Advertisement

Robust Solvers for Symmetric Positive Definite Operators and Weighted Poincaré Inequalities

  • Yalchin Efendiev
  • Juan Galvis
  • Raytcho Lazarov
  • Joerg Willems
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7116)

Abstract

An abstract setting for robustly preconditioning symmetric positive definite (SPD) operators is presented. The term “robust” refers to the property of the condition numbers of the preconditioned systems being independent of mesh parameters and problem parameters. Important instances of such problem parameters are in particular (highly varying) coefficients. The method belongs to the class of additive Schwarz preconditioners. The paper gives an overview of the results obtained in a recent paper by the authors. It, furthermore, focuses on the importance of weighted Poincaré inequalities, whose notion is extended to general SPD operators, for the analysis of stable decompositions. To demonstrate the applicability of the abstract preconditioner the scalar elliptic equation and the stream function formulation of Brinkman’s equations in two spatial dimensions are considered. Several numerical examples are presented.

Keywords

domain decomposition robust additive Schwarz preconditioner spectral coarse spaces high contrast Brinkman’s problem generalized weighted Poincaré inequalities 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chartier, T., Falgout, R.D., Henson, V.E., Jones, J., Manteuffel, T., McCormick, S., Ruge, J., Vassilevski, P.S.: Spectral AMGe (ρAMGe). SIAM J. Sci. Comput. 25(1), 1–26 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Efendiev, Y., Galvis, J., Lazarov, R., Willems, J.: Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. Technical Report 2011-05, RICAM (2011), submitted to Math. Model. Numer. Anal.Google Scholar
  3. 3.
    Efendiev, Y., Hou, T.Y.: Multiscale finite element methods. Surveys and Tutorials in the Applied Mathematical Sciences, vol. 4. Springer, New York (2009); Theory and applicationszbMATHGoogle Scholar
  4. 4.
    Galvis, J., Efendiev, Y.: Domain decomposition preconditioners for multiscale flows in high-contrast media. Multiscale Model. Simul. 8(4), 1461–1483 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Galvis, J., Efendiev, Y.: Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces. Multiscale Model. Simul. 8(5), 1621–1644 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Graham, I.G., Lechner, P.O., Scheichl, R.: Domain decomposition for multiscale PDEs. Numer. Math. 106(4), 589–626 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Hou, T.Y., Wu, X.-H., Cai, Z.: Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp. 68(227), 913–943 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Mathew, T.P.A.: Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2008)zbMATHCrossRefGoogle Scholar
  9. 9.
    Nepomnyaschikh, S.V.: Mesh theorems on traces, normalizations of function traces and their inversion. Sov. J. Numer. Anal. Math. Modelling 6(2), 151–168 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Pechstein, C., Scheichl, R.: Weighted Poincaré inequalities. Technical Report 2010-10, Inst. of Comp. Math., Johannes Kepler University (2010)Google Scholar
  11. 11.
    Sarkis, M.: Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements. Numer. Math. 77(3), 383–406 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Toselli, A., Widlund, O.: Domain Decomposition Methods – Algorithms and Theory. Springer Series in Computational Mathematics. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  13. 13.
    Van Lent, J., Scheichl, R., Graham, I.G.: Energy-minimizing coarse spaces for two-level Schwarz methods for multiscale PDEs. Numer. Linear Algebra Appl. 16(10), 775–799 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Vassilevski, P.S.: Multilevel block-factorization preconditioners. Matrix-based analysis and algorithms for solving finite element equations. Springer, New York (2008)zbMATHGoogle Scholar
  15. 15.
    Willems, J.: Numerical Upscaling for Multiscale Flow Problems. PhD thesis, University of Kaiserslautern (2009)Google Scholar
  16. 16.
    Xu, J., Zikatanov, L.T.: On an energy minimizing basis for algebraic multigrid methods. Comput. Vis. Sci. 7(3-4), 121–127 (2004)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Yalchin Efendiev
    • 1
  • Juan Galvis
    • 1
  • Raytcho Lazarov
    • 1
  • Joerg Willems
    • 2
  1. 1.Dept. MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.RICAMLinzAustria

Personalised recommendations