# The PPS method-based constraint preconditioners for generalized saddle point problems

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## Abstract

For the large sparse generalized saddle point problems with non-Hermitian (1,1) blocks, we introduce a constraint preconditioner, which is based on the positive definite and semidefinite splitting (PPS) iteration method. Then we discuss one trait of the PPS-based constraint preconditioner, such as invertibility. We give the convergence of conditions of the preconditioning iteration method. Moreover, numerical experiments are given to illustrate that PPS-based constraint preconditioner has an obvious advantage of efficiency.

## Keywords

Generalized saddle point problems Non-Hermitian matrix Positive definite matrix Constraint Preconditioner## Mathematics Subject Classification

65F10 65F08## Notes

### Acknowledgements

The authors would like to express their great thankfulness to the referees for the comments and constructive suggestions, which were valuable in improving the quality of the original paper. The project was supported by the National Natural Science Foundation of China (No. 11371081) and Liaoning Natural Science Foundation (No. 20170540323).

## References

- Bai Z-Z, Golub GH, Lu L-Z, Yin J-F (2005) Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J SciComput 26:844–863MathSciNetCrossRefGoogle Scholar
- Bai Z-Z, Ng MK, Wang Z-Q (2009) Constraint preconditioners for symmetric indefinite matrices. SIAM J Matrix Anal Appl 31:410–433MathSciNetCrossRefGoogle Scholar
- Beik FPA, Benzi M, Chaparpordi SHA (2017) On block diagonal and block triangular iterative schemes and preconditioners for stabilized saddle point problems. J ComputAppl Math 326:15–30MathSciNetCrossRefGoogle Scholar
- Benzi M, Golub GH, Liesen J (2005) Numerical solution of saddle point problems. Acta Numer 14:1–137MathSciNetCrossRefGoogle Scholar
- Botchev MA, Golub GH (2006) A class of nonsymmetric preconditioners for saddle point problems. SIAM J Matrix Anal Appl 27:1125–1149MathSciNetCrossRefGoogle Scholar
- Cao Z-H (2006) A note on block diagonal and constraint preconditioners for non-symmetric indefinite linear systems. Int J Comput Math 83:383–395MathSciNetCrossRefGoogle Scholar
- Cao Z-H (2008) Augmentation block preconditioners for saddle point-type matrices with singular (1,1) blocks. Numer Linear Algebra Appl 15:515–533MathSciNetCrossRefGoogle Scholar
- Cao Y, Jiang M-Q, Zheng Y-L (2011) A splitting preconditioner for saddle point problems. Numer Linear Algebra Appl 18:875–895MathSciNetCrossRefGoogle Scholar
- Cao Y, Niu Q, Jiang M-Q (2012) On PSS-based constraint preconditioners for generalized saddle point problems. Math NumerSinica 34:183–194MathSciNetzbMATHGoogle Scholar
- Cao Y, Yao L-Q, Jiang M-Q, Niu Q (2013) A relaxed HSS preconditioner for saddle point problems from meshfree discretization. J Comput Math 31:398–421MathSciNetCrossRefGoogle Scholar
- Cao Y, Du J, Niu Q (2014) Shift-splitting preconditioners for saddle point problems. J ComputAppl Math 272:239–250MathSciNetCrossRefGoogle Scholar
- Dollar HS (2007) Constraint-style preconditioners for regularized saddle point problems. SIAM J Matrix Anal Appl 29:672–684MathSciNetCrossRefGoogle Scholar
- Elman HC (2002) Preconditioners for saddle point problems arising in computational fluid dynamics. ApplNumer Math 43:75–89MathSciNetzbMATHGoogle Scholar
- Golub GH, Wathen AJ (1998) An iteration for indefinite systems and its application to the Navier-Stokes equations. SIAM J Sci Comput 19:530–539MathSciNetCrossRefGoogle Scholar
- Huang N, Ma C-F (2016) Positive definite and semi-definite splitting methods for non-Hermitian positive definite linear systems. J Comput Math 34:300–316MathSciNetCrossRefGoogle Scholar
- Li C-L and Ma C-F (2017) The Uzawa-PPS iteration methods for nonsingular and singular non-Hermitian saddle point problems. Comput Math ApplGoogle Scholar
- Notay Y (2014) A new analysis of block preconditioners for saddle point problems. SIAM J Matrix Anal Appl 35:143–173MathSciNetCrossRefGoogle Scholar
- Peng X-F, Li W, Xiang S-H (2011) New preconditioners based on symmetric-triangular decomposition for saddle point problems. Comput 93:27–46MathSciNetCrossRefGoogle Scholar
- Simoncini V, Benzi M (2005) Spectral Properties of the Hermitian and Skew-Hermitian splitting preconditioner for saddle point problems. SIAM J Matrix Anal Appl 26:377–389MathSciNetCrossRefGoogle Scholar
- Wu X-N, Golub GH, Cuminato JA, Yuan J-Y (2008) Symmetric-triangular decomposition and its applications part II: preconditioners for indefinite systems. BIT Numer Math 48:139–162CrossRefGoogle Scholar
- Zhang G-F, Ren Z-R, Zhou Y-Y (2011) On HSS-based constraint preconditioners for generalized saddle-point problems. Numer Algorithms 57:273–287MathSciNetCrossRefGoogle Scholar
- Zhu M-Z, Zhang G-F, Zheng Z, Liang Z-Z (2014) On HSS-based sequential two-stage method for non-Hermitian saddle point problems. Appl Math Comput 242:907–916MathSciNetzbMATHGoogle Scholar