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

John W. Barrett; Endre Süli

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2011)

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

Abstract

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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 β L ( · ) : = min ( · , L ) in the drag and convective terms, whereL ≫ 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.

How to cite

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Barrett, John W., and 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 45.1 (2011): 39-89. <http://eudml.org/doc/273175>.

@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, whereL ≫ 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 - Modélisation Mathématique et Analyse Numérique},
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},
language = {eng},
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/273175},
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 - Modélisation Mathématique et Analyse Numérique
PY - 2011
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, whereL ≫ 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
UR - http://eudml.org/doc/273175
ER -

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