Advertisement

Computational and Applied Mathematics

, Volume 37, Issue 3, pp 3256–3266 | Cite as

The generalized double shift-splitting preconditioner for nonsymmetric generalized saddle point problems from the steady Navier–Stokes equations

  • Hong-Tao Fan
  • Xin-Yun Zhu
  • Bing Zheng
Article
  • 178 Downloads

Abstract

In this paper, a generalized double shift-splitting (GDSS) preconditioner induced by a new matrix splitting method is proposed and implemented for nonsymmetric generalized saddle point problems having a nonsymmetric positive definite (1,1)-block and a positive definite (2,2)-block. Detailed theoretical analysis of the iteration matrix is provided to show the GDSS method, which corresponds to the GDSS preconditioner, is unconditionally convergent. Additionally, a deteriorated GDSS (DGDSS) method is proposed. It is shown that, with suitable choice of parameter matrix, the DGDSS preconditioned matrix has an eigenvalue at 1 with multiplicity n, and the other m eigenvalues are of the form \(1-\lambda \) with \(|\lambda |<1\), independently of the Schur complement matrix related. Finally, numerical experiments arising from a model Navier–Stokes problem are provided to validate and illustrate the effectiveness of the proposed preconditioner, with which a faster convergence for Krylov subspace iteration methods can be achieved.

Keywords

Nonsymmetric generalized saddle point problem Generalized double shift-splitting Krylov subspace method Convergence 

Mathematics Subject Classification

65F10 65F50 

Notes

Acknowledgements

The authors are very much indebted to the anonymous referees for their constructive suggestions and insightful comments. The incorporation of these suggestions has greatly improved the original manuscript of this paper. This work is supported by the National Natural Science Foundation of China (nos. 11571004, 11701456).

