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

^{1}
Department of Mathematics, Imperial College London,
London
SW7 2AZ,
UK

j.barrett@ imperial.ac.uk

^{2}
Mathematical Institute, University of Oxford,
24–29 St. Giles’, OX1
3 LB Oxford,
UK

endre.suli@ maths.ox.ac.uk

Received:
19
March
2011

Revised:
3
August
2011

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* ⊂ ℝ^{d}, *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 for the Navier–Stokes equation and a nonnegative initial probability
density function *ψ*_{0} for the Fokker–Planck equation, which has
finite relative entropy with respect to the Maxwellian *M*.

Mathematics Subject Classification: 35Q30 / 35K65 / 65M12 / 65M60 / 76A05 / 82D60

Key words: Finite element method / convergence analysis / existence of weak solutions / kinetic polymer models / FENE dumbbell / Navier–Stokes equations / Fokker–Planck equations

*© EDP Sciences, SMAI, 2012*