Free access article

Issue M2AN Volume 35, Number 6, November-December 2001 Page(s) 1007 - 1053 DOI 10.1051/m2an:2001147

DOI: 10.1051/m2an:2001147


M2AN, Vol. 35, N°6, pp. 1007-1053

Finite-element discretizations of a two-dimensional grade-two fluid model

Vivette Girault¹ and Larkin Ridgway Scott<sup>2

1</sup>  Laboratoire d'Analyse Numérique, Université Pierre et Marie Curie, 75252 Paris Cedex 05, France.
²  Department of Mathematics, University of Chicago, Chicago, Illinois 60637-1581, USA.

(Received: May 5, 2000. Revised: May 3, 2001.)

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.

AMS Subject: 65D30, 65N15, 65N30.

Key words: Mixed formulation, divergence-zero finite elements, inf-sup condition, uniform W^1,p-stability, Hood-Taylor method, streamline diffusion.


© EDP Sciences, SMAI 2001

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