Robust Solvers for Symmetric Positive Definite Operators and Weighted Poincaré Inequalities

Handle URI:
http://hdl.handle.net/10754/599528
Title:
Robust Solvers for Symmetric Positive Definite Operators and Weighted Poincaré Inequalities
Authors:
Efendiev, Yalchin; Galvis, Juan; Lazarov, Raytcho; Willems, Joerg
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. © 2012 Springer-Verlag.
Citation:
Efendiev Y, Galvis J, Lazarov R, Willems J (2012) Robust Solvers for Symmetric Positive Definite Operators and Weighted Poincaré Inequalities. Lecture Notes in Computer Science: 43–51. Available: http://dx.doi.org/10.1007/978-3-642-29843-1_4.
Publisher:
Springer Science + Business Media
Journal:
Lecture Notes in Computer Science
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
2012
DOI:
10.1007/978-3-642-29843-1_4
Type:
Book Chapter
ISSN:
0302-9743; 1611-3349
Sponsors:
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.
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Full metadata record

DC FieldValue Language
dc.contributor.authorEfendiev, Yalchinen
dc.contributor.authorGalvis, Juanen
dc.contributor.authorLazarov, Raytchoen
dc.contributor.authorWillems, Joergen
dc.date.accessioned2016-02-28T05:52:49Zen
dc.date.available2016-02-28T05:52:49Zen
dc.date.issued2012en
dc.identifier.citationEfendiev Y, Galvis J, Lazarov R, Willems J (2012) Robust Solvers for Symmetric Positive Definite Operators and Weighted Poincaré Inequalities. Lecture Notes in Computer Science: 43–51. Available: http://dx.doi.org/10.1007/978-3-642-29843-1_4.en
dc.identifier.issn0302-9743en
dc.identifier.issn1611-3349en
dc.identifier.doi10.1007/978-3-642-29843-1_4en
dc.identifier.urihttp://hdl.handle.net/10754/599528en
dc.description.abstractAn 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. © 2012 Springer-Verlag.en
dc.description.sponsorshipThe 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.en
dc.publisherSpringer Science + Business Mediaen
dc.subjectBrinkman's problemen
dc.subjectdomain decompositionen
dc.subjectgeneralized weighted Poincaré inequalitiesen
dc.subjecthigh contrasten
dc.subjectrobust additive Schwarz preconditioneren
dc.subjectspectral coarse spacesen
dc.titleRobust Solvers for Symmetric Positive Definite Operators and Weighted Poincaré Inequalitiesen
dc.typeBook Chapteren
dc.identifier.journalLecture Notes in Computer Scienceen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
dc.contributor.institutionJohann Radon Institute for Computational and Applied Mathematics, Linz, Austriaen
kaust.grant.numberKUS-C1-016-04en
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