Skip to main content Accessibility help

A New Uzawa-Type Iteration Method for Non-Hermitian Saddle-Point Problems

  • Yan Dou (a1), Ai-Li Yang (a1) and Yu-Jiang Wu (a1)


Based on a preconditioned shift-splitting of the (1,1)-block of non-Hermitian saddle-point matrix and the Uzawa iteration method, we establish a new Uzawa-type iteration method. The convergence properties of this iteration method are analyzed. In addition, based on this iteration method, a preconditioner is proposed. The spectral properties of the preconditioned saddle-point matrix are also analyzed. Numerical results are presented to verify the robustness and the efficiency of the new iteration method and the preconditioner.


Corresponding author

*Corresponding author. Email address: (Y.-J. Wu)


Hide All
[1] Arrow, K. J., Hurwicz, L., and Uzawa, H., Studies in Linear and Nonlinear Programming, Stanford University Press, Stanford, 1958.
[2] Bai, Z.-Z., Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput., 75 (2006), pp. 791815.
[3] Bai, Z.-Z., Optimal parameters in the HSS-like methods for saddle-point problems, Numer. Linear Algebra Appl., 16 (2009), pp. 447479.
[4] Bai, Z.-Z., Block preconditioners for elliptic PDE-constrained optimization problems, Computing, 91 (2011), pp. 379395.
[5] Bai, Z.-Z., Block alternating splitting implicit iteration methods for saddle-point problems from time-harmonic eddy current models, Numer. Linear Algebra Appl., 19 (2012), pp. 914936.
[6] Bai, Z.-Z., and Golub, G. H., Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27 (2007), pp. 123.
[7] Bai, Z.-Z., Golub, G. H., and Pan, J.-Y., Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), pp. 132.
[8] Bai, Z.-Z., and Hadjidimos, A., Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix, Linear Algebra Appl., 463 (2014), pp. 322339.
[9] Bai, Z.-Z., Ng, M. K., and Wang, Z.-Q., Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 410433.
[10] Bai, Z.-Z., Parlett, B. N., and Wang, Z.-Q., On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102 (2005), pp. 138.
[11] Bai, Z.-Z., and Wang, Z.-Q., On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl., 428 (2008), pp. 29002932.
[12] Bai, Z.-Z., Yin, J.-F., and Su, Y.-F., A shift-splitting preconditioner for non-Hermitian positive definite matrices, J. Comput. Math., 24 (2006), pp. 539552.
[13] Benzi, M., Golub, G. H., and Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1137.
[14] Bramble, J. H., Pasciak, J. E., and Vassilev, A. T., Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 10721092.
[15] Brezzi, F., and Fortin, M., Mixed and hybrid finite elements methods, Springer Series in Computational Mathematics, Vol. 15, Springer-Verlag, New York, 2012.
[16] Cao, Y., Du, J., and Niu, Q., Shift-splitting preconditioners for saddle point problems, J. Comput. Appl. Math., 272 (2014), pp. 239250.
[17] Cao, Y., Yao, L.-Q., Jiang, M.-Q., and Niu, Q., A relaxed HSS preconditioner for saddle point problems from meshfree discretization, J. Comput. Math., 31 (2013), pp. 398421.
[18] Cao, Y., and Yi, S.-C., A class of Uzawa–PSS iteration methods for nonsingular and singular non-Hermitian saddle point problems, Appl. Math. Comput., 275 (2016), pp. 4149.
[19] Elman, H. C., and Golub, G. H., Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal., 31 (1994), pp. 16451661.
[20] Fischer, B., Ramage, A., Silvester, D. J., and Wathen, A. J., Minimum residual methods for augmented systems, BIT Numer. Math., 38 (1998), pp. 527543.
[21] Golub, G. H., and Wathen, A. J., An iteration for indefinite systems and its application to the Navier-Stokes equations, SIAM J. Sci. Comput., 19 (1998), pp. 530539.
[22] Jiang, M.-Q., and Cao, Y., On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 231 (2009), pp. 973982.
[23] Keller, C., Gould, N. I. M., and Wathen, A. J., Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 13001317.
[24] Leem, K. H., Oliveira, S. P., and Stewart, D. E., Algebraic multigrid (AMG) for saddle point systems from meshfree discretizations, Numer. Linear Algebra Appl., 11 (2004), pp. 293308.
[25] Miller, J. J. H., On the location of zeros of certain classes of polynomials with applications to numerical analysis, J. Inst. Math. Appl., 8 (1971), pp. 397406.
[26] Saad, Y., Iterative Methods for Sparse Linear Systems, Second Edition, Society for Industrial and Applied Mathematics, Philadelphia, 2003.
[27] Santos, C. H., Silva, B. P. B., and Yuan, J.-Y., Block SOR methods for rank-deficient least-squares problems, J. Comput. Appl. Math., 100 (1998), pp. 19.
[28] Wu, X., Silva, B. P. B., and Yuan, J.-Y., Conjugate gradient method for rank deficient saddle point problems, Numer. Algor., 35 (2004), pp. 139154.
[29] Yang, A.-L., Li, X., and Wu, Y.-J., On semi-convergence of the Uzawa–HSS method for singular saddle-point problems, Appl. Math. Comput., 252 (2015), pp. 8898.
[30] Yang, A.-L., and Wu, Y.-J., The Uzawa–HSSmethod for saddle-point problems, Appl. Math. Lett., 38 (2014), pp. 3842.
[31] Yun, J.-H., Variants of the Uzawa method for saddle point problems, Comput. Math. Appl., 65 (2013), pp. 10371046.
[32] Zhang, J.-J., and Shang, J.-J., A class of Uzawa–SOR methods for saddle point problems, Appl. Math. Comput., 216 (2010), pp. 21632168.


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed