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 ⊂ , 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 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.
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 },
pages = {39--89},
year = {2011},
publisher = {EDP Sciences},
volume = {45},
number = {1},
doi = {10.1051/m2an/2010030},
mrnumber = {2781131},
zbl = {1291.35170},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2010030/}
}
TY - JOUR AU - Barrett, John W. AU - Süli, Endre TI - Finite element approximation of kinetic dilute polymer models with microscopic cut-off JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2011 SP - 39 EP - 89 VL - 45 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2010030/ DO - 10.1051/m2an/2010030 LA - en ID - M2AN_2011__45_1_39_0 ER -
%0 Journal Article %A Barrett, John W. %A Süli, Endre %T Finite element approximation of kinetic dilute polymer models with microscopic cut-off %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2011 %P 39-89 %V 45 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2010030/ %R 10.1051/m2an/2010030 %G en %F 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 , Tome 45 (2011) no. 1, pp. 39-89. doi: 10.1051/m2an/2010030
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