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

¹ : 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
² Faculté des Sciences, Université Saint-Joseph, B.P. 11-514 Riad El Solh, Beyrouth 1107 2050, Liban.

Received: 6 June 2006
Revised: 12 December 2006

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 u_H computed in the first 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 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.