Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, p. 949-978

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 $\Omega \subset {ℝ}^{d},d=2$ or $3$, 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 ${u}_{0}$ for the Navier-Stokes equation and a nonnegative initial probability density function ${\psi }_{0}$ for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian $M$.

DOI : https://doi.org/10.1051/m2an/2011062
Classification:  35Q30,  35K65,  65M12,  65M60,  76A05,  82D60
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {46},
number = {4},
year = {2012},
pages = {949-978},
doi = {10.1051/m2an/2011062},
mrnumber = {2891476},
language = {en},
url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, pp. 949-978. doi : 10.1051/m2an/2011062. http://www.numdam.org/item/M2AN_2012__46_4_949_0/

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