Total overlapping Schwarz' preconditioners for elliptic problems

¹ LMAC, Université de Technologie de Compiègne, BP 20529, 60205 Compiègne cedex, France.
² LAMSIN, École Nationale d'Ingénieurs de Tunis, B.P. 37, 1002 Le Belvédère, Tunisia. nabil.gmati@ipein.rnu.tn
³ LMAC, Université de Technologie de Compiègne, EA 2222, BP 20529, 60205 Compiègne cedex, France.
⁴ LAMSIN, Faculté des Sciences de Bizerte, Jarzouna, 7021 Bizerte, Tunisia.

Received: 3 January 2009
Revised: 12 September 2009

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.