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Computational Science, Engineering & Technology Series
ISSN 17593158 CSETS: 21
PARALLEL, DISTRIBUTED AND GRID COMPUTING FOR ENGINEERING Edited by: B.H.V. Topping, P. Iványi
Chapter 11
Meshfree Adaptative AitkenSchwarz Domain Decomposition with application to Darcy Flow D. TromeurDervout
University of Lyon, University Lyon 1, Institute Camille Jordan, UMR5208Lyon1ECLINSACNRS, Villeurbanne, France D. TromeurDervout, "Meshfree Adaptative AitkenSchwarz Domain Decomposition with application to Darcy Flow", in B.H.V. Topping, P. Iványi, (Editors), "Parallel, Distributed and Grid Computing for Engineering", SaxeCoburg Publications, Stirlingshire, UK, Chapter 11, pp 217260, 2009. doi:10.4203/csets.21.11
Keywords: partial differential equation of elliptic type, Aitken acceleration of convergence,
Schwarz domain decomposition, parallel computing, Darcy, Stokes.
Summary
This chapter is devoted to the numerical development of the Schwarz domain decomposition
for non separable operators where the convergence is accelerated by the
Aitken process which is not based on the mesh property. We demonstrate the linear
divergence/convergence of the Schwarz method on the coupling of Darcy equation
and Stokes vectorial equation. We derive explicit formula for the coefficient of amplification
of the error. Then we recall the good property of this twolevel domain
decomposition especially in a metacomputing context. When the operator is non separable
and/or the mesh is not regular, the technique for the acceleration cannot take
advantage of a decomposition of the solution in "Fourier" or "sinus" basis leading to
a diagonal or block diagonal acceleration. We propose three algorithms based on the
singular value decomposition of the iterates produced by iterative methods exhibiting
a pure linear convergence. Evidence of numerically efficient acceleration of convergence,
based on a posteriori estimates provided by the singular value decomposition
(SVD), are given for the Jacobi and Schwarz methods on Darcy problem where the
permeability exhibits strong contrasts.
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