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

Published online by Cambridge University Press:  12 January 2008

Hyam Abboud  and
Toni Sayah
Hyam Abboud
Affiliation:
: Faculté des Sciences et de Génie Informatique, Université Saint-Esprit de Kaslik, B.P. 446 Jounieh, Liban. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris 6), Boîte Courrier 187, 4, place Jussieu, 75252 Paris Cedex 05, France. abboud@ann.jussieu.fr Faculté des Sciences, Université Saint-Joseph, B.P. 11-514 Riad El Solh, Beyrouth 1107 2050, Liban.
Toni Sayah
Affiliation:
Faculté des Sciences, Université Saint-Joseph, B.P. 11-514 Riad El Solh, Beyrouth 1107 2050, Liban.
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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.

Keywords

Two-grid schemenon-linear problemincompressible flowtime and space discretizationsduality argument“superconvergence”.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 1 , January 2008 , pp. 141 - 174
Copyright
© EDP Sciences, SMAI, 2008

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References

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. I 341 (2005).
H. Abboud, V. Girault and T. Sayah, Second-order two-grid finite element scheme for the fully discrete transient Navier-Stokes equations. Preprint, http://www.ann.jussieu.fr/publications/2007/R07040.html.
R.-A. Adams, Sobolev Spaces. Academic Press, New York (1975).
Arnold, D., Brezzi, F. and Fortin, M., A stable finite element for the Stokes equations. Calcolo 21 (1984) 337–344. CrossRef
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).
Girault, V. and Lions, J.-L., Two-grid finite-element schemes for the steady Navier-Stokes problem in polyhedra. Portugal. Math. 58 (2001) 25–57.
Girault, V. and Lions, J.-L., Two-grid finite-element schemes for the transient Navier-Stokes equations. ESAIM: M2AN 35 (2001) 945–980. CrossRef
V. Girault and P.-A. Raviart, Finite Element Methods for the Navier-Stokes Equations. Theory and Algorithms, in Springer Series in Computational Mathematics 5, Springer-Verlag, Berlin (1986).
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Monographs and Studies in Mathematics 24. Pitman, Boston, (1985).
F. Hecht and O. Pironneau, FreeFem++. See: http://www.freefem.org.
O.A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow. (In Russian, 1961), First English translation, Gordon & Breach, New York (1963).
Layton, W., A two-level discretization method for the Navier-Stokes equations. Computers Math. Applic. 26 (1993) 33–38. CrossRef
Layton, W. and Lenferink, W., Two-level Picard-defect corrections for the Navier-Stokes equations at high Reynolds number. Applied Math. Comput. 69 (1995) 263–274.
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969).
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications I. Dunod, Paris (1968).
J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967).
Temam, R., Une méthode d'approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 98 (1968) 115–152. CrossRef
Wheeler, M.F., A priori L 2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM. J. Numer. Anal. 10 (1973) 723–759. CrossRef
J. Xu, Some Two-Grid Finite Element Methods. Tech. Report, P.S.U. (1992).
Xu, J., A novel two-grid method of semilinear elliptic equations. SIAM J. Sci. Comput. 15 (1994) 231–237. CrossRef
Two-grid, J. Xu finite element discretization techniques for linear and nonlinear PDE. SIAM J. Numer. Anal. 33 (1996) 1759–1777.