A Comparison of Preconditioning Methods for Saddle Point Problems with an Application to Porous Media Flow Problems

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9611)

Abstract

The paper overviews and compares some block preconditioners for the solution of saddle point systems, especially systems arising from the Brinkman model of porous media flow. The considered preconditioners involve different Schur complements as inverse free Schur complement in HSS (Hermitian - Skew Hermitian Splitting preconditioner), Schur complement to the velocity matrix and finally Schur complement to a regularization block in the augmented matrix preconditioner. The inverses appearing in most of the considered Schur complements are approximated by simple sparse approximation techniques as element-by-element and Frobenius norm minimization approaches. A special interest is devoted to problems involving various Darcy, Stokes and Brinkman flow regions, the efficiency of preconditioners in this case is demonstrated by some numerical experiments.

References

  1. 1.
    Axelsson, O., Blaheta, R.: Preconditioning methods for saddle point problems arising in porous media flow (submitted)Google Scholar
  2. 2.
    Axelsson, O.: Iterative Solution Methods. Cambridge University Press, Cambridge (1996)MATHGoogle Scholar
  3. 3.
    Axelsson, O., Blaheta, R.: Preconditioning of matrices partitioned in \(2\times 2\) block form: Eigenvalue estimates and Schwarz DD for mixed FEM. Numer. Linear Algebra Appl. 17(5), 787–810 (2010). http://dx.org/10.1002/nla.728
  4. 4.
    Axelsson, O., Blaheta, R., Neytcheva, M.: Preconditioning of boundary value problems using elementwise Schur complements. SIAM J. Matrix Anal. Appl. 31(2), 767–789 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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(3), 603–626 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numerica 14(1), 1–137 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford (2014)CrossRefMATHGoogle Scholar
  8. 8.
    Formaggia, L., Gerbeau, J.F., Nobile, F., Quarteroni, A.: Numerical treatment of defective boundary conditions for the Navier-Stokes equations. SIAM J. Numer. Anal. 40(1), 376–401 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Greenbaum, A., Strakos, Z.: Predicting the behavior of finite precision Lanczos and conjugate gradient computations. SIAM J. Matrix Anal. Appl. 13(1), 121–137 (1992)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Greif, C., Schötzau, D.: Preconditioners for saddle point linear systems with highly singular (1,1) blocks. Electron. Tran. Numer. Anal. ETNA 22, 114–121 (2006)MathSciNetMATHGoogle Scholar
  11. 11.
    Kolotilina, L.Y., Yeremin, A.Y.: Factorized sparse approximate inverse preconditionings I. Theory. SIAM J. Matrix Anal. Appl. 14(1), 45–58 (1993)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kraus, J.: Algebraic multilevel preconditioning of finite element matrices using local Schur complements. Numer. Linear Algebra Appl. 13(1), 49–70 (2006)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mardal, K.A., Winther, R.: Uniform preconditioners for the time dependent Stokes problem. Numer. Math. 98(2), 305–327 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Neytcheva, M.: On element-by-element Schur complement approximations. Linear Algebra Appl. 434(11), 2308–2324 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Neytcheva, M., Bängtsson, E., Linnér, E.: Finite-element based sparse approximate inverses for block-factorized preconditioners. Adv. Comput. Math. 35(2), 323–355 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)CrossRefMATHGoogle Scholar
  17. 17.
    Simoncini, V., Benzi, M.: Spectral properties of the Hermitian and skew-Hermitian splitting preconditioner for saddle point problems. SIAM J. Matrix Anal. Appl. 26(2), 377–389 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Taylor, C., Hood, P.: A numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids 1(1), 73–100 (1973)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Geonics AS CROstrava-PorubaCzech Republic

Personalised recommendations