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Theses

Schémas à deux-grilles pour la résolution du problème de Navier-Stokes instationnaire incompressible

Abstract : This thesis focuses on the time-dependent incompressible Navier-Stokes problem totally discretized in time and space, in two dimensions, by a two-grid method.
In the first part, we extend the two-grid method, applied
by V. Girault and J.-L. Lions to the transient semi-discretized Navier-Stokes problem, to the totally discretized Navier-Stokes problem in time (by a first order scheme) and in space (by a first order finite element method).
In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step Delta t. In the second step, the problem is linearized around the velocity u_H computed in the first step and is discretized in space on a fine grid with mesh-size h and the same time step.
The two-grid strategy is motivated by the fact that under suitable
assumptions, the contribution of u_H to the error in the non-linear term, is measured in the L^2 norm in space and time, and thus has hopefully a higher-order than if it were measured in the H^1 norm in space.
In the second part, since our objectif is to gain in order of convergence and in complexity of the scheme, we study a second
order time accuracy two-grid scheme for the totally discretized in
time and space Navier-Stokes problem.
We present the following results: for the first order scheme, if h = H^2 = Delta t, then the global error of the two-grid algorithm is of the order of h. For the second order scheme, if h^2 = (\Delta t)^2 = H^3, then the global error of the two-grid algorithm is of the order of h^2: the same results as would have been obtained if the non-linear problem had been solved directly on the fine grid.
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https://tel.archives-ouvertes.fr/tel-00132658
Contributor : Hyam Abboud <>
Submitted on : Thursday, February 22, 2007 - 12:10:17 PM
Last modification on : Wednesday, December 9, 2020 - 3:09:21 PM
Long-term archiving on: : Wednesday, April 7, 2010 - 12:30:06 AM

Identifiers

  • HAL Id : tel-00132658, version 1

Citation

Hyam Abboud. Schémas à deux-grilles pour la résolution du problème de Navier-Stokes instationnaire incompressible. Mathématiques [math]. Université Pierre et Marie Curie - Paris VI; Université Saint-Joseph, Beyrouth, 2006. Français. ⟨tel-00132658⟩

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