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Computational and Applied Mathematics

, Volume 37, Issue 3, pp 3580–3592 | Cite as

GSTS-Uzawa method for a class of complex singular saddle point problems

  • Jin-Song Xiong
  • Xing-Bao Gao
Article

Abstract

In this paper, we propose GSTS–Uzawa method for solving a class of complex singular saddle point problems based on generalized skew-Hermitian triangular splitting (GSTS) iteration method and classical Uzawa method. We research on its semi-convergence properties and the eigenvalues distributions of its preconditioned matrix. The resulting GSTS–Uzawa preconditioner is used to precondition Krylov subspace methods such as the restarted generalized minimal residual (GMRES) method for solving the equivalent formulation of the complex singular saddle point problems. The theoretical results and effectiveness of the GSTS–Uzawa method are supported by a numerical example.

Keywords

Complex singular saddle point problems GSTS-Uzawa method Iterative method Semi-convergence 

Mathematics Subject Classification

65F10 65F08 

Notes

Acknowledgements

The authors would like to express our sincere thanks to the anonymous reviewers for their valuable suggestions which greatly improved the presentation of this paper. The authors are supported by the National Natural Science Foundation of China under Grant no. 61273311.

References

  1. Arrow K, Hurwicz L, Uzawa H (1958) A studies in linear and nonlinear programming. Stanford University Press, StanfordzbMATHGoogle Scholar
  2. Bai ZZ, Wang ZQ (2008) On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl 428(11–12):2900–2932MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bai ZZ, Parlett BN, Wang ZQ (2005) On generalized successive overrelaxation methods for augmented linear systems. Numerische Mathematik 102(1):1–38MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bai Z-Z, Parlett BN, Wang Z-Q (2005) On generalized successive overrelaxation methods for augmented linear systems. Numer Math 102:1–38MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bai Z-Z, Benzi M, Chen F (2010) Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87:93–111MathSciNetCrossRefzbMATHGoogle Scholar
  6. Benzi M, Liu J (2007) An efficient solver for the incompressible Navier–Stokes equations in rotation form. SIAM J Sci Comput 29(5):1959–1981MathSciNetCrossRefzbMATHGoogle Scholar
  7. Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numerica 14(2):1–137MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bertaccini D (2004) Efficient solvers for sequences of complex symmetric linear systems. Electron Trans Numer Anal 18:49–64MathSciNetzbMATHGoogle Scholar
  9. Betts JT (2001) Practical methods for optimal control using nonlinear programming. SIAM, PhiladelphiazbMATHGoogle Scholar
  10. Bramble JH, Pasciak JE, Vassilev AT (2000) Uzawa type algorithms for nonsymmetric saddle point problems. Am Math Soc 69(230):667–689MathSciNetzbMATHGoogle Scholar
  11. Cao ZH (2003) Fast Uzawa algorithm for generalized saddle point problems. Appl Numer Math 46(2):157–171MathSciNetCrossRefzbMATHGoogle Scholar
  12. Cao ZH (2004) Fast Uzawa algorithm for solving non-symmetric stabilized saddle point problems. Numer Linear Algebra Appl 11(1):1–24MathSciNetCrossRefzbMATHGoogle Scholar
  13. Cao Y, Yi SC (2016) A class of Uzawa-PSS iteration methods for nonsingular and singular non-Hermitian saddle point problems. Appl Math Comput 275:41–49MathSciNetGoogle Scholar
  14. Chao Z, Chen GL (2014) Semi-convergence analysis of the Uzawa-SOR methods for singular saddle point problems. Appl Math Lett 35:52–57MathSciNetCrossRefzbMATHGoogle Scholar
  15. Elman HC, Golub GH (1994) Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J Numer Anal 31:1645–1661MathSciNetCrossRefzbMATHGoogle Scholar
