# Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

- Volume: 46, Issue: 4, page 949-978
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topBarrett, John W., and Süli, Endre. "Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers." ESAIM: Mathematical Modelling and Numerical Analysis 46.4 (2012): 949-978. <http://eudml.org/doc/276384>.

@article{Barrett2012,

abstract = {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
Ω ⊂ ℝd, 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 \hbox\{$\absundertilde$\} for the Navier–Stokes equation and a nonnegative initial probability
density function ψ0 for the Fokker–Planck equation, which has
finite relative entropy with respect to the Maxwellian M.},

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

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

keywords = {Finite element method; convergence analysis; existence of weak solutions; kinetic polymer models; FENE dumbbell; Navier–Stokes equations; Fokker–Planck equations; finite element method; Navier-Stokes equations; Fokker-Planck equations},

language = {eng},

month = {2},

number = {4},

pages = {949-978},

publisher = {EDP Sciences},

title = {Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers},

url = {http://eudml.org/doc/276384},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Barrett, John W.

AU - Süli, Endre

TI - Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2012/2//

PB - EDP Sciences

VL - 46

IS - 4

SP - 949

EP - 978

AB - 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
Ω ⊂ ℝd, 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 \hbox{$\absundertilde$} for the Navier–Stokes equation and a nonnegative initial probability
density function ψ0 for the Fokker–Planck equation, which has
finite relative entropy with respect to the Maxwellian M.

LA - eng

KW - Finite element method; convergence analysis; existence of weak solutions; kinetic polymer models; FENE dumbbell; Navier–Stokes equations; Fokker–Planck equations; finite element method; Navier-Stokes equations; Fokker-Planck equations

UR - http://eudml.org/doc/276384

ER -

## References

top- L. Ambrosio, Transport equation and Cauchy problem for BV vector fields. Invent. Math.158 (2004) 227–260.
- J.W. Barrett and R. Nürnberg, Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces. IMA J. Numer. Anal.24 (2004) 323–363.
- J.W. Barrett and E. Süli, Existence of global weak solutions to some regularized kinetic models of dilute polymers. Multiscale Model. Simul.6 (2007) 506–546.
- J.W. Barrett and E. Süli, Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off. Math. Models Methods Appl. Sci.18 (2008) 935–971.
- J.W. Barrett and E. Süli, Numerical approximation of corotational dumbbell models for dilute polymers. IMA J. Numer. Anal.29 (2009) 937–959.
- J.W. Barrett and E. Süli, Existence and equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers. Available as arXiv:1004.1432v2 [math.AP] from (2010). URIhttp://arxiv.org/abs/1004.1432
- J.W. Barrett and E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers I : Finitely extensible nonlinear bead-spring chains. Math. Models Methods Appl. Sci.21 (2011) 1211–1289.
- J.W. Barrett and E. Süli, Finite element approximation of kinetic dilute polymer models with microscopic cut-off. ESAIM : M2AN45 (2011) 39–89.
- J.W. Barrett and E. Süli, Existence and equilibration of global weak solutions to kinetic models for dilute polymers II : Hookean bead-spring chains. Math. Models Methods Appl. Sci.22 (2012), to appear. Extended version available as arXiv:1008.3052 [math.AP] from . URIhttp://arxiv.org/abs/1008.3052
- A.V. Bhave, R.C. Armstrong and R.A. Brown, Kinetic theory and rheology of dilute, nonhomogeneous polymer solutions. J. Chem. Phys.95 (1991) 2988–3000.
- J. Brandts, S. Korotov, M. Křížek and J. Šolc, On acute and nonobtuse simplicial partitions. Helsinki University of Technology, Institute of Mathematics, Research Reports, A503 (2006).
- F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin (1991).
- Ph. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
- P. Degond and H. Liu, Kinetic models for polymers with inertial effects. Netw. Heterog. Media4 (2009) 625–647.
- R.J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math.98 (1989) 511–547.
- D. Eppstein, J.M. Sullivan and A. Üngör, Tiling space and slabs with acute tetrahedra. Comput. Geom.27 (2004) 237–255.
- G. Grün and M. Rumpf, Nonnegativity preserving numerical schemes for the thin film equation. Numer. Math.87 (2000) 113–152.
- J.-I. Itoh and T. Zamfirescu, Acute triangulations of the regular dodecahedral surface. Eur. J. Comb.28 (2007) 1072–1086.
- D.J. Knezevic and E. Süli, A deterministic multiscale approach for simulating dilute polymeric fluids, in BAIL 2008 – boundary and interior layers. Lect. Notes Comput. Sci. Eng.69 (2009) 23–38.
- D.J. Knezevic and E. Süli, A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model. ESAIM : M2AN43 (2009) 1117–1156.
- D.J. Knezevic and E. Süli, Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift. ESAIM : M2AN43 (2009) 445–485.
- S. Korotov and M. Křížek, Acute type refinements of tetrahedral partitions of polyhedral domains. SIAM J. Numer. Anal.39 (2001) 724–733.
- S. Korotov and M. Křížek, Global and local refinement techniques yielding nonobtuse tetrahedral partitions. Comput. Math. Appl.50 (2005) 1105–1113.
- P.-L. Lions and N. Masmoudi, Global existence of weak solutions to some micro-macro models. C. R. Math. Acad. Sci. Paris345 (2007) 15–20.
- N. Masmoudi, Well posedness of the FENE dumbbell model of polymeric flows. Comm. Pure Appl. Math.61 (2008) 1685–1714.
- N. Masmoudi, Global existence of weak solutions to the FENE dumbbell model of polymeric flows. Preprint (2010).
- R.H. Nochetto, Finite element methods for parabolic free boundary problems, in Advances in Numerical Analysis I. Lancaster (1990); Oxford Sci. Publ., Oxford Univ. Press, New York (1991) 34–95.
- J.D. Schieber, Generalized Brownian configuration field for Fokker–Planck equations including center-of-mass diffusion. J. Non-Newtonian Fluid Mech.135 (2006) 179–181.
- W.H.A. Schilders and E.J.W. ter Maten, Eds., Numerical Methods in Electromagnetics, Handbook of Numerical AnalysisXIII. North-Holland, Amsterdam (2005).
- R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications 2. North-Holland, Amsterdam (1984).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.