# On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations

Xuejun Xu; C. O. Chow; S. H. Lui

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 39, Issue: 6, page 1251-1269
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topXu, Xuejun, Chow, C. O., and Lui, S. H.. "On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations." ESAIM: Mathematical Modelling and Numerical Analysis 39.6 (2010): 1251-1269. <http://eudml.org/doc/194303>.

@article{Xu2010,

abstract = {
In this paper, a Dirichlet-Neumann substructuring domain
decomposition method is presented for a finite element
approximation to the nonlinear Navier-Stokes equations. It is
shown that the Dirichlet-Neumann domain decomposition sequence
converges geometrically to the true solution provided the Reynolds
number is sufficiently small. In this method, subdomain problems
are linear. Other version where the subdomain problems are linear
Stokes problems is also presented.
},

author = {Xu, Xuejun, Chow, C. O., Lui, S. H.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Nonoverlapping domain decomposition; incompressible Navier-Stokes equations; finite elements; nonlinear problems.; convergence; Dirichlet-Neumann substructuring; finite element approximation},

language = {eng},

month = {3},

number = {6},

pages = {1251-1269},

publisher = {EDP Sciences},

title = {On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations},

url = {http://eudml.org/doc/194303},

volume = {39},

year = {2010},

}

TY - JOUR

AU - Xu, Xuejun

AU - Chow, C. O.

AU - Lui, S. H.

TI - On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 39

IS - 6

SP - 1251

EP - 1269

AB -
In this paper, a Dirichlet-Neumann substructuring domain
decomposition method is presented for a finite element
approximation to the nonlinear Navier-Stokes equations. It is
shown that the Dirichlet-Neumann domain decomposition sequence
converges geometrically to the true solution provided the Reynolds
number is sufficiently small. In this method, subdomain problems
are linear. Other version where the subdomain problems are linear
Stokes problems is also presented.

LA - eng

KW - Nonoverlapping domain decomposition; incompressible Navier-Stokes equations; finite elements; nonlinear problems.; convergence; Dirichlet-Neumann substructuring; finite element approximation

UR - http://eudml.org/doc/194303

ER -

## References

top- J. Cahouet, On some difficulties occurring in the simulation of incompressible fluid flows by domain decomposition methods, in Proc. of the First International Symposium On Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. Golub, G. Meurant and J. Periaux Eds., SIAM, Philadelphia, PA (1988). Zbl0654.76020
- X.C Cai, D.E. Keyes and V. Venkatakrishnan, Newton-Krylov-Schwarz: An implicit solver for CFD, in Proc. of the Eighth International Conference on Domain Decomposition Methods in Science and Engineering, R. Glowinski, J. Periaux, Z.C. Shi and O.B. Widlund Eds., Wiley, Strasbourg (1997).
- T.F. Chan and T.P. Mathew, Domain decomposition algorithm. Acta Numerica (1994) 61–143. Zbl0809.65112
- P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). Zbl0383.65058
- Q.V. Dinh, R. Glowinski, J. Periaux and G. Terrasson, On the coupling of viscous and inviscid models for incompressible fluid flows via domain decomposition, in Proc. the First International Symposium On Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. Golub, G. Meurant and J. Periaux Eds., SIAM, Philadelphia, PA (1988). Zbl0652.76023
- L. Fatone, P. Gervasio and A. Quarteroni, Multimodels for incompressible flows. J. Math. Fluid Dynamics2 (2000) 126–150. Zbl0962.76021
- M. Fortin and R. Aboulaich, Schwarz's Decomposition Method for Incompressible Flow Problems, in Proc. of the First International Symposium On Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. Golub, G. Meurant and J. Periaux Eds., SIAM, Philadelphia, PA (1988). Zbl0652.76022
- V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Spring-Verlag, Berlin (1986). Zbl0585.65077
- M. Gunzburger and H.K. Lee, An optimization-based domain decomposition method for the Navier-Stokes equations. SIAM J. Numer. Anal.37 (2000) 1455–1480. Zbl1003.76024
- M. Gunzburger and R. Nicolaides, On substructuring algorithms and solution techniques for numerical approximation of partial differential equations. Appl. Numer. Math.2 (1986) 243–256. Zbl0645.65066
- P. Le Tallec, Domain decomposition methods in computational mechanics. Comput. Mech. Adv.1 (1994) 121–220. Zbl0802.73079
- P.L. Lions, On the Schwarz alternating method, in Proc. of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G.H. Golub, G.A. Meurant and J. Periaux Eds., SIAM, Philadelphia (1988) 1–42.
- S.H. Lui, On Schwarz alternating methods for nonlinear PDEs. SIAM J. Sci. Comput.21 (2000) 1506–1523. Zbl0959.65140
- S.H. Lui, On Schwarz alternating methods for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput.22 (2001) 1974–1986. Zbl1008.76077
- S.H. Lui, On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs. Numer. Math.93 (2002) 109–129. Zbl1010.65052
- L.D. Marini and A. Quarteroni, A relaxation procedure for domain decomposition methods using finite elements. Numer. Math.55 (1989) 575–598. Zbl0661.65111
- A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999). Zbl0931.65118
- B.F. Smith, P.E. Bjorstad and W.D. Gropp, Domain Decomposition: Parallel Multilevel Algorithms for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge, UK (1996). Zbl0857.65126
- R. Teman, The Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1977).
- J. Xu and J. Zou, Some nonoverlapping domain decomposition methods. SIAM Rev.40 (1998) 867–914. Zbl0913.65115

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.