Robust solvers for symmetric positive definite operators and weighted Poincaré inequalities

Yalchin Efendiev*, Juan Galvis, Raytcho Lazarov, Joerg Willems

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations


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.

Original languageEnglish (US)
Title of host publicationLarge-Scale Scientific Computing - 8th International Conference, LSSC 2011, Revised Selected Papers
Number of pages9
StatePublished - 2012
Event8th International Conference on Large-Scale Scientific Computations,LSSC 2011 - Sozopol, Bulgaria
Duration: Jun 6 2011Jun 10 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume7116 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference8th International Conference on Large-Scale Scientific Computations,LSSC 2011

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: The research of Y. Efendiev was partially supported bythe DOE and NSF (DMS 0934837, DMS 0724704, and DMS 0811180). The re-search of Y. Efendiev, J. Galvis, and R. Lazarov was supported in parts by awardKUS-C1-016-04, made by King Abdullah University of Science and Technology(KAUST). The research of R. Lazarov and J. Willems was supported in partsby NSF Grant DMS-1016525.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


Dive into the research topics of 'Robust solvers for symmetric positive definite operators and weighted Poincaré inequalities'. Together they form a unique fingerprint.

Cite this