Advertisement

BIT Numerical Mathematics

, Volume 51, Issue 4, pp 865–888 | Cite as

On an augmented Lagrangian-based preconditioning of Oseen type problems

  • Xin HeEmail author
  • Maya Neytcheva
  • Stefano Serra Capizzano
Article

Abstract

The paper deals with a general framework for constructing preconditioners for saddle point matrices, in particular as arising in the discrete linearized Navier-Stokes equations (Oseen’s problem). We utilize the so-called augmented Lagrangian framework, where the original linear system of equations is first transformed to an equivalent one, which latter is then solved by a preconditioned iterative solution method.

The matrices in the linear systems, arising after the discretization of Oseen’s problem, are of two-by-two block form as are the best known preconditioners for these. In the augmented Lagrangian formulation, a scalar regularization parameter is involved, which strongly influences the quality of the block-preconditioners for the system matrix (referred to as outer), as well as the conditioning and the solution of systems with the resulting pivot block (referred to as inner) which, in the case of large scale numerical simulations has also to be solved using an iterative method. We analyse the impact of the value of the regularization parameter on the convergence of both outer and inner solution methods.

The particular preconditioner used in this work exploits the inverse of the pressure mass matrix. We study the effect of various approximations of that inverse on the performance of the preconditioners, in particular that of a sparse approximate inverse, computed in an element-by-element fashion. We analyse and compare the spectra of the preconditioned matrices for the different approximations and show that the resulting preconditioner is independent of problem, discretization and method parameters, namely, viscosity, mesh size, mesh anisotropy.

We also discuss possible approaches to solve the modified pivot matrix block.

Keywords

Navier-Stokes equations Saddle point systems Augmented Lagrangian Finite elements Approximation of mass matrix Iterative methods Preconditioning 

Mathematics Subject Classification (2000)

65F10 65F08 65N30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Axelsson, O.: Preconditioning of indefinite problems by regularization. SIAM J. Numer. Anal. 16, 58–69 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Axelsson, O.: Iterative Solution Methods. Oxford University Press, Oxford (1994) zbMATHGoogle Scholar
  3. 3.
    Axelsson, O., Blaheta, R.: Preconditioning of matrices partitioned in two-by-two block form: eigenvalue estimates and Schwarz DD for mixed FEM. Numer. Linear Algebra Appl. 17, 787–810 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Axelsson, O., Neytcheva, M.: A general approach to analyse preconditioners for two-by-two block matrices. TR 2010-029, Department of Information Technology, Uppsala University (2010) Google Scholar
  5. 5.
    Axelsson, O., Padiy, A.: On the additive version of the algebraic multilevel iteration method for anisotropic elliptic problems. SIAM J. Sci. Comput. 20, 1807–1830 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Axelsson, O., Blaheta, R., Neytcheva, M.: Preconditioning for boundary value problems using elementwise Schur complement. SIAM J. Matrix Anal. Appl. 31, 767–789 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Benzi, M., Olshanskii, M.A.: An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput. 28, 2095–2113 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Benzi, M., Tůma, M.: A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. 19, 968–994 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Benzi, M., Olshanskii, M.A., Wang, Z.: Modified augmented Lagrangian preconditioners for the incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids (2010). Published online in Wiley InterScience (www.interscience.wiley.com)
  11. 11.
    Bru, R., Cerdán, J., Marín, J., Mas, J.: Preconditioning sparse nonsymmetric linear systems with the Sherman-Morrison formula. SIAM J. Sci. Comput. 25, 701–715 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Bru, R., Marín, J., Mas, J., Tůma, M.: Balanced incomplete factorization. SIAM J. Sci. Comput. 30, 2302–2318 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Cerdán, J., Faraj, T., Malla, N., Marín, J., Mas, J.: Block approximate inverse preconditioners for sparse nonsymmetric linear systems. Electron. Trans. Numer. Anal. 37, 23–40 (2010) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chow, E., Saad, Y.: Approximate inverse techniques for block-partitioned matrices. SIAM J. Sci. Comput. 18, 1657–1675 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    de Niet, A., Wubs, F.: Two preconditioners for saddle point problems in fluid flows. Int. J. Numer. Methods Fluids 54, 355–377 (2007) zbMATHCrossRefGoogle Scholar
  16. 16.
    Elman, H.C.: Preconditioning for the steady-state Navier-Stokes equations with low viscosity. SIAM J. Sci. Comput. 20, 1299–1316 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Elman, H.C., Silvester, D.: Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations. SIAM J. Sci. Comput. 17, 33–46 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Element and Fast Iterative Solvers: With Application in Incompressible Fluid Dynamics. Oxford University Press, Oxford (2005) Google Scholar
  19. 19.
    Fortin, M., Glowinski, R.: Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problems. Stud. Math. Appl., vol. 15. North-Holland, Amsterdam (1983) Google Scholar
  20. 20.
    Fried, I.: Bounds on the spectral and maximum norms of the finite element stiffness, flexibility and mass matrices. Int. J. Solids Struct. 9, 1013–1034 (1973) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Fried, I., Coleman, M.: Improvable bounds on the largest eigenvalue of a completely positive finite element flexibility matrix. J. Sound Vib. 283, 487–494 (2005) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Grote, M., Huckle, T.: Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput. 18, 838–853 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    He, X., Neytcheva, M., Capizzano, S.C.: On an augmented Lagrangian-based preconditioning of Oseen type problems. TR 2010-026, Department of Information Technology, Uppsala University (2010) Google Scholar
  24. 24.
    Kay, D., Loghin, D., Wathen, A.J.: A preconditioner for the steady-state Navier-Stokes equations. SIAM J. Sci. Comput. 24, 237–256 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kolotilina, L., Yeremin, Yu.: Factorized sparse approximate inverse preconditionings. I. Theory. SIAM J. Matrix Anal. Appl. 14, 45–58 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Kraus, J.: Algebraic multilevel preconditioning of finite element matrices using local Schur complements. Numer. Linear Algebra Appl. 13, 49–70 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Neytcheva, M., Bängtsson, E.: Preconditioning of nonsymmetric saddle point systems as arising in modelling of visco-elastic problems. Electron. Trans. Numer. Anal. 29, 193–211 (2008) Google Scholar
  28. 28.
    Neytcheva, M., He, X., Do-Quang, M.: Element-by-element Schur complete approximations for general nonsymmetric matrices of two-by-two block form. Lect. Notes Comput. Sci. 108, 108–115 (2010) CrossRefGoogle Scholar
  29. 29.
    Olshanskii, M.A., Benzi, M.: An augmented Lagrangian approach to linearized problems in hydrodynamic stability. SIAM J. Sci. Comput. 30, 1459–1473 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Olshanskii, M.A., Vassilevski, Y.: Pressure Schur complement preconditioners for the discrete Oseen problem. SIAM J. Sci. Comput. 29, 2686–2704 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Sun Performance Library Reference Manual. http://docs.sun.com/app/docs/doc/820-2171
  32. 32.
    Wathen, A.J.: Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J. Numer. Anal. 7, 449–457 (1987) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  • Xin He
    • 1
    Email author
  • Maya Neytcheva
    • 1
  • Stefano Serra Capizzano
    • 2
  1. 1.Department of Information TechnologyUppsala UniversityUppsalaSweden
  2. 2.Department of Physics and MathematicsUniversity of InsubriaComoItaly

Personalised recommendations