Finite element approximation of kinetic dilute polymer models with microscopic cut-off
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 1, p. 39-89

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω ⊂ ${ℝ}^{d}$, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as 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, crucial features of which are the presence of a centre-of-mass diffusion term and a cut-off function ${\beta }^{L}\left(·\right):=min\left(·,L\right)$ in the drag and convective terms, where L ≫ 1. We focus on finitely-extensible nonlinear elastic, FENE-type, dumbbell models. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters 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. Our arguments therefore also provide a new proof of global existence of weak solutions to Fokker-Planck-Navier-Stokes systems with centre-of-mass diffusion and microscopic cut-off. The convergence proof rests on several auxiliary technical results including the stability, in the Maxwellian-weighted H1 norm, of the orthogonal projector in the Maxwellian-weighted L2 inner product onto finite element spaces consisting of continuous piecewise linear functions. We establish optimal-order quasi-interpolation error bounds in the Maxwellian-weighted L2 and H1 norms, and prove a new elliptic regularity result in the Maxwellian-weighted H2 norm.

DOI : https://doi.org/10.1051/m2an/2010030
Classification:  35Q30,  35J70,  35K65,  65M12,  65M60,  76A05,  82D60
Keywords: finite element method, polymeric flow models, convergence analysis, existence of weak solutions, Navier-Stokes equations, Fokker-Planck equations, FENE
@article{M2AN_2011__45_1_39_0,
author = {Barrett, John W. and S\"uli, Endre},
title = {Finite element approximation of kinetic dilute polymer models with microscopic cut-off},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {45},
number = {1},
year = {2011},
pages = {39-89},
doi = {10.1051/m2an/2010030},
zbl = {1291.35170},
mrnumber = {2781131},
language = {en},
url = {http://www.numdam.org/item/M2AN_2011__45_1_39_0}
}

Barrett, John W.; Süli, Endre. Finite element approximation of kinetic dilute polymer models with microscopic cut-off. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 1, pp. 39-89. doi : 10.1051/m2an/2010030. http://www.numdam.org/item/M2AN_2011__45_1_39_0/

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