Skip to main content

A Bramble-Pasciak Conjugate Gradient Method for Discrete Stokes Problems with Lognormal Random Viscosity

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 124)

Abstract

We study linear systems of equations arising from a stochastic Galerkin finite element discretization of saddle point problems with random data and its iterative solution. We consider the Stokes flow model with random viscosity described by the exponential of a correlated random process and shortly discuss the discretization framework and the representation of the emerging matrix equation. Due to the high dimensionality and the coupling of the associated symmetric, indefinite, linear system, we resort to iterative solvers and problem-specific preconditioners. As a standard iterative solver for this problem class, we consider the block diagonal preconditioned MINRES method and further introduce the Bramble-Pasciak conjugate gradient method as a promising alternative. This special conjugate gradient method is formulated in a non-standard inner product with a block triangular preconditioner. From a structural point of view, such a block triangular preconditioner enables a better approximation of the original problem than the block diagonal one. We derive eigenvalue estimates to assess the convergence behavior of the two solvers with respect to relevant physical and numerical parameters and verify our findings by the help of a numerical test case. We model Stokes flow in a cavity driven by a moving lid and describe the viscosity by the exponential of a truncated Karhunen-Loève expansion. Regarding iteration numbers, the Bramble-Pasciak conjugate gradient method with block triangular preconditioner is superior to the MINRES method with block diagonal preconditioner in the considered example.

Keywords

  • Uncertainty quantification
  • PDEs with random data
  • Stokes flow
  • Preconditioning
  • Stochastic Galerkin
  • Lognormal data
  • Mixed finite elements
  • Conjugate gradient method
  • Saddle point problems

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-93891-2_5
  • Chapter length: 25 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   119.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-93891-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   159.99
Price excludes VAT (USA)
Hardcover Book
USD   159.99
Price excludes VAT (USA)
Fig. 1
Fig. 2
Fig. 3

References

  1. Ashby, S.F., Manteuffel, T.A., Saylor, P.E.: A taxonomy for conjugate gradient methods. SIAM J. Numer. Anal. 27(6), 1542–1568 (1990). https://doi.org/10.1137/0727091

    MathSciNet  CrossRef  Google Scholar 

  2. Babuška, I., Tempone, R., Zouraris G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004). https://doi.org/https://doi.org/10.1137/S0036142902418680

    MathSciNet  CrossRef  Google Scholar 

  3. Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007). https://doi.org/10.1137/050645142

    MathSciNet  CrossRef  Google Scholar 

  4. Bachmayr, M., Cohen, A., DeVore, R., Migliorati, G.: Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients. ESAIM: M2AN 51(1), 341–363 (2017). https://doi.org/10.1051/m2an/2016051

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

    MathSciNet  CrossRef  Google Scholar 

  6. Bespalov, A., Powell, C.E., Silvester, D.: A priori error analysis of stochastic Galerkin mixed approximations of elliptic PDEs with random data. SIAM J. Numer. Anal. 50(4), 2039–2063 (2012). https://doi.org/10.1137/110854898

    MathSciNet  CrossRef  Google Scholar 

  7. Bramble, J.H., Pasciak, J.E.: A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Math. Comput. 50(181), 1–17 (1988). https://doi.org/10.1090/S0025-5718-1988-0917816-8

    MathSciNet  CrossRef  Google Scholar 

  8. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2014)

    CrossRef  Google Scholar 

  9. Elman, H.C., Ramage, A., Silvester, D.J.: IFISS: A computational laboratory for investigating incompressible flow problems. SIAM Rev. 56(2), 261–273 (2014). https://doi.org/10.1137/120891393

    MathSciNet  CrossRef  Google Scholar 

  10. Ernst, O.G., Ullmann, E.: Stochastic Galerkin matrices. SIAM J. Matrix Anal. Appl. 31(4), 1848–1872 (2010). https://doi.org/10.1137/080742282

    MathSciNet  CrossRef  Google Scholar 

  11. Ernst, O.G., Powell, C.E., Silvester, D.J., Ullmann, E.: Efficient solvers for a linear stochastic Galerkin mixed formulation of diffusion problems with random data. SIAM J. Sci. Comput. 31(2), 1424–1447 (2009). https://doi.org/10.1137/070705817

    MathSciNet  CrossRef  Google Scholar 

  12. Faber, V., Manteuffel, T.: Necessary and sufficient conditions for the existence of a conjugate gradient method. SIAM J. Numer. Anal. 21(2), 352–362 (1984). https://doi.org/10.1137/0721026

    MathSciNet  CrossRef  Google Scholar 

  13. Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements—A Spectral Approach. Springer, New York (1991)

    CrossRef  Google Scholar 

  14. Gunzburger, M.D., Webster, C.G., Zhang, G.: Stochastic finite element methods for partial differential equations with random input data. Acta Numer. 23, 521–650 (2014). https://doi.org/10.1017/S0962492914000075

