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Total overlapping Schwarz'preconditioners for elliptic problems

Published online by Cambridge University Press:  15 April 2010

Faker Ben Belgacem,
Nabil Gmati and
Faten Jelassi
Faker Ben Belgacem
Affiliation:
LMAC, Université de Technologie de Compiègne, BP 20529, 60205 Compiègne cedex, France.
Nabil Gmati
Affiliation:
LAMSIN, École Nationale d'Ingénieurs de Tunis, B.P. 37, 1002 Le Belvédère, Tunisia. nabil.gmati@ipein.rnu.tn
Faten Jelassi
Affiliation:
LMAC, Université de Technologie de Compiègne, EA 2222, BP 20529, 60205 Compiègne cedex, France.LAMSIN, Faculté des Sciences de Bizerte, Jarzouna, 7021 Bizerte, Tunisia.

Abstract

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A variant of the Total OverlappingSchwarz (TOS) method has been introduced in [Ben Belgacem et al., C. R. Acad. Sci., Sér. 1Math.336(2003) 277–282] as an iterative algorithm to approximate the absorbing boundary condition, in unbounded domains.That same method turns to be an efficient toolto make numerical zoomsin regions of a particular interest. The TOS method enjoys, then, the ability to compute small structures one wants to capture andthe reliability to obtain the behavior of the solution at infinity, when handling exterior problems. The main aim of the paper isto use this modified Schwarz procedure as a preconditioner to Krylovsubspaces methods so to accelerate the calculations. A detailed study concludes toa super-linear convergence ofGMRES and enables us to state accurate estimates on the convergence speed. Afterward, some implementation hints are discussed. Analytical and numericalexamples are also provided and commented that demonstrate the reliability of theTOS-preconditioner.

Keywords

Total Overlapping Schwarz methodminimum residual Krylov methodsnumerical zooms

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 1 , January 2011, pp. 91 - 113
Copyright
© EDP Sciences, SMAI, 2010

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