  16. Elman HC, Silvester DJ, Wathen AJ (2002) Performance and analysis of saddle point preconditioners for the discrete steady-state Navier–Stokes equations. Numer Math 90:665–688MathSciNetCrossRefzbMATHGoogle Scholar
  17. Elman HC, Ramage A, Silvester DJ (2007) A MatLab toolbox for modelling incompressible flow. ACM Trans Math Softw 33:1–18CrossRefzbMATHGoogle Scholar
  18. Huang ZH, Su H (2017) A modified shift-splitting method for nonsymmetric saddle point problems. J Comput Appl Math 317:535–546MathSciNetCrossRefzbMATHGoogle Scholar
  19. Krukier LA, Krukier BL, Ren ZR (2014) Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems. Linear Algebra Appl 21(1):152–170MathSciNetCrossRefzbMATHGoogle Scholar
  20. Krukier LA, Krukier BL, Ren Z-R (2014) Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems. Numer Linear Algebra Appl 21:152–170MathSciNetCrossRefzbMATHGoogle Scholar
  21. Li X, Wu YJ, Yang AL, Yuan JY (2014) Modified accelerated parameterized inexact Uzawa method for singular and nonsingular saddle point problems. Appl Math Comput 244:552–560MathSciNetzbMATHGoogle Scholar
  22. Liang ZZ, Zhang GF (2014) On semi-convergence of a class of Uzawa methods for singular saddle-point problems. Appl Math Comput 247:397–409MathSciNetzbMATHGoogle Scholar
  23. Liang ZZ, Zhang GF (2017) Convergence behavior of generalized parameterized Uzawa method for singular saddle-point problems. J Comput Appl Math 311:293–305MathSciNetCrossRefzbMATHGoogle Scholar
  24. Liesen J, de Sturler E, Sheffer A, Aydin Y, Siefert C (2001) Preconditioners for indefinite linear systems arising in surface parameterization. In: Proc. 10th international meshing round table, pp 71–81Google Scholar
  25. Ma HF, Zhang N-M (2011) A note on block-diagonally preconditioned PIU methods for singular saddle point problems. Int J Comput Math 88(16):3448–3457MathSciNetCrossRefzbMATHGoogle Scholar
  26. Miller JJH (1971) On the location of zeros of certain classes of polynomials with applications to numerical analysis. J Inst Math Appl 8:397–406MathSciNetCrossRefzbMATHGoogle Scholar
  27. Perugia I, Simoncini V (2000) Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations. Numer Linear Algebra Appl 7:585–616MathSciNetCrossRefzbMATHGoogle Scholar
  28. Shao XH, Li C (2015) A generalization of preconditioned parameterized inexact Uzawa method for indefinite saddle point problems. Appl Math Comput 269(15):691–698MathSciNetGoogle Scholar
  29. Song Y-Z (2001) Semiconvergence of block SOR method for singular linear systems with \(p\)-cyclic matrices. J Comput Appl Math 130:217–229MathSciNetCrossRefzbMATHGoogle Scholar
  30. Xiong JS, Gao XB (2017) Semi-convergence analysis of Uzawa-AOR method for singular saddle point problems. Comput Appl Math 36(1):383–395MathSciNetCrossRefzbMATHGoogle Scholar
  31. Yang AL, Li X, Wu YJ (2015) On semi-convergence of the Uzawa-HSS method for singular saddle-point problems. Appl Math Comput 252:88–98MathSciNetzbMATHGoogle Scholar
  32. Yuan JY (1993) Iterative methods for generalized least squares problems. Ph.D. Thesis, IMPA, BrazilGoogle Scholar
  33. Yuan JY, Iusem AN (1996) Preconditioned conjugate gradient method for generalized least squares problems. J Comput Appl Math 71:287–297MathSciNetCrossRefzbMATHGoogle Scholar
  34. Yuan JY, Sampaio RJB, Sun W (1996) Algebraic relationships between updating and downdating least squares problems. Numer Math J Chin Univ 3:203–210 (in Chinese)zbMATHGoogle Scholar
  35. Zhang N-M, Wei Y-M (2010) On the convergence of general stationary iterative methods for range-Hermitian singular linear systems. Numer Linear Algebra Appl 17:139–154MathSciNetCrossRefzbMATHGoogle Scholar
  36. Zheng B, Bai ZZ, Yang X (2009) On semi-convergence of parameterized Uzawa methods for singular saddle point problems. Linear Algebra Appl 431(5–7):808–817MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina

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