A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 1, p. 141-174

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 L 2 norm in space and time, and thus has a higher-order than if it were measured in the H 1 norm in space. We present the following results: if h=H 2 =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.

DOI : https://doi.org/10.1051/m2an:2007056
Classification:  35Q30,  74S10,  76D05
Keywords: two-grid scheme, non-linear problem, incompressible flow, time and space discretizations, duality argument, “superconvergence”
@article{M2AN_2008__42_1_141_0,
     author = {Abboud, Hyam and Sayah, Toni},
     title = {A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {1},
     year = {2008},
     pages = {141-174},
     doi = {10.1051/m2an:2007056},
     zbl = {1137.76032},
     mrnumber = {2387425},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2008__42_1_141_0}
}
Abboud, Hyam; Sayah, Toni. A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 42 (2008) no. 1, pp. 141-174. doi : 10.1051/m2an:2007056. http://www.numdam.org/item/M2AN_2008__42_1_141_0/

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