EDP Sciences Journals List
Home Document
Subscriber Authentication Point
Free access article

Issue ESAIM: 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 Scott2

1  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 W1,p-stability, Hood-Taylor method, streamline diffusion.


© EDP Sciences, SMAI 2001


What is OpenURL?

The OpenURL standard is a protocol for transmission of metadata describing the resource that you wish to access. An OpenURL link contains article metadata and directs it to the OpenURL server of your choice. The OpenURL server can provide access to the resource and also offer complementary services (specific search engine, export of references...). The OpenURL link can be generated by different means.
  • If your librarian has set up your subscription with an OpenURL resolver, OpenURL links appear automatically on the abstract pages.
  • You can define your own OpenURL resolver with your EDPS Account. In this case your choice will be given priority over that of your library.
  • You can use an add-on for your browser (Firefox or I.E.) to display OpenURL links on a page (see http://www.openly.com/openurlref/). You should disable this module if you wish to use the OpenURL server that you or your library have defined.