Total overlapping Schwarz' preconditioners for elliptic problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 1, p. 91-113

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.

DOI : https://doi.org/10.1051/m2an/2010032
Classification:  65F08,  65N12,  65N80
Keywords: total overlapping Schwarz method, minimum residual Krylov methods, numerical zooms
@article{M2AN_2011__45_1_91_0,
     author = {Ben Belgacem, Faker and Gmati, Nabil and Jelassi, Faten},
     title = {Total overlapping Schwarz' preconditioners for elliptic problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {1},
     year = {2011},
     pages = {91-113},
     doi = {10.1051/m2an/2010032},
     zbl = {1270.65073},
     mrnumber = {2781132},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2011__45_1_91_0}
}
Ben Belgacem, Faker; Gmati, Nabil; Jelassi, Faten. Total overlapping Schwarz' preconditioners for elliptic problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 1, pp. 91-113. doi : 10.1051/m2an/2010032. http://www.numdam.org/item/M2AN_2011__45_1_91_0/

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