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.
Mots-clés : mixed formulation, divergence-zero finite elements, inf-sup condition, uniform $W^{1,p}$-stability, Hood-Taylor method, streamline diffusion
@article{M2AN_2001__35_6_1007_0,
author = {Girault, Vivette and Scott, Larkin Ridgway},
title = {Finite-element discretizations of a two-dimensional grade-two fluid model},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1007--1053},
publisher = {EDP-Sciences},
volume = {35},
number = {6},
year = {2001},
mrnumber = {1873516},
zbl = {1032.76033},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_6_1007_0/}
}
TY - JOUR AU - Girault, Vivette AU - Scott, Larkin Ridgway TI - Finite-element discretizations of a two-dimensional grade-two fluid model JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2001 SP - 1007 EP - 1053 VL - 35 IS - 6 PB - EDP-Sciences UR - http://www.numdam.org/item/M2AN_2001__35_6_1007_0/ LA - en ID - M2AN_2001__35_6_1007_0 ER -
%0 Journal Article %A Girault, Vivette %A Scott, Larkin Ridgway %T Finite-element discretizations of a two-dimensional grade-two fluid model %J ESAIM: Modélisation mathématique et analyse numérique %D 2001 %P 1007-1053 %V 35 %N 6 %I EDP-Sciences %U http://www.numdam.org/item/M2AN_2001__35_6_1007_0/ %G en %F M2AN_2001__35_6_1007_0
Girault, Vivette; Scott, Larkin Ridgway. Finite-element discretizations of a two-dimensional grade-two fluid model. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 6, pp. 1007-1053. http://www.numdam.org/item/M2AN_2001__35_6_1007_0/
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