Citation: | Chengliang Li, Changfeng Ma. THE MODIFIED ASSOR-LIKE METHOD FOR SADDLE POINT PROBLEMS[J]. Journal of Applied Analysis & Computation, 2021, 11(4): 1718-1730. doi: 10.11948/20200039 |
In this paper, we established the modified accelerated symmetric SOR-like (MASSOR) method for solving the large sparse saddle point systems of linear equations. The convergence of the MASSOR method for solving saddle point problems is analyzed. Numerical examples are presented to show the effectiveness of the proposed method.
[1] | F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York and London, 1991. |
[2] | Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 2003, 24, 603-626. doi: 10.1137/S0895479801395458 |
[3] | Z. Bai, G. H. Golub and J. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semi-definite linear systems, Numer. Math. 2004, 98, 1-32. doi: 10.1007/s00211-004-0521-1 |
[4] | Z. Bai and G. H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal. 2007, 27, 1-23. doi: 10.1093/imanum/drl017 |
[5] | Z. Bai, Optimal parameters in the HSS-like methods for saddle-point problems, Numer. Linear Algebra Appl. 2009, 16, 447-479. doi: 10.1002/nla.626 |
[6] | Z. Bai, G. H. Golub, L. Lu and J. Yin, Block Triangular and Skew-Hermitian Splitting Methods for Positive-Definite Linear Systems, SIAM J. Sci. Comput. 2005, 26, 844-863. doi: 10.1137/S1064827503428114 |
[7] | Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing, 2010, 89, 171-197. doi: 10.1007/s00607-010-0101-4 |
[8] | Z. Bai and M. Benzi, Regularized HSS iteration methods for saddle-point linear systems, BIT Numer. Math. 2017, 57, 287-311. doi: 10.1007/s10543-016-0636-7 |
[9] | Z. Bai, Regularized HSS iteration methods for stabilized saddle-point problems, IMA J. Numer. Anal. 2018, 00, 1-36. |
[10] | J. H. Bramble, J. E. Pasciak and A. T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 1997, 34, 1072-1092. doi: 10.1137/S0036142994273343 |
[11] | Z. Bai and Z. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008) 2900-2932. |
[12] | M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl. 2004, 26, 20-41. doi: 10.1137/S0895479802417106 |
[13] | M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numer. 2005, 14, 1-137. doi: 10.1017/S0962492904000212 |
[14] | F. P. A. Beik and M. Benzi, Iterative methods for double saddle point systems, SIAM J. Matrix Anal. Appl. 2018, 39, 602-621. |
[15] | Z. Bai, B. N. Parlett and Z. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 2005, 102, 1-38. doi: 10.1007/s00211-005-0643-0 |
[16] | Y. Cao, J. Du and Q. Niu, Shift-splitting preconditioners for saddle point problems, J. Comput. Appl. Math. 2014, 272, 239-250. doi: 10.1016/j.cam.2014.05.017 |
[17] | C. Chen and C. Ma, A generalized shift-splitting preconditioner for saddle point problems, Appl. Math. Lett. 2015, 43, 49-55. doi: 10.1016/j.aml.2014.12.001 |
[18] | N. Dyn and W. E. Ferguson, The numerical solution of equality constrained quadratic programming problems, Math. Comput. 1983, 41, 165-170. |
[19] | M. T. Darvishi and P. Hessari, Symmetric SOR method for augmented systems, Appl. Math. Comput. 2006, 183, 409-415. |
[20] | M. T. Darvishi and P. Hessari, A modified symmetric successive overrelaxation method for augmented systems, Comput. Math. Appl. 2011, 61, 3128-3135. doi: 10.1016/j.camwa.2011.03.103 |
[21] | H. Elman and D. Silvester, Fast nonsymmetric iteration and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comput. 1996, 17, 33-46. doi: 10.1137/0917004 |
[22] | H. Elman and D. Silvester, Fast nonsymmetric iteration and preconditioning for Navier-Stokes equations, SIAM J. Sci. Comput. 1996, 17, 33-46. doi: 10.1137/0917004 |
[23] | H. C. Elman and G. H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Appl. 1994, 31 1645-1661. doi: 10.1137/0731085 |
[24] | H. C. Elman, Preconditioners for saddle-point problems arising in computational fluid dynamics, Appl. Numer. Math. 2002, 43, 75-89. doi: 10.1016/S0168-9274(02)00118-6 |
