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

Published online by Cambridge University Press:  12 January 2008

Hyam Abboud  and
Toni Sayah
Hyam Abboud
Affiliation:
: Faculté des Sciences et de Génie Informatique, Université Saint-Esprit de Kaslik, B.P. 446 Jounieh, Liban. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (Paris 6), Boîte Courrier 187, 4, place Jussieu, 75252 Paris Cedex 05, France. abboud@ann.jussieu.fr Faculté des Sciences, Université Saint-Joseph, B.P. 11-514 Riad El Solh, Beyrouth 1107 2050, Liban.
Toni Sayah
Affiliation:
Faculté des Sciences, Université Saint-Joseph, B.P. 11-514 Riad El Solh, Beyrouth 1107 2050, Liban.
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Abstract

We study a two-grid scheme fully discrete in time andspace for solving the Navier-Stokes system. In the first step, thefully non-linear problem is discretized in space on a coarse gridwith mesh-size H and time step k. In the second step, theproblem is discretized in space on a fine grid with mesh-size hand the same time step, and linearized around the velocity u H computed in the first step. The two-grid strategy is motivated bythe fact that under suitable assumptions, the contribution ofu H to the error in the non-linear term, is measured in theL 2 norm in space and time, and thus has a higher-order than ifit were measured in the H 1 norm in space. We present thefollowing results: if h = H2 = k, then the global error ofthe two-grid algorithm is of the order of h, the same as wouldhave been obtained if the non-linear problem had been solveddirectly on thefine grid.

Keywords

Two-grid schemenon-linear problemincompressible flowtime and space discretizationsdualityargument“superconvergence”.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 1 , January 2008 , pp. 141 - 174
Copyright
© EDP Sciences, SMAI, 2008

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