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

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- by Victorita Dolean, Frédéric Nataf and Gerd Rapin PDF
- Math. Comp.
**78**(2009), 789-814 Request permission

## 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
- Received by editor(s): October 17, 2006
- Received by editor(s) in revised form: October 29, 2007
- Published electronically: November 24, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp.
**78**(2009), 789-814 - MSC (2000): Primary 65-xx
- DOI: https://doi.org/10.1090/S0025-5718-08-02172-8
- MathSciNet review: 2476560