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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.
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Abstract

A variant of the Total Overlapping Schwarz (TOS) method has been introduced in [Ben Belgacem et al., C. R. Acad. Sci., Sér. 1 Math.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 tool to make numerical zooms in regions of a particular interest. The TOS method enjoys, then, the ability to compute small structures one wants to capture and the reliability to obtain the behavior of the solution at infinity, when handling exterior problems. The main aim of the paper is to use this modified Schwarz procedure as a preconditioner to Krylov subspaces methods so to accelerate the calculations. A detailed study concludes to a super-linear convergence of GMRES and enables us to state accurate estimates on the convergence speed. Afterward, some implementation hints are discussed. Analytical and numerical examples are also provided and commented that demonstrate the reliability of the TOS-preconditioner.

Keywords

Total Overlapping Schwarz methodminimum residual Krylov methodsnumerical zooms
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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