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

• Volume: 35, Issue: 5, page 945-980
• ISSN: 0764-583X

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## Abstract

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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 uH 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 uH to the error analysis is measured in the L2 norm in space and time, and thus, for the lowest-degree elements, is of the order of H2. Hence, an error of the order of h can be recovered at the second step, provided h = H2.

## How to cite

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Girault, Vivette, and Lions, Jacques-Louis. "Two-grid finite-element schemes for the transient Navier-Stokes problem." ESAIM: Mathematical Modelling and Numerical Analysis 35.5 (2010): 945-980. <http://eudml.org/doc/197497>.

@article{Girault2010,
abstract = { 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 uH 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 uH to the error analysis is measured in the L2 norm in space and time, and thus, for the lowest-degree elements, is of the order of H2. Hence, an error of the order of h can be recovered at the second step, provided h = H2. },
author = {Girault, Vivette, Lions, Jacques-Louis},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Two grids; a priori estimates; duality.; three-dimensional polyhedron; a priori estimates; duality; coarse grid; linearized problem; fine grid; error analysis; lowest-degree elements},
language = {eng},
month = {3},
number = {5},
pages = {945-980},
publisher = {EDP Sciences},
title = {Two-grid finite-element schemes for the transient Navier-Stokes problem},
url = {http://eudml.org/doc/197497},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Girault, Vivette
AU - Lions, Jacques-Louis
TI - Two-grid finite-element schemes for the transient Navier-Stokes problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 5
SP - 945
EP - 980
AB - 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 uH 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 uH to the error analysis is measured in the L2 norm in space and time, and thus, for the lowest-degree elements, is of the order of H2. Hence, an error of the order of h can be recovered at the second step, provided h = H2.
LA - eng
KW - Two grids; a priori estimates; duality.; three-dimensional polyhedron; a priori estimates; duality; coarse grid; linearized problem; fine grid; error analysis; lowest-degree elements
UR - http://eudml.org/doc/197497
ER -

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