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Finite-element discretizations of a two-dimensional grade-two fluidmodel

Published online by Cambridge University Press:  15 April 2002

Vivette Girault  and
Larkin Ridgway Scott
Vivette Girault
Affiliation:
Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, 75252 Paris Cedex 05, France.
Larkin Ridgway Scott
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637-1581, USA.
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Abstract

We propose and analyze several finite-element schemes for solving a grade-two fluid model, with a tangential boundary condition, in a two-dimensional polygon. The exact problem is split into a generalized Stokes problem and a transport equation, in such a way that it always has a solution without restriction on the shape of the domain and on the size of the data. The first scheme uses divergence-free discrete velocities and a centered discretization of the transport term, whereas the other schemes use Hood-Taylor discretizations for the velocity and pressure, and either a centered or an upwind discretization of the transport term. One facet of our analysis is that, without restrictions on the data, each scheme has a discrete solution and all discrete solutions converge strongly to solutions of the exact problem. Furthermore, if the domain is convex and the data satisfy certain conditions, each scheme satisfies error inequalities that lead to error estimates.

Keywords

Mixed formulationdivergence-zero finite elementsinf-sup conditionuniform W1,p-stabilityHood-Taylor methodstreamline diffusion.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 6 , November 2001 , pp. 1007 - 1053
Copyright
© EDP Sciences, SMAI, 2001

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