    MathSciNet  CrossRef  Google Scholar 

  15. Hackbusch, W.: Iterative Solution of Large Sparse Systems of Equations, 1st edn. Springer, New York (1994)

    CrossRef  Google Scholar 

  16. Hoang, V.H., Schwab, C.: N-term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs. Math. Models Methods Appl. Sci. 24(4), 797–826 (2014). https://doi.org/10.1142/S0218202513500681

    MathSciNet  CrossRef  Google Scholar 

  17. John, V.: Finite Element Methods for Incompressible Flow Problems. Springer International Publishing, Cham (2016)

    CrossRef  Google Scholar 

  18. Lord, G.J., Powell, C.E., Shardlow, R.: An Introduction to Computational Stochastic PDEs. Cambridge University Press, New York (2014)

    CrossRef  Google Scholar 

  19. Müller, C., Ullmann, S., Lang, J.: A Bramble-Pasicak conjugate gradient method for discrete Stokes equations with random viscosity. Preprint. arXiv:1801.01838 (2018). https://arxiv.org/abs/1801.01838

  20. Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 5th edn. Springer, Berlin (1998)

    CrossRef  Google Scholar 

  21. Peters, J., Reichelt, V., Reusken, A.: Fast iterative solvers for discrete Stokes equations, SIAM J. Sci. Comput. 27(2), 646–666 (2005). https://doi.org/10.1137/040606028

    MathSciNet  CrossRef  Google Scholar 

  22. Powell, C.E., Elman, H.C.: Block-diagonal preconditioning for spectral stochastic finite-element systems. IMA J. Numer. Anal. 29(2), 350–375 (2009). https://doi.org/10.1093/imanum/drn014

    MathSciNet  CrossRef  Google Scholar 

  23. Powell, C.E., Silvester, D.: Optimal preconditioning for Raviart–Thomas mixed formulation of second-order elliptic problems. SIAM J. Matrix Anal. Appl. 25(3), 718–738 (2003). https://doi.org/10.1137/S0895479802404428

    MathSciNet  CrossRef  Google Scholar 

  24. Powell, C.E., Ullmann, E.: Preconditioning stochastic Galerkin saddle point systems. SIAM J. Matrix Anal. Appl. 31(5), 2813–2840 (2010). https://doi.org/10.1137/090777797

    MathSciNet  CrossRef  Google Scholar 

  25. Schwab, C., Gittelson, C. J.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer. 20, 291–467 (2011). https://doi.org/10.1017/S0962492911000055

    MathSciNet  CrossRef  Google Scholar 

  26. Ullmann, E.: Solution strategies for stochastic finite element discretizations. PhD thesis, Bergakademie Freiberg University of Technology (2008)

    Google Scholar 

  27. Ullmann, E.: A Kronecker product preconditioner for stochastic Galerkin finite element discretizations. SIAM J. Sci. Comput. 32(2), 923–946 (2010). https://doi.org/10.1137/080742853

    MathSciNet  CrossRef  Google Scholar 

  28. Ullmann S.: Triangular Taylor Hood finite elements, version 1.4. Retrieved: 06 October 2017. www.mathworks.com/matlabcentral/fileexchange/49169

  29. Ullmann, E., Elman, H.C., Ernst, O.G.: Efficient iterative solvers for stochastic Galerkin discretizations of log-transformed random diffusion problems. SIAM J. Sci. Comput. 34(2), A659–A682 (2012). https://doi.org/10.1137/110836675

    MathSciNet  CrossRef  Google Scholar 

  30. Wathen, A.J.: On relaxation of Jacobi iteration for consistent and generalized mass matrices. Commun. Appl. Numer. Methods 7(2), 93–102 (1991). https://doi.org/10.1002/cnm.1630070203

    MathSciNet  CrossRef  Google Scholar 

  31. Xiu, D., Karniadakis, G.E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002). https://doi.org/10.1137/S1064827501387826

    MathSciNet  CrossRef  Google Scholar 

  32. Zulehner, W.: Analysis of iterative methods for saddle point problems: a unified approach. Math. Comput. 71(238), 479–505 (2001). http://www.jstor.org/stable/2698830

    MathSciNet  CrossRef  Google Scholar 

Download references

Acknowledgements

This work is supported by the ‘Excellence Initiative’ of the German federal and state governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Christopher Müller .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Müller, C., Ullmann, S., Lang, J. (2018). A Bramble-Pasciak Conjugate Gradient Method for Discrete Stokes Problems with Lognormal Random Viscosity. In: Schäfer, M., Behr, M., Mehl, M., Wohlmuth, B. (eds) Recent Advances in Computational Engineering. ICCE 2017. Lecture Notes in Computational Science and Engineering, vol 124. Springer, Cham. https://doi.org/10.1007/978-3-319-93891-2_5

Download citation