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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-twofluid model, with atangential boundary condition, in a two-dimensional polygon. The exactproblem is split into ageneralized Stokes problem and a transport equation, in such a way that italways has a solutionwithout restriction on the shape of the domain and on the size of the data.The first scheme usesdivergence-free discrete velocities and a centered discretization of thetransport term, whereas theother schemes use Hood-Taylor discretizations for the velocity andpressure, and either a centered or an upwinddiscretization of the transport term. One facet of our analysis isthat, without restrictions on the data,each scheme has a discrete solution and all discrete solutions convergestrongly to solutions of theexact problem. Furthermore, if the domain is convex and the data satisfycertain conditions, eachscheme satisfies error inequalities that lead to error estimates.

Keywords

Mixed formulationdivergence-zerofinite 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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