## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2012, Special Issue (2012), Article ID 365124, 9 pages.

### A Modified SSOR Preconditioning Strategy for Helmholtz Equations

#### Abstract

The finite difference method discretization of Helmholtz equations usually leads to the large spare linear systems. Since the coefficient matrix is frequently indefinite, it is difficult to solve iteratively. In this paper, a modified symmetric successive overrelaxation (MSSOR) preconditioning strategy is constructed based on the coefficient matrix and employed to speed up the convergence rate of iterative methods. The idea is to increase the values of diagonal elements of the coefficient matrix to obtain better preconditioners for the original linear systems. Compared with SSOR preconditioner, MSSOR preconditioner has no additional computational cost to improve the convergence rate of iterative methods. Numerical results demonstrate that this method can reduce both the number of iterations and the computational time significantly with low cost for construction and implementation of preconditioners.

#### Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 365124, 9 pages.

Dates
First available in Project Euclid: 3 January 2013

https://projecteuclid.org/euclid.jam/1357179947

Digital Object Identifier
doi:10.1155/2012/365124

Mathematical Reviews number (MathSciNet)
MR2861936

Zentralblatt MATH identifier
1235.65126

#### Citation

Wu, Shi-Liang; Li, Cui-Xia. A Modified SSOR Preconditioning Strategy for Helmholtz Equations. J. Appl. Math. 2012, Special Issue (2012), Article ID 365124, 9 pages. doi:10.1155/2012/365124. https://projecteuclid.org/euclid.jam/1357179947

#### References

• C.-H. Guo, “Incomplete block factorization preconditioning for linear systems arising in the numerical solution of the Helmholtz equation,” Applied Numerical Mathematics, vol. 19, no. 4, pp. 495–508, 1996.
• Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, “On a class of preconditioners for solving the Helmholtz equation,” Applied Numerical Mathematics, vol. 50, no. 3-4, pp. 409–425, 2004.
• A. Bayliss, C. I. Goldstein, and E. Turkel, “An iterative method for the Helmholtz equation,” Journal of Computational Physics, vol. 49, no. 3, pp. 443–457, 1983.
• G. Bao and W. W. Sun, “A fast algorithm for the electromagnetic scattering from a large cavity,” SIAM Journal on Scientific Computing, vol. 27, no. 2, pp. 553–574, 2005.
• Y. Wang, K. Du, and W. W. Sun, “Preconditioning iterative algorithm for the electromagnetic scattering from a large cavity,” Numerical Linear Algebra with Applications, vol. 16, no. 5, pp. 345–363, 2009.
• Y. Saad, Iterative Methods for Sparse Linear Systems, PWS, Boston, Mass, USA, 1996.
• J. Gozani, A. Nachshon, and E. Turkel, “Conjugate gradient comipled with multi-grid for an indefinite problem,” in Advances in Computer Methods for Partial Differential Equations, R. Vichnevestsky and R. S. Tepelman, Eds., vol. 5, pp. 425–427, IMACS, New Brunswick, NJ, USA, 1984.
• A. L. Laird, “Preconditioned iterative solution of the 2D Helmholtz equation,” First Year's Report 02/12, Hugh's College, Oxford, UK, 2002.
• Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, “Comparison of multigrid and incomplete LU shifted-Laplace preconditioners for the inhomogeneous Helmholtz equation,” Applied Numerical Mathematics, vol. 56, no. 5, pp. 648–666, 2006.
• M. Gander, “AILU for Helmholtz problems: a new preconditioner based on the analytic parabolic factorization,” Journal of Computational Acoustics, vol. 9, no. 4, pp. 1499–1506, 2001.
• R. E. Plessix and W. A. Mulder, “Separation-of-variables as a preconditioner for an iterative Helmholtz solver,” Applied Numerical Mathematics, vol. 44, no. 3, pp. 385–400, 2003.
• M. M. M. Made, R. Beauwens, and G. Warzée, “Preconditioning of discrete Helmholtz operators perturbed by a diagonal complex matrix,” Communications in Numerical Methods in Engineering, vol. 16, no. 11, pp. 801–817, 2000.
• M. Benzi, “Preconditioning techniques for large linear systems: a survey,” Journal of Computational Physics, vol. 182, no. 2, pp. 418–477, 2002.
• F. Mazzia and R. Alan McCoy, “Numerical experimetns with a shifted SSOR preconditioner for symmetric matrices,” type TR/PA/98/12, CERFACS, 1998.
• Z.-Z. Bai, “Mofidied block SSOR preconditioners for symmetric positive definite linear systems,” Annals of Operations Research, vol. 103, pp. 263–282, 2001.
• X. Chen, K. C. Toh, and K. K. Phoon, “A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations,” International Journal for Numerical Methods in Engineering, vol. 65, no. 6, pp. 785–807, 2006.
• Y.-H. Ran and L. Yuan, “On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3050–3068, 2010.
• O. Axelsson, “A generalized SSOR method,” BIT, vol. 18, pp. 443–467, 1972.
• A. Hadjidimos, “Successive overrelaxation (SOR) and related methods,” Journal of Computational and Applied Mathematics, vol. 123, no. 1-2, pp. 177–199, 2000.
• O. Axelsson, Iterative Solution Methods, Cambridge University Press, New York, NY, USA, 1995.
• M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” Journal of Research of the National Bureau of Standards, vol. 49, pp. 409–436, 1952.
• H. A. van der Vorst and J. B. M. Melissen, “A Petrov-Galerkin type method for solving $Ax=b$, where A is symmetric complex,” IEEE Transactions on Magnetics, vol. 26, pp. 706–708, 1990.
• R. W. Freund, “Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices,” SIAM Journal on Scientific Computing, vol. 13, no. 1, pp. 425–448, 1992.
• Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM Journal on Scientific Computing, vol. 7, no. 3, pp. 856–869, 1986.
• S.-L. Wu, T.-Z. Huang, L. Li, and L.-L. Xiong, “Positive stable preconditioners for symmetric indefinite linear systems arising from Helmholtz equations,” Physics Letters A, vol. 373, no. 29, pp. 2401–2407, 2009.