# Finite element approximation of kinetic dilute polymer models with microscopic cut-off

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

- Volume: 45, Issue: 1, page 39-89
- ISSN: 0764-583X

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topBarrett, John W., and Süli, Endre. "Finite element approximation of kinetic dilute polymer models with microscopic cut-off." ESAIM: Mathematical Modelling and Numerical Analysis 45.1 (2011): 39-89. <http://eudml.org/doc/197392>.

@article{Barrett2011,

abstract = {
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 ⊂ $\mathbb\{R\}^d$, 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 $\beta^L(\cdot) :=\min(\cdot,L)$
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.
},

author = {Barrett, John W., Süli, Endre},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Finite element method; polymeric flow models; convergence analysis; existence of weak solutions;
Navier–Stokes equations; Fokker–Planck equations; FENE; polymeric flow model; incompressible Navier-Stokes equations; weak solutions; finite element method},

language = {eng},

month = {1},

number = {1},

pages = {39-89},

publisher = {EDP Sciences},

title = {Finite element approximation of kinetic dilute polymer models with microscopic cut-off},

url = {http://eudml.org/doc/197392},

volume = {45},

year = {2011},

}

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

DA - 2011/1//

PB - EDP Sciences

VL - 45

IS - 1

SP - 39

EP - 89

AB -
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 ⊂ $\mathbb{R}^d$, 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 $\beta^L(\cdot) :=\min(\cdot,L)$
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.

LA - eng

KW - Finite element method; polymeric flow models; convergence analysis; existence of weak solutions;
Navier–Stokes equations; Fokker–Planck equations; FENE; polymeric flow model; incompressible Navier-Stokes equations; weak solutions; finite element method

UR - http://eudml.org/doc/197392

ER -

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