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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