A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme

Hyam Abboud; Toni Sayah

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 1, page 141-174
  • ISSN: 0764-583X

Abstract

top
We study a two-grid scheme fully discrete in time and space for solving the Navier-Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH to the error in the non-linear term, is measured in the L2 norm in space and time, and thus has a higher-order than if it were measured in the H1 norm in space. We present the following results: if h = H2 = k, then the global error of the two-grid algorithm is of the order of h, the same as would have been obtained if the non-linear problem had been solved directly on the fine grid.

How to cite

top

Abboud, Hyam, and Sayah, Toni. "A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme." ESAIM: Mathematical Modelling and Numerical Analysis 42.1 (2008): 141-174. <http://eudml.org/doc/250283>.

@article{Abboud2008,
abstract = { We study a two-grid scheme fully discrete in time and space for solving the Navier-Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH to the error in the non-linear term, is measured in the L2 norm in space and time, and thus has a higher-order than if it were measured in the H1 norm in space. We present the following results: if h = H2 = k, then the global error of the two-grid algorithm is of the order of h, the same as would have been obtained if the non-linear problem had been solved directly on the fine grid. },
author = {Abboud, Hyam, Sayah, Toni},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Two-grid scheme; non-linear problem; incompressible flow; time and space discretizations; duality argument; “superconvergence”.; global error estimate; coarse grid; fine grid},
language = {eng},
month = {1},
number = {1},
pages = {141-174},
publisher = {EDP Sciences},
title = {A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme},
url = {http://eudml.org/doc/250283},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Abboud, Hyam
AU - Sayah, Toni
TI - A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/1//
PB - EDP Sciences
VL - 42
IS - 1
SP - 141
EP - 174
AB - We study a two-grid scheme fully discrete in time and space for solving the Navier-Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and linearized around the velocity uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH to the error in the non-linear term, is measured in the L2 norm in space and time, and thus has a higher-order than if it were measured in the H1 norm in space. We present the following results: if h = H2 = k, then the global error of the two-grid algorithm is of the order of h, the same as would have been obtained if the non-linear problem had been solved directly on the fine grid.
LA - eng
KW - Two-grid scheme; non-linear problem; incompressible flow; time and space discretizations; duality argument; “superconvergence”.; global error estimate; coarse grid; fine grid
UR - http://eudml.org/doc/250283
ER -

References

top
  1. H. Abboud, V. Girault and T. Sayah, Two-grid finite element scheme for the fully discrete time-dependent Navier-Stokes problem. C. R. Acad. Sci. Paris, Ser. I341 (2005).  
  2. H. Abboud, V. Girault and T. Sayah, Second-order two-grid finite element scheme for the fully discrete transient Navier-Stokes equations. Preprint, .  URIhttp://www.ann.jussieu.fr/publications/2007/R07040.html
  3. R.-A. Adams, Sobolev Spaces. Academic Press, New York (1975).  
  4. D. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo21 (1984) 337–344.  
  5. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).  
  6. V. Girault and J.-L. Lions, Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. Portugal. Math.58 (2001) 25–57.  
  7. V. Girault and J.-L. Lions, Two-grid finite-element schemes for the transient Navier-Stokes equations. ESAIM: M2AN35 (2001) 945–980.  
  8. V. Girault and P.-A. Raviart, Finite Element Methods for the Navier-Stokes Equations. Theory and Algorithms, in Springer Series in Computational Mathematics5, Springer-Verlag, Berlin (1986).  
  9. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Monographs and Studies in Mathematics24. Pitman, Boston, (1985).  
  10. F. Hecht and O. Pironneau, FreeFem++. See: .  URIhttp://www.freefem.org
  11. O.A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow. (In Russian, 1961), First English translation, Gordon & Breach, New York (1963).  
  12. W. Layton, A two-level discretization method for the Navier-Stokes equations. Computers Math. Applic.26 (1993) 33–38.  
  13. W. Layton and W. Lenferink, Two-level Picard-defect corrections for the Navier-Stokes equations at high Reynolds number. Applied Math. Comput.69 (1995) 263–274.  
  14. J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969).  
  15. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications I. Dunod, Paris (1968).  
  16. J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967).  
  17. R. Temam, Une méthode d'approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France98 (1968) 115–152.  
  18. M.F. Wheeler, A priori L2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM. J. Numer. Anal.10 (1973) 723–759.  
  19. J. Xu, Some Two-Grid Finite Element Methods. Tech. Report, P.S.U. (1992).  
  20. J. Xu, A novel two-grid method of semilinear elliptic equations. SIAM J. Sci. Comput.15 (1994) 231–237.  
  21. J. Xu, Two-grid finite element discretization techniques for linear and nonlinear PDE. SIAM J. Numer. Anal.33 (1996) 1759–1777.  

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.