Abstract
We present a preconditioner for saddle point problems. The proposed preconditioner is extracted from a stationary iterative method which is convergent under a mild condition. Some properties of the preconditioner as well as the eigenvalues distribution of the preconditioned matrix are presented. The preconditioned system is solved by a Krylov subspace method like restarted GMRES. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioner.
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Bai, Z.-Z.: A class of modified block SSOR preconditioners for symmetric positive definite systems of linear equations. Adv Comput. Math 10, 169–186 (1999)
Bai, Z.-Z.: Modified block SSOR preconditioners for symmetric positive definite linear systems. Ann. Oper. Res 103, 263–282 (2001)
Bai, Z.-Z.: Structured preconditioners for nonsingular matrices of block two-by-two structures. Math. Comput 75, 791–815 (2006)
Bai, Z.-Z., Golub, G.H.: Accelerated and Hermitian skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal 27, 1–23 (2007)
Bai, Z.-Z., Golub, G.H., Li, C.-K.: Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math. Comput. 76, 287–298 (2007)
Bai, Z.-Z., Golub, G.H., Lu, L.-Z., Yin, J.-F.: Block triangular skew-Hermitian splitting methods for positive-definite linear systems. SIAM J. Sci. Comput 26, 844–863 (2004)
Bai, Z.-Z., Golub, G.H., Ng, M.K.: Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl 24, 603–626 (2003)
Bai, Z.-Z., Golub, G.H., Pan, J.-Y.: Preconditioned Hermitian skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math 98, 1–32 (2004)
Bai, Z.-Z., Ng, M.K., Wang, Z.-Q.: Constraint preconditioners for symmetric indefinite matrices. SIAM J. Matrix Anal. Appl 31, 410–433 (2009)
Bai, Z.-Z., Parlett, B.N., Wang, Z.-Q.: On generalized successive overrelaxation methods for augmented linear systems. Numer. Math 102, 1–38 (2005)
Bai, Z.-Z., Wang, Z.-Q.: On parametrized inexact Uzawa methods for generalized saddle point problems. Linear Algebra Appl 428, 2900–2932 (2008)
Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182, 418–477 (2002)
Benzi, M., Golub, G.H.: A preconditioner for generalized saddle point problems. SIAM J. Matrix Anal. Appl. 26, 20–41 (2004)
Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer 14, 1–137 (2005)
Benzi, M., Guo, X.-P.: A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations. Appl. Numer. Math 61, 66–76 (2011)
Benzi, M., Ng, M.K., Niu, Q., Wang, Z.: A relaxed dimensional factorization preconditioner for the incompressible Navier-Stokes equations. J. Comput. Phys 230, 6185–6202 (2011)
Cao, Y., Yao, L.-Q., Jiang, M.-Q.: A modified dimensional split preconditioner for generalized saddle point problems. J. Comput. Appl. Math 250, 70–82 (2013)
Cao, Z.-H.: Positive stable block triangular preconditioners for symmetric saddle point problems. Appl. Numer. Math 57, 899–910 (2007)
Cao, Y., Yao, L.-Q., Jiang, M.-Q., Niu, Q.: A relaxed HSS preconditioner for saddle point problems from mesh free discretization. J. Comput. Math 31, 398–421 (2013)
Elman, H.C., Ramage, A., Silvester, D.J.: IFISS: a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Software 33, Article 14 (2007)
Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite elements and fast iterative solvers, oxford university press oxford (2003)
Fan, H.-T., Zheng, B., Zhu, X.-Y.: A relaxed positive semi-definite and skew-Hermitian splitting preconditioner for non-Hermitian generalied saddle point problems. Numer. Algor. doi:10.1007/s11075-015-0068-5. In press
Jiang, M.-Q., Cao, Y., Yao, L.-Q.: On parametrized block triangular preconditioners for generalized saddle point problems. Appl. Math. Comput 216, 1777–1789 (2010)
Golub, G.H., Wu, X., Yuan, J.-Y.: SOR-Like methods for augmented systems. BIT 55, 71–85 (2001)
Keller, C., Gould, N.I.M., Wathen, A.J.: Constraint preconditioning for indenite linear systems. SIAM J. Matrix Anal. Appl 21, 1300–1317 (2000)
Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indenite linear systems. SIAM J. Sci. Comput 21, 1969–1972 (2000)
Pan, J.-Y., Ng, M.K., Bai, Z.-Z.: New preconditioners for saddle point problems. Appl. Math. Comput 172, 762–771 (2006)
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)
Simoncini, V.: Block triangular preconditioners for symmetric saddle-point problems. Appl. Numer. Math 49, 63–80 (2004)
Sturler, E.D., Liesen, J.: Block-diagonal and constraint preconditioners for nonsymmetric indenite linear systems. SIAM J. Sci. Comput 26, 1598–619 (2005)
Wu, X.-N., Golub, G.H., Cuminato, J.A., Yuan, J.-Y.: Symmetric-triangular decomposition and its applications Part II: preconditioners for inde?nite systems. BIT 48, 139–162 (2008)
Xie, Y.-J., Ma, C.-F.: A modified positive-definite and skew-Hermitian splitting preconditioner for generalized saddle point problems from the Navier-Stokes equation. Numer. Algor. doi:10.1007/s11075-015-0043-1. In press
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Salkuyeh, D.K., Masoudi, M. A new relaxed HSS preconditioner for saddle point problems. Numer Algor 74, 781–795 (2017). https://doi.org/10.1007/s11075-016-0171-2
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DOI: https://doi.org/10.1007/s11075-016-0171-2