A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme
: 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,
2 Faculté des Sciences, Université Saint-Joseph, B.P. 11-514 Riad El Solh, Beyrouth 1107 2050, Liban.
Revised: 12 December 2006
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.
Mathematics Subject Classification: 35Q30 / 74S10 / 76D05
Key words: Two-grid scheme / non-linear problem / incompressible flow / time and space discretizations / duality argument / “superconvergence”.
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