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)


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.


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


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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

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