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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Deriving a new domain decomposition method for the Stokes equations using the Smith factorization


Authors: Victorita Dolean, Frédéric Nataf and Gerd Rapin
Journal: Math. Comp. 78 (2009), 789-814
MSC (2000): Primary 65-xx
Published electronically: November 24, 2008
MathSciNet review: 2476560
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Abstract: In this paper the Smith factorization is used systematically to derive a new domain decomposition method for the Stokes problem. In two dimensions the key idea is the transformation of the Stokes problem into a scalar bi-harmonic problem. We show, how a proposed domain decomposition method for the bi-harmonic problem leads to a domain decomposition method for the Stokes equations which inherits the convergence behavior of the scalar problem. Thus, it is sufficient to study the convergence of the scalar algorithm. The same procedure can also be applied to the three-dimensional Stokes problem.

As transmission conditions for the resulting domain decomposition method of the Stokes problem we obtain natural boundary conditions. Therefore it can be implemented easily.

A Fourier analysis and some numerical experiments show very fast convergence of the proposed algorithm. Our algorithm shows a more robust behavior than Neumann-Neumann or FETI type methods.


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Additional Information

Victorita Dolean
Affiliation: Laboratoire J.A. Dieudonné, CNRS UMR 6621, Université de Nice Sophia-Antipolis, 06108 Nice Cedex 02, France
Email: dolean@math.unice.fr

Frédéric Nataf
Affiliation: Laboratoire J.L. Lions, CNRS UMR 7598, Université Pierre et Marie Curie, 75252 Paris Cedex 05, France
Email: nataf@ann.jussieu.fr

Gerd Rapin
Affiliation: Department of Mathematics, NAM, University of Göttingen, D-37083, Germany
Email: grapin@math.uni-goettingen.de

DOI: http://dx.doi.org/10.1090/S0025-5718-08-02172-8
PII: S 0025-5718(08)02172-8
Received by editor(s): October 17, 2006
Received by editor(s) in revised form: October 29, 2007
Published electronically: November 24, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.