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

Hyam Abboud; Toni Sayah

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 1, page 141-174
  • ISSN: 0764-583X

Abstract

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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 uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH to the error in the non-linear term, is measured in the L2 norm in space and time, and thus has a higher-order than if it were measured in the H1 norm in space. We present the following results: if h = H2 = 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.

How to cite

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Abboud, Hyam, and Sayah, Toni. "A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme." ESAIM: Mathematical Modelling and Numerical Analysis 42.1 (2008): 141-174. <http://eudml.org/doc/250283>.

@article{Abboud2008,
abstract = { 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 uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH to the error in the non-linear term, is measured in the L2 norm in space and time, and thus has a higher-order than if it were measured in the H1 norm in space. We present the following results: if h = H2 = 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. },
author = {Abboud, Hyam, Sayah, Toni},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Two-grid scheme; non-linear problem; incompressible flow; time and space discretizations; duality argument; “superconvergence”.; global error estimate; coarse grid; fine grid},
language = {eng},
month = {1},
number = {1},
pages = {141-174},
publisher = {EDP Sciences},
title = {A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme},
url = {http://eudml.org/doc/250283},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Abboud, Hyam
AU - Sayah, Toni
TI - A full discretization of the time-dependent Navier-Stokes equations by a two-grid scheme
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/1//
PB - EDP Sciences
VL - 42
IS - 1
SP - 141
EP - 174
AB - 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 uH computed in the first step. The two-grid strategy is motivated by the fact that under suitable assumptions, the contribution of uH to the error in the non-linear term, is measured in the L2 norm in space and time, and thus has a higher-order than if it were measured in the H1 norm in space. We present the following results: if h = H2 = 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.
LA - eng
KW - Two-grid scheme; non-linear problem; incompressible flow; time and space discretizations; duality argument; “superconvergence”.; global error estimate; coarse grid; fine grid
UR - http://eudml.org/doc/250283
ER -

References

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