128.Google Scholar
[11]Hemmingsson, L. and Otto, K., “Analysis of semi-Toeplitz preconditioners for first-order PDE”, SIAM J. Sci. Comput. 17 (1996) 47–64.CrossRefGoogle Scholar
[12]Ho, A. C. and Ng, M., “Iterative methods for Robbins problems”, Appl. Math. Comput. 165 (2005) 103–125.Google Scholar
[13]Hu, X. and Ling, X., “Preconditioners for elliptic problems via non-uniform meshes”, Appl. Math. Comput. 181 (2006) 1182–1198.Google Scholar
[14]Joubert, W. D. and Manteuffel, T. A., “Iterative methods for non-symmetric linear systems”, in: Iterative methods for large linear systems (eds Kincaid, D. R. and Hayes, L. J.), (Academic Press, New York, 1990) 348–358.Google Scholar
[15]Kechroud, R., Soulaimani, A. and Saad, Y., “Preconditioning techniques for the solution of the Helmholtz equation by the finite element method”, 2003 Workshop in Wave Phenomena in Physics and Engineering: New Models, Algorithms, and Applications, May 18–21, 2003 (eds V. Kumar et al.), (Springer, Berlin, 2003).Google Scholar
[16]Laird, A. L. and Giles, M. B., “Preconditioned iterative solution of the 2D Helmholtz equation”, Technical Report NA 02-12, Computing Laboratory, Oxford University, 2002.Google Scholar
[17]Laub, A. J., Matrix analysis for scientists and engineers (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2005).Google Scholar
[18]Manteuffel, T. A. and Parter, S. V., “Preconditioning and boundary conditions”, SIAM J. Numer. Anal. 27 (1990) 656–694.CrossRefGoogle Scholar
[19]Monga Made, M. M., “Incomplete factorization-based preconditionings for solving the Helmholtz equation”, Internat. J. Numer. Methods Engrg. 50 (2001) 1077–1101.Google Scholar
[20]Otto, K., “Construction and analysis of preconditioners for first-order PDE”, Ph. D. Thesis, Department of Scientific Computing, Uppsala University, Uppsala, Sweden, 1993.Google Scholar
[21]Pickering, W. and Harley, P., “FFT solution of the Robbins problem”, IMA J. Numer. Anal. 13 (1993) 215–233.CrossRefGoogle Scholar
[22]Pickering, W. and Harley, P., “On Robbins boundary conditions, elliptic equations, and FFT methods”, J. Comput. Phys. 122 (1995) 380–383.CrossRefGoogle Scholar
[23]Plessix, R. E. and Mulder, W. A., “Separation-of-variables as a preconditioner for an iterative Helmholtz solver”, Appl. Numer. Math. 44 (2003) 385–400.CrossRefGoogle Scholar
[24]Saad, Y., Iterative methods for sparse linear system (PWS Press, New York, NY, 1996).Google Scholar
[25]Urbach, H. P. and Merkx, R. T. M., “Finite element simulation of electromagnetic plane wave diffraction at gratings for arbitrary angles of incidence”, in: Mathematical and numerical aspects of wave propagation phenomena, Proceedings of SIAM First Conference on Mathematical and Numerical Aspects of Wave Propagation (eds Cohen, G., Halpern, L. and Joly, P.), (Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992) 89–99.Google Scholar
[26]Van Loan, C. and Pitsianis, N. P., “Approximation with Kronecker products”, in: Linear algebra for large scale and real time applications (eds Moonen, M. S. and Golub, G. H.), (Kluwer Publications, Dordrecht, 1992) 293–314.Google Scholar
[27]Vuik, C., Erlangga, Y. A. and Oosterlee, C. W., “Shifted Laplace preconditioners for the Helmholtz equations”, Report 03-18, Delft University of Technology, ISSN 1389-6520, 2003.Google Scholar