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 - Modélisation Mathématique et Analyse Numérique (2005)

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

Abstract

top
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.

How to cite

top

Xu, 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 - Modélisation Mathématique et Analyse Numérique 39.6 (2005): 1251-1269. <http://eudml.org/doc/245765>.

@article{Xu2005,
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 - Modélisation Mathématique et Analyse Numérique},
keywords = {nonoverlapping domain decomposition; incompressible Navier-Stokes equations; finite elements; nonlinear problems; convergence; Dirichlet-Neumann substructuring; finite element approximation},
language = {eng},
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/245765},
volume = {39},
year = {2005},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2005
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/245765
ER -

References

top
  1. [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). Zbl0654.76020MR972509
  2. [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. [3] T.F. Chan and T.P. Mathew, Domain decomposition algorithm. Acta Numerica (1994) 61–143. Zbl0809.65112
  4. [4] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). Zbl0383.65058MR520174
  5. [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). Zbl0652.76023MR972518
  6. [6] L. Fatone, P. Gervasio and A. Quarteroni, Multimodels for incompressible flows. J. Math. Fluid Dynamics 2 (2000) 126–150. Zbl0962.76021
  7. [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). Zbl0652.76022
  8. [8] V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Theory and Algorithms. Spring-Verlag, Berlin (1986). Zbl0585.65077MR851383
  9. [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. Zbl1003.76024
  10. [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. Zbl0645.65066
  11. [11] P. Le Tallec, Domain decomposition methods in computational mechanics. Comput. Mech. Adv. 1 (1994) 121–220. Zbl0802.73079
  12. [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. Zbl0658.65090
  13. [13] S.H. Lui, On Schwarz alternating methods for nonlinear PDEs. SIAM J. Sci. Comput. 21 (2000) 1506–1523. Zbl0959.65140
  14. [14] S.H. Lui, On Schwarz alternating methods for the incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 22 (2001) 1974–1986. Zbl1008.76077
  15. [15] S.H. Lui, On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs. Numer. Math. 93 (2002) 109–129. Zbl1010.65052
  16. [16] L.D. Marini and A. Quarteroni, A relaxation procedure for domain decomposition methods using finite elements. Numer. Math. 55 (1989) 575–598. Zbl0661.65111
  17. [17] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications (1999). Zbl0931.65118MR1857663
  18. [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). Zbl0857.65126MR1410757
  19. [19] R. Teman, The Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland, Amsterdam (1977). Zbl0383.35057
  20. [20] J. Xu and J. Zou, Some nonoverlapping domain decomposition methods. SIAM Rev. 40 (1998) 867–914. Zbl0913.65115

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.