\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A new Bramble-Pasciak-like preconditioner for saddle point problems

Abstract / Introduction Related Papers Cited by
  • A new Bramble-Pasciak-like preconditioner with parameter is proposed for solving a linear system in the saddle point form. The system can be reformulated as a symmetric positive definite system with respect to some inner product and thus can be solved by the Bramble-Pasciak conjugate gradient (BPCG) method. Based on the spectral condition number of the associated system, the quasi-optimal parameters can be obtained to improve the convergence rate of the BPCG method. Numerical experiments on the Stokes problem are given to illustrate our theoretical results.
    Mathematics Subject Classification: Primary: 65F10, 65F15.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    O. Axelsson and G. Lindskog, On the rate of convergence of the preconditioned conjugate gradient method, Numer. Math., 48 (1986), 499-523.doi: 10.1007/BF01389448.

    [2]

    Z. Z. Bai and G. Q. Li, Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal., 23 (2003), 561-580.doi: 10.1093/imanum/23.4.561.

    [3]

    J. H. Bramble and J. E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of ellipstic problems, Math. Comp., 50 (1988), 1-17.doi: 10.1090/S0025-5718-1988-0917816-8.

    [4]

    M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), 1-137.doi: 10.1017/S0962492904000212.

    [5]

    M. Benzi and V. Simoncini, On the eigenvalues of a class of saddle point matrices, Numer. Math., 103 (2006), 173-196.doi: 10.1007/s00211-006-0679-9.

    [6]

    H. S. Dollar, N. I. M Gould, M. Stoll and A. J. Wathen, Preconditioning saddle-point systems with applications in optimization, SIAM J. Sci. Comput., 32 (2010), 249-270.doi: 10.1137/080727129.

    [7]

    H. C. Elman, D. J. Silvester and Andrew J. Wathen, "Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics," in "Numerical Mathematics and Scientific Computation," Oxford University Press, Oxford, 2005.

    [8]

    H. C. Elman, A. Ramage and D. J. Silvester, Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Softw., 33 (2007), 251-268.

    [9]

    G. N. Gatica and N. Heuer, Conjugate gradient method for dual-dual mixed formulations, Math. Comp., 71 (2002), 1455-1472.doi: 10.1090/S0025-5718-01-01394-1.

    [10]

    M. R. Hestenes and E. Stiefel, Methods of conjugate gradient for solving linear systems, J. Res. Nat. Bur. Stand., 49 (1952), 409-436.

    [11]

    A. Klawonn, Block-triangular preconditioners for saddle point problems with a penalty term, SIAM J. Sci. Comput., 19 (1998), 172-184.doi: 10.1137/S1064827596303624.

    [12]

    J. Liesen and B. N. Parlett, On nonsymmetric saddle point matrices that allow conjugate gradient iterations, Numer. Math., 108 (2008), 605-624.doi: 10.1007/s00211-007-0131-9.

    [13]

    J. A. Meijerink and H. A. Van der vorst, An iterative solution method for linear equation systems of which the coefficient matrix is symmetric M-matrix, Math. Comp., 31 (1977), 148-162.

    [14]

    X. F. Peng, W. Li and S. H. Xiang, New preconditioners based on symmetric-triangular decomposition for saddle point problems, Computing, 93 (2011), 27-46.doi: 10.1007/s00607-011-0150-3.

    [15]

    J. Schöberl and W. Zulehner, Symmetric indefinite preconditioners for saddle point problems with applications to pde-constrained optimization problems, SIAM J. Matrix Anal. Appl., 29 (2007), 752-773.doi: 10.1137/060660977.

    [16]

    M. Stoll and A. J. Wathen, Combination preconditioning and the Bramble-Pasciak+ preconditioner, SIAM J. Matrix Anal. Appl., 30 (2008), 582-608.doi: 10.1137/070688961.

    [17]

    X. N. Wu, G. H. Golub, J. A. Cuminato and J. Y. Yuan, Symmetric-triangular decomposition and its applications - Part Π: Preconditioners for indefinite systems, BIT Numer. Math., 48 (2008), 139-162.doi: 10.1007/s10543-008-0160-5.

    [18]

    W. Zulehner, Analysis of iterative methods for saddle point problems: A unified approach, Math. Comp., 71 (2002), 479-505.doi: 10.1090/S0025-5718-01-01324-2.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(84) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return