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

• Volume: 35, Issue: 6, page 1007-1053
• ISSN: 0764-583X

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## Abstract

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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.

## How to cite

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Girault, Vivette, and Scott, Larkin Ridgway. "Finite-element discretizations of a two-dimensional grade-two fluid model." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.6 (2001): 1007-1053. <http://eudml.org/doc/194085>.

@article{Girault2001,
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.},
author = {Girault, Vivette, Scott, Larkin Ridgway},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {mixed formulation; divergence-zero finite elements; inf-sup condition; uniform $W^\{1,p\}$-stability; Hood-Taylor method; streamline diffusion; tangential boundary condition; two-dimensional polygon; generalized-Stokes problem; -stability; transport equation; divergence-free discrete velocities; centered discretization; Hood-Taylor discretizations; error estimates},
language = {eng},
number = {6},
pages = {1007-1053},
publisher = {EDP-Sciences},
title = {Finite-element discretizations of a two-dimensional grade-two fluid model},
url = {http://eudml.org/doc/194085},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Girault, Vivette
AU - Scott, Larkin Ridgway
TI - Finite-element discretizations of a two-dimensional grade-two fluid model
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 6
SP - 1007
EP - 1053
AB - 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.
LA - eng
KW - mixed formulation; divergence-zero finite elements; inf-sup condition; uniform $W^{1,p}$-stability; Hood-Taylor method; streamline diffusion; tangential boundary condition; two-dimensional polygon; generalized-Stokes problem; -stability; transport equation; divergence-free discrete velocities; centered discretization; Hood-Taylor discretizations; error estimates
UR - http://eudml.org/doc/194085
ER -

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