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Computational and Applied Mathematics

, Volume 37, Issue 3, pp 3385–3398 | Cite as

The parameterized preconditioner for the generalized saddle point problems from the incompressible Navier–Stokes equations

  • Yi-Fen Ke
  • Chang-Feng Ma
Article
  • 134 Downloads

Abstract

A parameterized preconditioner is proposed for the generalized saddle point problems arising from the incompressible Navier–Stokes equations. The eigenvalues and eigenvectors of the preconditioned matrix are analyzed. Numerical results show that the proposed preconditioner is efficient to accelerate the convergence rate of Krylov subspace methods, such as GMRES method.

Keywords

Generalized saddle point problem Eigenvalue Preconditioner Navier–Stokes equation 

Mathematics Subject Classification

65F10 65F08 65F50 

Notes

Acknowledgements

The authors would like to express their sincere thanks to the anonymous referees for useful comments, which greatly improved the original manuscript of this paper.

References

  1. Bai Z-Z (2006) Structured preconditioners for nonsigular matrices of block two-by-two structures. Math Comput 75:791–815CrossRefzbMATHGoogle Scholar
  2. Bai Z-Z, Golub GH (2007) Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J Numer Anal 27:1–23MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bai Z-Z, Ng MK (2005) On inexact preconditioners for nonsymmetric matrices. SIAM J Sci Comput 26:1710–1724MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bai Z-Z, Wang Z-Q (2008) On parameterized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl 428:2900–2932MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bai Z-Z, Golub GH, Li CK (2007) Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math Comput 76:287–298MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bai Z-Z, Ng MK, Wang Z-Q (2009) Constraint preconditioners for symmetric indefinite matrices. SIAM J Matrix Anal Appl 31:410–433MathSciNetCrossRefzbMATHGoogle Scholar
  7. Benzi M, Golub GH (2004) A preconditioner for generalized saddle point problems. SIAM J Matrix Anal Appl 26:20–41MathSciNetCrossRefzbMATHGoogle Scholar
  8. Benzi M, Guo X-P (2011) A dimensional split preconditioner for Stokes and linearized Navier–Stokes equations. Appl Numer Math 61:66–76MathSciNetCrossRefzbMATHGoogle Scholar
  9. Benzi M, Wang Z (2011) Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier–Stokes equations. SIAM J Sci Comput 33:2761–2784MathSciNetCrossRefzbMATHGoogle Scholar
  10. Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14:1–137MathSciNetCrossRefzbMATHGoogle Scholar
  11. Benzi M, Ng M, Niu Q, Wang Z (2011) A relaxed dimensional factorization preconditioner for the incompressible Navier–Stokes equations. J Comput Phys 230:6185–6202MathSciNetCrossRefzbMATHGoogle Scholar
  12. Cao Z-H (2003) Fast Uzawa algorithm for generalized saddle point problems. Appl Numer Math 46:157–171MathSciNetCrossRefzbMATHGoogle Scholar
  13. Cao Y, Jiang M-Q, Zheng Y-L (2011) A splitting preconditioner for saddle point problems. Numer Linear Algebra Appl 18:875–895MathSciNetCrossRefzbMATHGoogle Scholar
  14. Cao Y, Yao L-Q, Jiang M-Q (2013) A modified dimensional split preconditioner for generalized saddle point problems. J Comput Appl Math 250:70–82MathSciNetCrossRefzbMATHGoogle Scholar
  15. Cao Y, Dong J-L, Wang Y-M (2015) A relaxed deteriorated PSS preconditioner for nonsymmetric saddle point problems from the steady Navier–Stokes equation. J Comput Appl Math 273:41–60MathSciNetCrossRefzbMATHGoogle Scholar
  16. Cao Y, Miao S-X, Cui Y-S (2015) A relaxed splitting preconditioner for generalized saddle point problems. Comput Appl Math 34:865–879MathSciNetCrossRefzbMATHGoogle Scholar
  17. Elman H, Silvester D, Wathen A (2014) Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. Oxford University Press, OxfordCrossRefzbMATHGoogle Scholar
  18. 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
  19. Fan H-T, Zheng B, Zhu X-Y (2016) A relaxed positive semi-definite and skew-Hermitian splitting preconditioner for non-Hermitian generalized saddle point problems. Numer Algorithms 72:813–834MathSciNetCrossRefzbMATHGoogle Scholar
  20. Huang Z-H, Huang T-Z (2010) Spectral properties of the preconditioned AHSS iteration method for generalized saddle point problems. Comput Appl Math 29:269–295MathSciNetzbMATHGoogle Scholar
  21. Huang T-Z, Wu S-L, Li C-X (2009) The spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for generalized saddle point problems. J Comput Appl Math 229:37–46MathSciNetCrossRefzbMATHGoogle Scholar
  22. Ke Y-F, Ma C-F (2016) Spectrum analysis of a more general augmentation block preconditioner for generalized saddle point matrices. BIT Numer Math 56:489–500MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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
  24. Lu J-F, Jin Y-M (2015) Eigenvalue analysis of a generalized indefinite block triangular preconditioner for generalized saddle point problems. Appl Math Comput 271:389–399MathSciNetGoogle Scholar
  25. Miao S-X, Cao Y (2014) A note on GPIU method for generalized saddle point problems. Appl Math Comput 230:27–34MathSciNetzbMATHGoogle Scholar
  26. Shen H-L, Xiang H (2015) Uzawa algorithms with variable relaxation for nonsymmetric generalized saddle point problems. Numer Linear Algebra Appl 22:1020–1038MathSciNetCrossRefzbMATHGoogle Scholar
  27. Tan N-B, Huang T-Z, Hu Z-J (2012) A relaxed splitting preconditioner for the incompressible Navier–Stokes equations. J Appl Math 1–12, Art ID 402490Google Scholar
  28. Wu X-N, Golub GH, Cuminato JA, Yuan J-Y (2008) Symmetric-triangular decomposition and its applications part II: preconditioners for indefinite systems. BIT 48:139–162MathSciNetCrossRefzbMATHGoogle Scholar
  29. Zhang G-F, Ren Z-R, Zhou Y-Y (2011) On HSS-based constraint preconditioners for generalized saddle-point problems. Numer Algorithms 57:273–287MathSciNetCrossRefzbMATHGoogle Scholar
  30. Zheng L, Zhang T, Li C-J (2011) Parameterized preconditioning for generalized saddle point problems arising from the Stokes equation. J Comput Appl Math 236:1511–1520MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  1. 1.College of Mathematics and Informatics and FJKLMAAFujian Normal UniversityFuzhouPeople’s Republic of China
  2. 2.Key Laboratory of Computational GeodynamicsChinese Academy of SciencesBeijingPeople’s Republic of China

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