159
Views
2
CrossRef citations to date
0
Altmetric
Original Article

Optimization of a parameterized inexact Uzawa method for saddle point problems

&
Pages 2539-2548 | Received 05 Apr 2017, Accepted 25 Oct 2017, Published online: 13 Dec 2017

References

  • Z.-Z. Bai, Optimal parameters in the HSS-like methods for saddle-point problems, Numer. Linear Algebra Appl. 16 (2009), pp. 447–479. doi: 10.1002/nla.626
  • Z.-Z. Bai and G.H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal. 27 (2007), pp. 1–23. doi: 10.1093/imanum/drl017
  • Z.-Z. Bai and Z.-Q. Wang, On parameterized inexact Uzawa methods for generalized saddle point problems, Linear Algebra Appl. 428 (2008), pp. 2900–2932. doi: 10.1016/j.laa.2008.01.018
  • Z.-Z. Bai, G.H. Golub, and M.K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003), pp. 603–626. doi: 10.1137/S0895479801395458
  • Z.-Z. Bai, G.H. Golub, and J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math. 98 (2004), pp. 1–32. doi: 10.1007/s00211-004-0521-1
  • Z.-Z. Bai, B.N. Parlett, and Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math. 102 (2005), pp. 1–38. doi: 10.1007/s00211-005-0643-0
  • Z.-Z. Bai, G.H. Golub, and M.K. Ng, On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl. 14 (2007), pp. 319–335. doi: 10.1002/nla.517
  • Z.-Z. Bai, G.H. Golub, and M.K. Ng, On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl. 428 (2008), pp. 413–440. doi: 10.1016/j.laa.2007.02.018
  • M. Benzi, M.J. Gander, and G.H. Golub, Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems, BIT Numer. Math. 43 (2003), pp. 881–900. doi: 10.1023/B:BITN.0000014548.26616.65
  • J.H. Bramble, J.E. Pasciak, and A.T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34 (1997), pp. 1072–1092. doi: 10.1137/S0036142994273343
  • Y. Cao, M.Q. Jiang, and L.Q. Yao, New choices of preconditioning matrices for generalized inexact parameterized iterative methods, J. Comput. Appl. Math. 235 (2010), pp. 263–269. doi: 10.1016/j.cam.2010.05.054
  • Y. Cao, M.-Q. Jiang, and Y.-L. Zheng, A splitting preconditioner for saddle point problems, Numer. Linear Algebra Appl. 18 (2011), pp. 875–895. doi: 10.1002/nla.772
  • F. Chen and Y.L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206 (2008), pp. 765–771.
  • L. Dai, M. Liang, and H. Fan, A new generalized parameterized inexact Uzawa method for solving saddle point problems, Appl. Math. Comput. 265 (2015), pp. 414–430.
  • H.C. Elman and G.H. Golub, Inexact and preconditioned Uzawa algorithms for saddle point problems, SIAM J. Numer. Anal. 31 (1994), pp. 1645–1661. doi: 10.1137/0731085
  • H.C. Elman, D.J. Silvester, and A.J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd ed., Oxford University Press, Oxford, 2014.
  • M.J. Gander, Q. Niu, and Y.X. Xu, Analysis of a new dimension-wise splitting iteration with selective relaxation for saddle point problems, BIT Numer. Math. 56 (2016), pp. 441–465. doi: 10.1007/s10543-016-0606-0
  • N. Gao and X. Kong, Block diagonally preconditioned PIU methods of saddle point problem, Appl. Math. Comput. 216 (2010), pp. 1880–1887.
  • G.H. Golub, X. Wu, and J.-Y. Yuan, SOR-like methods for augmented systems, BIT Numer. Math. 41 (2001), pp. 71–85. doi: 10.1023/A:1021965717530
  • F.H. Harlow and J.E. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluids 8 (1965), pp. 2182–2189. doi: 10.1063/1.1761178
  • Z. Huang, L. Wang, Z. Xu, and J. Cui, Generalized ASOR and modified ASOR methods for saddle point problems, Math. Probl. Eng. 2016 (2016), pp. 1–18.
  • Q.T. Le Gia, I.H. Sloan, and A.J. Wathen, Stability and preconditioning for a hybrid approximation on the sphere, Numer. Math. 118 (2011), pp. 695–711. doi: 10.1007/s00211-011-0369-0
  • P.N. Njeru and X.-P. Guo, Accelerated SOR-like method for augmented linear systems, BIT Numer. Math. 56 (2016), pp. 557–571. doi: 10.1007/s10543-015-0571-z
  • R.H. Nochetto and J.-H. Pyo, Optimal relaxation parameter for the Uzawa method, Numer. Math. 98 (2004), pp. 695–702. doi: 10.1007/s00211-004-0522-0
  • J. Pestana and A.J. Wathen, Natural preconditioning and iterative methods for saddle point systems, SIAM Rev. 57 (2015), pp. 71–91. doi: 10.1137/130934921
  • H. Uzawa, Iterative methods for concave programming in Studies in Linear and Nonlinear Programming, K.J. Arrow, L. Hurwicz, and H. Uzawa, eds., Stanford University Press, Stanford, CA, 6, 1958, pp. 154–165.
  • Q. Zheng and C. Ma, A new SOR-like method for the saddle point problems, Appl. Math. Comput. 233 (2014), pp. 421–429.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.