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.
Similar content being viewed by others
References
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)
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)
Beik, F.P.A., Benzi, M.: Iterative methods for double saddle point systems. SIAM J. Matrix Anal. Appl. 39, 902–921 (2018)
Beik, F.P.A., Benzi, M.: Block preconditioners for saddle point systems arising from liquid crystal directors modeling. CALCOLO 55, 29 (2018)
Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182, 418–477 (2002)
Benzi, M., Golub, G.H., Liesen, J.: Numerical Solution of Saddle Point Problems. Acta Numer. 14, 1–137 (2005)
Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientic, Nashua (1999)
Cao, Z.-H.: Positive stable block triangular preconditioners for symmetric saddle point problems. Appl. Numer. Math. 57, 899–910 (2007)
Cao, Y.: Shift-splitting preconditioners for a class of block three-by-three saddle point problems. Appl. Math. Lett. 96, 40–46 (2019)
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)
Cao, Y., Du, J., Niu, Q.: Shift-splitting preconditioners for saddle point problems. J. Comput. Appl. Math. 270, 239–250 (2014)
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)
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)
Chen, C.-R., Ma, C.-F.: A generalized shift-splitting preconditioner for singular saddle point problems. Appl. Math. Comput. 269, 947–955 (2015)
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)
Ciarlet, P., Zou, J.: Finite element convergence for the Darwin model to Maxwell’s equations. RAIRO Math. Modelling Numer. Anal. 31, 213–249 (1997)
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)
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)
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)
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)
Huang, N.: Variable parameter Uzawa method for solving a class of block three-by-three saddle point problems. Numer. Algor. 85, 1233–1254 (2020)
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)
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)
Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–629 (1975)
Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 461–469 (1993)
Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS Press, New York (1995)
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)
Salkuyeh, D.K., Masoudi, M., Hezari, D.: On the generalized shift-splitting preconditioner for saddle point problems. Appl. Math. Lett. 48, 55–61 (2015)
Shen, Q.-Q., Shi, Q.: Generalized shift-splitting preconditioners for nonsingular and singular generalized saddle point problems. Comput. Math. Appl. 72, 632–641 (2016)
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)
Young, D.M.: Iterative Solution or Large Linear Systems. Academic Press, New York (1971)
Yuan, J.-Y.: Numerical methods for generalized least squares problems. J. Comput. Appl. Math. 66, 571–584 (1996)
Acknowledgements
The authors would like to thank the referees for their useful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13160-022-00561-8