Skip to main content
Log in

A block triangular preconditioner for a class of three-by-three block saddle point problems

  • Original Paper
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

This paper deals with solving a class of three-by-three block saddle point problems. The systems are solved by preconditioning techniques. Based on an iterative method, we construct a block upper triangular preconditioner. The convergence of the presented method is studied in details. Finally, some numerical experiments are given to demonstrate the superiority of the proposed preconditioner over some existing ones.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Abdolmaleki, M., Karimi, S., Salkuyeh, D.K.: A new block-diagonal preconditioner for a class of 3\(\times \)3 block saddle point problems. Mediterr. J. Math. 19, 1–15 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  2. Assous, F., Degond, P., Heintze, E., Raviart, P.A., Segre, J.: On a finite-element method for solving the three-dimensional Maxwell equations. J. Comput. Phys. 109, 222–237 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beik, F.P.A., Benzi, M.: Iterative methods for double saddle point systems. SIAM J. Matrix Anal. Appl. 39, 902–921 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beik, F.P.A., Benzi, M.: Block preconditioners for saddle point systems arising from liquid crystal directors modeling. CALCOLO 55, 29 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182, 418–477 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benzi, M., Golub, G.H., Liesen, J.: Numerical Solution of Saddle Point Problems. Acta Numer. 14, 1–137 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientic, Nashua (1999)

    MATH  Google Scholar 

  8. Cao, Z.-H.: Positive stable block triangular preconditioners for symmetric saddle point problems. Appl. Numer. Math. 57, 899–910 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cao, Y.: Shift-splitting preconditioners for a class of block three-by-three saddle point problems. Appl. Math. Lett. 96, 40–46 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cao, Y.: A general class of shift-splitting preconditioners for non-Hermitian saddle point problems with applications to time-harmonic eddy current models. Comput. Math. Appl. 77, 1124–1143 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cao, Y., Du, J., Niu, Q.: Shift-splitting preconditioners for saddle point problems. J. Comput. Appl. Math. 270, 239–250 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cao, Y., Li, S., Yao, L.: A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems. Appl. Math. Lett. 49, 20–27 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cao, Y., Miao, S.-X., Ren, Z.-R.: On preconditioned generalized shift-splitting iteration methods for saddle point problems. Comput. Math. Appl. 74, 859–872 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, C.-R., Ma, C.-F.: A generalized shift-splitting preconditioner for singular saddle point problems. Appl. Math. Comput. 269, 947–955 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Chen, Z.-M., Du, Q., Zou, J.: Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer Anal. 37, 1542–1570 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ciarlet, P., Zou, J.: Finite element convergence for the Darwin model to Maxwell’s equations. RAIRO Math. Modelling Numer. Anal. 31, 213–249 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Elman, H.C., Silvester, D.J., Wathen, A.J.: Performance and analysis of saddle point preconditioners for the discrete steady-state Navier–Stokes equations. Numer. Math. 90, 665–688 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Estrin, R., Greif, C.: Towards an optimal condition number of certain augmented Lagrangian-type saddle-point matrices. Numer. Linear Algebra Appl. 23, 693–705 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gould, N.I.M., Orban, D., Toint, P.L.: CUTEr and SifDec, a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29, 373–394 (2003)

    Article  MATH  Google Scholar 

  20. Han, D.R., Yuan, X.M.: Local linear convergence of the alternating direction method of multipliers for quadratic programs. SIAM J. Numer. Anal. 51, 3446–3457 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huang, N.: Variable parameter Uzawa method for solving a class of block three-by-three saddle point problems. Numer. Algor. 85, 1233–1254 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huang, N., Ma, C.-F.: Spectral analysis of the preconditioned system for the 3\(\times \)3 block saddle point problem. Numer. Algor. 81, 421–444 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ke, Y.-F., Ma, C.-F.: The parameterized preconditioner for the generalized saddle point problems from the incompressible Navier–Stokes equations. J. Comput. Appl. Math. 37, 3385–3398 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–629 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  25. Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 461–469 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  26. Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Press, New York (1995)

    Google Scholar 

  27. Salkuyeh, D.K., Rahimian, M.: A modification of the generalized shift-splitting method for singular saddle point problems. Comput. Math. Appl. 74, 2940–2949 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  28. Salkuyeh, D.K., Masoudi, M., Hezari, D.: On the generalized shift-splitting preconditioner for saddle point problems. Appl. Math. Lett. 48, 55–61 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shen, Q.-Q., Shi, Q.: Generalized shift-splitting preconditioners for nonsingular and singular generalized saddle point problems. Comput. Math. Appl. 72, 632–641 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xie, X., Li, H.-B.: A note on preconditioning for the \(3 \times 3\) block saddle point problem. Comput. Math. Appl. 79, 3289–3296 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Young, D.M.: Iterative Solution or Large Linear Systems. Academic Press, New York (1971)

    Google Scholar 

  32. Yuan, J.-Y.: Numerical methods for generalized least squares problems. J. Comput. Appl. Math. 66, 571–584 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their useful comments and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Davod Khojasteh Salkuyeh.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aslani, H., Salkuyeh, D.K. A block triangular preconditioner for a class of three-by-three block saddle point problems. Japan J. Indust. Appl. Math. 40, 1015–1030 (2023). https://doi.org/10.1007/s13160-022-00561-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-022-00561-8

Keywords

Mathematics Subject Classification

Navigation