Total overlapping Schwarz' preconditioners for elliptic problems

Faker Ben Belgacem; Nabil Gmati; Faten Jelassi

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2011)

  • Volume: 45, Issue: 1, page 91-113
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Ben Belgacem, Faker, Gmati, Nabil, and Jelassi, Faten. "Total overlapping Schwarz' preconditioners for elliptic problems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 45.1 (2011): 91-113. <http://eudml.org/doc/273109>.

@article{BenBelgacem2011,
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.},
author = {Ben Belgacem, Faker, Gmati, Nabil, Jelassi, Faten},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {total overlapping Schwarz method; minimum residual Krylov methods; numerical zooms; cracked or perforated domains; Laplace boundary value problems; total overlapping Schwarz preconditioner; minimum residual Krylov algorithms; convergence; Laplace equation; generalized minimal residual (GMRES) method},
language = {eng},
number = {1},
pages = {91-113},
publisher = {EDP-Sciences},
title = {Total overlapping Schwarz' preconditioners for elliptic problems},
url = {http://eudml.org/doc/273109},
volume = {45},
year = {2011},
}

TY - JOUR
AU - Ben Belgacem, Faker
AU - Gmati, Nabil
AU - Jelassi, Faten
TI - Total overlapping Schwarz' preconditioners for elliptic problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2011
PB - EDP-Sciences
VL - 45
IS - 1
SP - 91
EP - 113
AB - 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.
LA - eng
KW - total overlapping Schwarz method; minimum residual Krylov methods; numerical zooms; cracked or perforated domains; Laplace boundary value problems; total overlapping Schwarz preconditioner; minimum residual Krylov algorithms; convergence; Laplace equation; generalized minimal residual (GMRES) method
UR - http://eudml.org/doc/273109
ER -

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