[25] | B. Fischer, A. Ramage, D. J. Silvester and A. J. Wathen, Minimum residual methods for augmented systems, BIT Numer. Math. 1998, 38, 527-543. doi: 10.1007/BF02510258 |
[26] | G. H. Golub, X. Wu and J. Yuan, SOR-like methods for augmented system, BIT Numer. Math. 2001, 41, 71-85. doi: 10.1023/A:1021965717530 |
[27] | P. Guo, C. Li and S. Wu, A modified SOR-like method for the augmented systems, J. Comput. Appl. Math. 2015, 274, 58-69. doi: 10.1016/j.cam.2014.07.002 |
[28] | Z. Huang, L. Wang, Z. Xu and J. Cui, Generalized ASOR and Modified ASOR methods for saddle point problems, Math. Probl. Eng. 2016, 2, 1-18. |
[29] | Z. Huang, L. Wang, Z. Xu and J. Cui, A modified generalized shift-splitting preconditioner for nonsymmetric saddle point problems, Numer. Algor. 2018, 78, 297-331. doi: 10.1007/s11075-017-0377-y |
[30] | Q. Hu and J. Zou, An iterative method with variable relaxation parameters for saddle-point problems, SIAM J. Matrix Anal. Appl. 2001, 23, 317-338. doi: 10.1137/S0895479899364064 |
[31] | A. Klawonn, Block-triangular preconditioners for saddle point problems with a penalty term, SIAM J. Sci. Comput. 1998, 19, 172-184. doi: 10.1137/S1064827596303624 |
[32] | C. Li, B. Li and D. J. Evans, A generalized successive overrelaxation method for the least squares problems, BIT Numer. Math. 1998, 38, 347-356. doi: 10.1007/BF02512371 |
[33] | J. Li and N. Zhang, A triple-parameter modified SSOR method for solving singular saddle point problems, BIT Numer. Math. 2016, 56, 1-21. doi: 10.1007/s10543-015-0558-9 |
[34] | Z. Liang and G. Zhang, Modified unsymmetric SOR method for saddle-point problems, Appl. Math. Comput. 234 (2014) 584-598. |
[35] | C. Li and C. Ma, An accelerated symmetric SOR-like method for augmented systems, Appl. Math. Comput. 2019, 341, 408-417. |
[36] | H. S. Najafi and S. A. Edalatpanah, A new modified SSOR iteration method for solving augmented linear systems, Int. J. Comput. Math. 2014, 91, 539-552. doi: 10.1080/00207160.2013.792923 |
[37] | P. N. Njeru and X. Guo, Accelerated SOR-like method for augmented linear systems, BIT Numer. Math. 2016, 56, 557-571. doi: 10.1007/s10543-015-0571-z |
[38] | I. Perugia and V. Simoncini, Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations, Numer. Linear Algebra Appl. 2000, 7, 585-616. doi: 10.1002/1099-1506(200010/12)7:7/8<585::AID-NLA214>3.0.CO;2-F |
[39] | V. Simoncini, Block triangular preconditioners for symmetric saddle-point problems, Appl. Numer. Math. 2004, 49, 63-80. doi: 10.1016/j.apnum.2003.11.012 |
[40] | X. Shao, L. Zheng and C. Li, Modified SOR-like method for augmented systems, Int. J. Comput. Math. 2007, 84, 1653-1662. |
[41] | S. Wu, T. Huang and X. Zhao, A modified SSOR iterative method for augmented systems, J. Comput. Appl. Math. 2009, 228, 424-433. doi: 10.1016/j.cam.2008.10.006 |
[42] | C. Wen and T. Huang, Modified SSOR-like method for augmented systems, Math. Model. Anal. 2011, 16, 475-487. doi: 10.3846/13926292.2011.603165 |
[43] | H. Wang and Z. Huang. On a new SSOR-like method with four parameters for the augmented systems, East Asian J. Appl. Math. 2017, 7, 82-100. doi: 10.4208/eajam.190716.081116a |
[44] | H. Wang and Z. Huang. On convergence and semi-convergence of SSOR-like methods for augmented linear systems, Appl. Math. Comput. 2018, 326, 87-104. |
[45] | D. M. Young, Iterative Solution for Large Linear Systems, Academic press. New York. 1971. |
[46] | J. Yuan, Numerical methods for generalized least squares problems, J. Comput. Appl. Math. 1996, 66, 571-584. doi: 10.1016/0377-0427(95)00167-0 |
[47] | B. Zheng, Z. Bai and X. Yang, On semi-convergence of parameterized Uzawa methods for singular saddle point problems, Linear Algebra Appl. 2009, 431, 808-817. doi: 10.1016/j.laa.2009.03.033 |
[48] | B. Zheng, K. Wang and Y. Wu, SSOR-like methods for saddle point problems, Int. J. Comput. Math. 2009, 86, 1405-1423. doi: 10.1080/00207160701871835 |