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.
Mots clés : nonoverlapping domain decomposition, incompressible Navier-Stokes equations, finite elements, nonlinear problems
@article{M2AN_2005__39_6_1251_0,
author = {Xu, Xuejun and Chow, C. O. and Lui, S. H.},
title = {On nonoverlapping domain decomposition methods for the incompressible {Navier-Stokes} equations},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1251--1269},
publisher = {EDP-Sciences},
volume = {39},
number = {6},
year = {2005},
doi = {10.1051/m2an:2005046},
mrnumber = {2195911},
zbl = {1085.76041},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an:2005046/}
}
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: Modélisation mathématique et analyse numérique PY - 2005 SP - 1251 EP - 1269 VL - 39 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2005046/ DO - 10.1051/m2an:2005046 LA - en ID - M2AN_2005__39_6_1251_0 ER -
%0 Journal Article %A Xu, Xuejun %A Chow, C. O. %A Lui, S. H. %T On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations %J ESAIM: Modélisation mathématique et analyse numérique %D 2005 %P 1251-1269 %V 39 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an:2005046/ %R 10.1051/m2an:2005046 %G en %F M2AN_2005__39_6_1251_0
Xu, Xuejun; Chow, C. O.; Lui, S. H. On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations. ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 6, pp. 1251-1269. doi : 10.1051/m2an:2005046. http://archive.numdam.org/articles/10.1051/m2an:2005046/
[1] , 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). | MR | Zbl
[2] , and, 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).
[3] and, Domain decomposition algorithm. Acta Numerica (1994) 61-143. | Zbl
[4] , The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl
[5] ,, and, 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). | MR | Zbl
[6] , and, Multimodels for incompressible flows. J. Math. Fluid Dynamics 2 (2000) 126-150. | Zbl
[7] and, 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). | Zbl
[8] and, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Spring-Verlag, Berlin (1986). | MR | Zbl
[9] and, An optimization-based domain decomposition method for the Navier-Stokes equations. SIAM J. Numer. Anal. 37 (2000) 1455-1480. | Zbl
[10] and, On substructuring algorithms and solution techniques for numerical approximation of partial differential equations. Appl. Numer. Math. 2 (1986) 243-256. | Zbl
[11] , Domain decomposition methods in computational mechanics. Comput. Mech. Adv. 1 (1994) 121-220. | Zbl
[12] , 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. | Zbl
[13] , On Schwarz alternating methods for nonlinear PDEs. SIAM J. Sci. Comput. 21 (2000) 1506-1523. | Zbl
[14] , On Schwarz alternating methods for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 22 (2001) 1974-1986. | Zbl
[15] , On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs. Numer. Math. 93 (2002) 109-129. | Zbl
[16] and, A relaxation procedure for domain decomposition methods using finite elements. Numer. Math. 55 (1989) 575-598. | Zbl
[17] and, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999). | MR | Zbl
[18] , and, Domain Decomposition: Parallel Multilevel Algorithms for Elliptic Partial Differential Equations. Cambridge University Press, Cambridge, UK (1996). | MR | Zbl
[19] , The Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1977). | Zbl
[20] and, Some nonoverlapping domain decomposition methods. SIAM Rev. 40 (1998) 867-914. | Zbl
Cité par Sources :