References

  1. Bai Z-Z, Li G-Q (2003) Restrictively preconditioned conjugate gradient methods for systems of linear equations. IMA J Numer Anal 23:561–580MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bai Z-Z, Zhang S-L (2004) A regular conjugate gradient method for symmetric positive definite system of linear equations. J Comput Math 20:338–437Google Scholar
  3. Bai Z-Z, Golub GH, Ng MK (2003) Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J Matrix Anal Appl 24:603–626MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bai Z-Z, Yin J-F, Su Y-F (2006) A shift-splitting preconditioner for non-Hermitian positive definite matrices. J Comput Math 24:539–552MathSciNetzbMATHGoogle Scholar
  5. Benzi M, Golub GH (2004) A precoditioner for generalized saddle point problems. SIAM J Matrix Anal Appl 26:20–41MathSciNetCrossRefzbMATHGoogle Scholar
  6. Benzi M, Simoncini V (2006) On the eigenvalues of a class of saddle point matrices. Numer. Math. 103:173–196MathSciNetCrossRefzbMATHGoogle Scholar
  7. Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14:1–137MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bergamaschi L, Gondzio J, Zilli G (2004) Preconditioning indefinite systems in interior point methods for optimization. Comput Optim Appl 28:149–171MathSciNetCrossRefzbMATHGoogle Scholar
  9. Botchev MA, Golub GH (2006) A class of nonsymmetric preconditioners for saddle point problems. SIAM J Matrix Anal Appl 27:1125–1149MathSciNetCrossRefzbMATHGoogle Scholar
  10. Bramble JH, Pasciak JE, Vassilev AT (1999) Uzawa type algorithms for nonsymmetric saddle point problems. Math Comput 69:667–689MathSciNetCrossRefzbMATHGoogle Scholar
  11. Brezzi F, Fortin M (1991) Mixed and hybrid finite element methods. Springer, New York and LondonCrossRefzbMATHGoogle Scholar
  12. Cao Z-H (2004) Fast Uzawa algorithms for solving non-symmetric stabilized saddle point problems. Numer Linear Algebra Appl 11:1–24MathSciNetCrossRefzbMATHGoogle Scholar
  13. Cao Y, Du J, Niu Q (2014) Shift-splitting preconditioner for saddle point problems. J Comput Appl Math 272:239–250MathSciNetCrossRefzbMATHGoogle Scholar
  14. Cao Y, Li S, Yao L-Q (2015) A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems. Appl Math Lett 49:20–27MathSciNetCrossRefzbMATHGoogle Scholar
  15. Chen C-R, Ma C-F (2015) A generalized shift-splitting preconditioner for saddle point problems. Appl Math Lett 43:49–55MathSciNetCrossRefzbMATHGoogle Scholar
  16. Dollar HS (2007) Constraint-style preconditioners for regularized saddle point problems. SIAM J Matrix Anal Appl 29:672–684MathSciNetCrossRefzbMATHGoogle Scholar
  17. Elman HC, Ramage A, Silvester DJ (2007) IFISS: a Matlab toolbox for modelling incompressible flow, ACM Trans. Math Softw 33. Article 14Google Scholar
  18. Elman HC, Silvester DJ, Wathen AJ (2005) Finite elements and fast iterative solvers. Numerical mathmatics and scientific computation. Oxford University, OxfordzbMATHGoogle Scholar
  19. Fan H-T, Zhu X-Y (2015) A generalized relaxed positive-definite and skew-Hermitian splitting preconditioner for non-Hermitian saddle point problems. Appl Math Comput 258:36–48MathSciNetzbMATHGoogle Scholar
  20. Fan H-T, Zhu X-Y (2016) A modified relaxed splitting preconditioner for generalized saddle point problems from the incompressible Navier-Stokes equations. Appl Math Lett 55:18–26MathSciNetCrossRefzbMATHGoogle Scholar
  21. Golub GH, Grief C (2003) On solving the block-structuered indefinite linear systems. SIAM J Sci Comput 24:2076–2092MathSciNetCrossRefGoogle Scholar
  22. Golub GH, Wu X, Yuan J-Y (2001) SOR-like methods for augmented systems. BIT Numer Math 41:71–85MathSciNetCrossRefzbMATHGoogle Scholar
  23. Keller C, Gould NI, Wathen AJ (2000) Constraint preconditioning for indefinite linear systems. SIAM J Matrix Anal Appl 21:1300–1317MathSciNetCrossRefzbMATHGoogle Scholar
  24. Krukier LA, Krukier BL, Ren Z-R (2014) Generalized skew-Hermitian triangular splitting iteration methodsfor saddle-point linear systems. Numer Linear Algebra Appl 21:152–170MathSciNetCrossRefzbMATHGoogle Scholar
  25. Pan J-Y, Ng MK, Bai Z-Z (2006) New preconditioners for saddle point problems. Appl Math Comput 172:762–771MathSciNetzbMATHGoogle Scholar
  26. Rusten T, Winther R (1992) A preconditioned iterative method for saddle point problems. SIAM J Matrix Anal Appl 13:887–904MathSciNetCrossRefzbMATHGoogle Scholar
  27. Salkuyeh DK, Masoudi M, Hezari D (2015) A preconditioner based on the shift-splitting method for generalized saddle point problems. arXiv:1506.04661v1 [math.NA]
  28. Salkuyeh DK, Masoudi M, Hezari D (2015) On the generalized shift-splitting preconditioner for saddle point problems. Appl Math Lett 48:55–61MathSciNetCrossRefzbMATHGoogle Scholar
  29. Silvester DJ, Wathen AJ (1994) Fast iterative solution of stabilised Stokes systems. Part II: using general block preconditioners. SIAM J Numer Anal 31:1352–1367MathSciNetCrossRefzbMATHGoogle Scholar
  30. Simoncini V (2004) Block triangular preconditioners for symmetric saddle-point problems. Appl Numer Math 49:63–80MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouChina
  2. 2.Institute of Applied MathematicsCollege of Science, Northwest A&F UniversityYanglingChina
  3. 3.Department of MathematicsUniversity of Texas of the Permian BasinOdessaUSA

Personalised recommendations