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

Barrett, John W.; Süli, Endre

doi:10.1051/m2an/2010030

Barrett, John W. ; Süli, Endre

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

Résumé

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 $β^{L} (\cdot) : = min (\cdot, L)$ 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.

MR Zbl | 2 citations dans Numdam

DOI : 10.1051/m2an/2010030

Classification : 35Q30, 35J70, 35K65, 65M12, 65M60, 76A05, 82D60
Mots clés : finite element method, polymeric flow models, convergence analysis, existence of weak solutions, Navier-Stokes equations, Fokker-Planck equations, FENE

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     author = {Barrett, John W. and S\"uli, Endre},
     title = {Finite element approximation of kinetic dilute polymer models with microscopic cut-off},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {39--89},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {1},
     year = {2011},
     doi = {10.1051/m2an/2010030},
     mrnumber = {2781131},
     zbl = {1291.35170},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2010030/}
}

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Barrett, John W.; Süli, Endre. Finite element approximation of kinetic dilute polymer models with microscopic cut-off. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 1, pp. 39-89. doi : 10.1051/m2an/2010030. http://www.numdam.org/articles/10.1051/m2an/2010030/

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