Preconditioners for Saddle-Point Problems

  • Miroslav Rozložník
Part of the Nečas Center Series book series (NECES)


Preconditioners for saddle-point problems exploit the 2-by-2 block structure of the saddle-point matrix \(\mathbb {A}\). They have been a subject of active research in a whole range of applications. For a current survey, we refer to [67].


  1. 20.
    C. Durazzi, V. Ruggiero, Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems. Numer. Linear Algebra Appl. 10(8), 673–688 (2002)MathSciNetCrossRefGoogle Scholar
  2. 24.
    H. Elman, D.J. Silvester, A.J. Wathen, Block preconditioners for the discrete incompressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids 40, 333–344 (2002)MathSciNetCrossRefGoogle Scholar
  3. 44.
    I.C.F. Ipsen, A note on preconditioning non-symmetric matrices. SIAM J. Sci. Comput. 23(3), 1050–1051 (2001)MathSciNetCrossRefGoogle Scholar
  4. 48.
    A. Klawonn, Block-triangular preconditioners for saddle point problems with a penalty term. SIAM J. Sci. Comput. 19(1), 172–184 (1998)MathSciNetCrossRefGoogle Scholar
  5. 49.
    A. Klawonn, An optimal preconditioner for a class of saddle point problems with a penalty term. SIAM J. Sci. Comput. 19(2), 540–552 (1998)MathSciNetCrossRefGoogle Scholar
  6. 55.
    L. Lukšan, J. Vlček, Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems. Numer. Linear Algebra Appl. 5(3), 1099–1506 (1999)MathSciNetzbMATHGoogle Scholar
  7. 61.
    M.F. Murphy, G.H. Golub, A.J. Wathen, A note on preconditioning for indefinite linear systems. SIAM J. Sci. Comput. 21, 1969–1972 (2000)MathSciNetCrossRefGoogle Scholar
  8. 67.
    J. Pestana, A.J. Wathen, Natural preconditioning and iterative methods for saddle point systems. SIAM Rev. 57, 71–91 (2015)MathSciNetCrossRefGoogle Scholar
  9. 71.
    M. Rozložník, V. Simoncini, Krylov subspace methods for saddle point problems with indefinite preconditioning. SIAM J. Matrix Anal. Appl. 24(2), 368–391 (2002)MathSciNetCrossRefGoogle Scholar
  10. 76.
    D.J. Silvester, A.J. Wathen, Fast iterative solution of stabilized Stokes systems, part II: using block preconditioners. SIAM J. Numer. Anal. 31, 1352–1367 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Miroslav Rozložník
    • 1
  1. 1.Institute of MathematicsCzech Academy of SciencesPragueCzech Republic

Personalised recommendations