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

Barrett, John W.; Süli, Endre

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 1, pp. 39-89.

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 ${\beta }^L(·):=min(·,L)$ 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.

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