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total overlapping schwarz method

ESAIM: Mathematical Modelling and Numerical Analysis (1)

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Total overlapping Schwarz'preconditioners for elliptic problems
Faker Ben Belgacem, Nabil Gmati, Faten Jelassi
Journal:

ESAIM: Mathematical Modelling and Numerical Analysis / Volume 45 / Issue 1 / January 2011

Published online by Cambridge University Press:

15 April 2010, pp. 91-113

Print publication:

January 2011
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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.