262
Views
5
CrossRef citations to date
0
Altmetric
Original Articles

A relaxed block-triangular splitting preconditioner for generalized saddle-point problems

, &
Pages 1609-1623 | Received 20 Feb 2016, Accepted 29 Jun 2016, Published online: 03 Sep 2016

References

  • Z.-Z. Bai, Structured preconditioners for nonsingular matrices of block two-by-two structures, Math. Comput. 75 (2006), pp. 791–816.
  • Z.-Z. Bai, Splitting iteration methods for non-Hermitian positive definite systems of linear equations, Hokkaido Math. J. 36 (2007), pp. 801–814.
  • Z.-Z. Bai, Optimal parameters in the HSS-like methods for saddle-point problems, Numer. Linear Algebra Appl. 16 (2009), pp. 447–479.
  • 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.
  • Z.-Z. Bai and M.K. Ng, On inexact preconditioners for nonsymmetric matrices, SIAM J. Sci. Comput. 26 (2005), pp. 1710–1724.
  • 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.
  • Z.-Z. Bai, G.H. Golub, and C.-K. Li, Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput. 28 (2006), pp. 583–603.
  • Z.-Z. Bai, G.H. Golub, and C.-K. Li, Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comput. 76 (2007), pp. 287–299.
  • 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.
  • 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(1) (2004), pp. 1–32.
  • Z.-Z. Bai, M.K. Ng, and Z.-Q. Wang, Constraint preconditioners for symmetric indefinite matrices, SIAM J. Matrix Anal. Appl. 31 (2009), pp. 410–433.
  • 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.
  • Z.-Z. Bai, G.H. Golub, L.-Z. Lu, and J.-F. Yin, Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput. 26(3) (2005), pp. 844–863.
  • Z.-Z. Bai, M. Benzi, F. Chen, and Z.-Q. Wang, Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems, IMA J. Numer. Anal. 33 (2013), pp. 343–369.
  • M. Benzi and G.H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix. Anal. Appl. 26 (2004), pp. 20–41.
  • M. Benzi and X.-P. Guo, A dimensional split preconditioner for Stokes and linearized Navier–Stokes equations, Appl. Numer. Math. 61 (2011), pp. 66–76.
  • M. Benzi and J. Liu, Block preconditioning for saddle point systems with indefinite (1,1) block, Int. J. Comput. Math. 84 (2007), pp. 1117–1129.
  • M. Benzi, G.H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numer. 14 (2005), pp. 1–137.
  • M. Benzi, M. Ng, Q. Niu, and Z. Wang, A relaxed dimensional factorization preconditioner for the incompressible Navier–Stokes equations, J. Comput. Phys. 230 (2011), pp. 6185–6202.
  • Z.-H. Cao, A note on constraint preconditioning for nonsymmetric indefinite matrices, SIAM J. Matrix Anal. Appl. 24 (2002), pp. 121–125.
  • Z.-H. Cao, A note on block diagonal and constraint preconditioners for non-symmetric indefinite linear systems, Int. J. Comput. Math. 13 (2006), pp. 383–395.
  • Z.-H. Cao, A class of constraint preconditioners for nonsymmetric saddle point matrices, Numer. Math. 103 (2006), pp. 47–61.
  • Z.-H. Cao, Positive stable block triangular preconditioners for symmetric saddle point problems, Appl. Numer. Math. 57 (2007), pp. 899–910.
  • Y. Cao, J.-L. Dong, and Y.-M. Wang, A relaxed deteriorated PSS preconditioner for nonsymmetric saddle-point problems from the steady Navier–Stokes equation, J. Comput. Appl. Math. 273 (2015), pp. 41–60.
  • Y. Cao, M.-Q. Jiang, and Y.-L. Zheng, A note on the positive stable block triangular preconditioner for generalized saddle-point problems, Appl. Math. Comput. 218(22) (2012), pp. 11075–11082.
  • Y. Cao, S.-X. Miao, and Y.-S. Cui, A relaxed splitting preconditioner for generalized saddle point problems, Comput. Appl. Math. 34 (2015), pp. 865–879.
  • Y. Cao, W.-W. Tan, and M.-Q. Jiang, A relaxed dimensional factorization preconditioner for generalized saddle-point problems, Math. Numer. Sinica 34 (2012), pp. 351–360.
  • Y. Cao, L.-Q. Yao, and M.-Q. Jiang, A modified dimensional split preconditioner for generalized saddle point problems, J. Comput. Appl. Math. 250 (2013), pp. 70–82.
  • Y. Cao, L.-Q. Yao, M.-Q. Jiang, and Q. Niu, A relaxed HSS preconditioner for saddle point problems from meshfree discretization, J. Comput. Math. 31 (2013), pp. 398–421.
  • H.S. Dollar, Constraint-style preconditioners for regularized saddle point problems, SIAM J. Matrix Anal. Appl. 29 (2006), pp. 672–684.
  • H.S. Dollar and A.J. Wathen, Approximate factorization constraint preconditioners for saddle-point matrices, SIAM J. Sci. Comput. 27 (2005), pp. 1555–1572.
  • H.C. Elman, Preconditioning for the steady-state Navier–Stokes equations with low viscosity, SIAM J. Sci. Comput. 20 (1999), pp. 1299–1316.
  • H.C. Elman, A. Ramage, and D.J. Silvester, Algorithm 866: IFISS, a matlab toolbox for modelling incompressible flow, ACM Trans. Math. Softw. 33 (2) (2007), Article 14. doi:10.1145/1236463.1236469.
  • G.H. Golub and A.J. Wathen, An iteration for indefinite systems and its application to the Navier–Stokes equations, SIAM J. Sci. Comput. 19 (1996), pp. 530–539.
  • M.-Q. Jiang and Y. Cao, On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math. 231 (2009), pp. 973–982.
  • M.-Q. Jiang, Y. Cao, and L.-Q. Yao, On parameterized block triangular preconditioners for generalized saddle-point problems, Appl. Math. Comput. 216 (2010), pp. 1777–1789.
  • J.-Y. Pan, M.K. Ng, and Z.-Z. Bai, New preconditioners for saddle-point problems, Appl. Math. Comput. 172 (2006), pp. 762–771.
  • I. Perugia and V. Simoncini, Block-diagonal and indefinite symmetric preconditioners for mixed finite element formulations, Numer. Linear Algebra Appl. 7 (2000), pp. 585–616.
  • Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 2003.
  • S.-Q. Shen, A note on PSS preconditioners for generalized saddle point problems, Appl. Math. Comput. 237 (2014), pp. 723–729.
  • V. Simoncini, Block triangular preconditioners for symmetric saddle-point problems, Appl. Numer. Math. 49 (2004), pp. 63–80.
  • N.-B. Tan, T.-Z. Huang, and Z.-J. Hu, A relaxed splitting preconditioner for the incompressible Navier–Stokes equations, J. Appl. Math. 2012 (2012), Article ID 402490.

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.