TY - CHAP

T1 - Robust Solvers for Symmetric Positive Definite Operators and Weighted Poincaré Inequalities

AU - Efendiev, Yalchin R.

AU - Galvis, Juan

AU - Lazarov, Raytcho

AU - Willems, Joerg

N1 - 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.

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/599528

UR - http://link.springer.com/10.1007/978-3-642-29843-1_4

UR - http://www.scopus.com/inward/record.url?scp=84861732957&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-29843-1_4

DO - 10.1007/978-3-642-29843-1_4

M3 - Chapter

AN - SCOPUS:84861732957

SN - 9783642298424

SP - 43

EP - 51

BT - Lecture Notes in Computer Science

PB - Springer Nature

ER -