We construct a Galerkin finite element method for the numerical approximation of weak solutions to a general class of coupled FENE-type finitely extensible nonlinear elastic dumbbell models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain or , for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. We perform a rigorous passage to the limit as first the spatial discretization parameter, and then the temporal discretization parameter tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier-Stokes-Fokker-Planck system. The passage to the limit is performed under minimal regularity assumptions on the data: a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian .
Mots clés : finite element method, convergence analysis, existence of weak solutions, kinetic polymer models, FENE dumbbell, Navier-Stokes equations, Fokker-Planck equations
@article{M2AN_2012__46_4_949_0,
author = {Barrett, John W. and S\"uli, Endre},
title = {Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {949--978},
publisher = {EDP-Sciences},
volume = {46},
number = {4},
year = {2012},
doi = {10.1051/m2an/2011062},
mrnumber = {2891476},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2011062/}
}
TY - JOUR AU - Barrett, John W. AU - Süli, Endre TI - Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 949 EP - 978 VL - 46 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2011062/ DO - 10.1051/m2an/2011062 LA - en ID - M2AN_2012__46_4_949_0 ER -
%0 Journal Article %A Barrett, John W. %A Süli, Endre %T Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 949-978 %V 46 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2011062/ %R 10.1051/m2an/2011062 %G en %F M2AN_2012__46_4_949_0
Barrett, John W.; Süli, Endre. Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 4, pp. 949-978. doi : 10.1051/m2an/2011062. http://www.numdam.org/articles/10.1051/m2an/2011062/
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