Two-grid finite-element schemes for the transient Navier-Stokes problem

Girault, Vivette; Lions, Jacques-Louis

ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 5, pp. 945-980

We semi-discretize in space a time-dependent Navier-Stokes system on a three-dimensional polyhedron by finite-elements schemes defined on two grids. In the first step, the fully non-linear problem is semi-discretized on a coarse grid, with mesh-size $H$. In the second step, the problem is linearized by substituting into the non-linear term, the velocity ${𝐮}_H$ computed at step one, and the linearized problem is semi-discretized on a fine grid with mesh-size $h$. This approach is motivated by the fact that, on a convex polyhedron and under adequate assumptions on the data, the contribution of ${𝐮}_H$ to the error analysis is measured in the $L^2$ norm in space and time, and thus, for the lowest-degree elements, is of the order of $H^2$. Hence, an error of the order of $h$ can be recovered at the second step, provided $h=H^2$.

Classification : 76D05, 65N15, 65N30, 65N55
Keywords: two grids, a priori estimates, duality

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