Two-grid finite-element schemes for the transient Navier-Stokes problem
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 5, p. 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
@article{M2AN_2001__35_5_945_0,
author = {Girault, Vivette and Lions, Jacques-Louis},
title = {Two-grid finite-element schemes for the transient Navier-Stokes problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {35},
number = {5},
year = {2001},
pages = {945-980},
zbl = {1032.76032},
mrnumber = {1866277},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_5_945_0}
}

Girault, Vivette; Lions, Jacques-Louis. Two-grid finite-element schemes for the transient Navier-Stokes problem. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 5, pp. 945-980. http://www.numdam.org/item/M2AN_2001__35_5_945_0/

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