Robust Solvers for Symmetric Positive Definite Operators and Weighted Poincaré Inequalities

Yalchin R. Efendiev, Juan Galvis, Raytcho Lazarov, Joerg Willems

Research output: Chapter in Book/Report/Conference proceedingChapter

1 Scopus citations

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.
Original languageEnglish (US)
Title of host publicationLecture Notes in Computer Science
PublisherSpringer Nature
Pages43-51
Number of pages9
ISBN (Print)9783642298424
DOIs
StatePublished - 2012
Externally publishedYes

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