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

Abboud, Hyam; Sayah, Toni

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

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

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