On nonoverlapping domain decomposition methods for the incompressible Navier-Stokes equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 6, pp. 1251-1269.

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.

DOI : https://doi.org/10.1051/m2an:2005046
Classification : 65F10,  65N30,  65N55
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: Mathematical Modelling and Numerical Analysis - 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},
zbl = {1085.76041},
mrnumber = {2195911},
language = {en},
url = {http://www.numdam.org/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 39 (2005) no. 6, pp. 1251-1269. doi : 10.1051/m2an:2005046. http://www.numdam.org/item/M2AN_2005__39_6_1251_0/

[1] 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). | MR 972509 | Zbl 0654.76020

[2] 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).

[3] T.F. Chan and T.P. Mathew, Domain decomposition algorithm. Acta Numerica (1994) 61-143. | Zbl 0809.65112

[4] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[5] 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). | MR 972518 | Zbl 0652.76023

[6] L. Fatone, P. Gervasio and A. Quarteroni, Multimodels for incompressible flows. J. Math. Fluid Dynamics 2 (2000) 126-150. | Zbl 0962.76021

[7] 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). | Zbl 0652.76022

[8] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Spring-Verlag, Berlin (1986). | MR 851383 | Zbl 0585.65077

[9] 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. | Zbl 1003.76024

[10] 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. | Zbl 0645.65066

[11] P. Le Tallec, Domain decomposition methods in computational mechanics. Comput. Mech. Adv. 1 (1994) 121-220. | Zbl 0802.73079

[12] 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. | Zbl 0658.65090

[13] S.H. Lui, On Schwarz alternating methods for nonlinear PDEs. SIAM J. Sci. Comput. 21 (2000) 1506-1523. | Zbl 0959.65140

[14] S.H. Lui, On Schwarz alternating methods for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 22 (2001) 1974-1986. | Zbl 1008.76077

[15] S.H. Lui, On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs. Numer. Math. 93 (2002) 109-129. | Zbl 1010.65052

[16] L.D. Marini and A. Quarteroni, A relaxation procedure for domain decomposition methods using finite elements. Numer. Math. 55 (1989) 575-598. | Zbl 0661.65111

[17] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999). | MR 1857663 | Zbl 0931.65118

[18] 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). | MR 1410757 | Zbl 0857.65126

[19] R. Teman, The Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1977). | Zbl 0383.35057

[20] J. Xu and J. Zou, Some nonoverlapping domain decomposition methods. SIAM Rev. 40 (1998) 867-914. | Zbl 0913.65115