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

Abboud, Hyam; Sayah, Toni

doi:10.1051/m2an:2007056

Abboud, Hyam ; Sayah, Toni

ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 1, pp. 141-174

Résumé

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.

MR Zbl

DOI : 10.1051/m2an:2007056

Classification : 35Q30, 74S10, 76D05
Keywords: two-grid scheme, non-linear problem, incompressible flow, time and space discretizations, duality argument, “superconvergence”

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     author = {Abboud, Hyam and Sayah, Toni},
     title = {A full discretization of the time-dependent {Navier-Stokes} equations by a two-grid scheme},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {141--174},
     year = {2008},
     publisher = {EDP Sciences},
     volume = {42},
     number = {1},
     doi = {10.1051/m2an:2007056},
     mrnumber = {2387425},
     zbl = {1137.76032},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an:2007056/}
}

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Abboud, Hyam; Sayah, Toni. A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 1, pp. 141-174. doi: 10.1051/m2an:2007056

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