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.

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

[1] F. Antoci, Some necessary and some sufficient conditions for the compactness of the embedding of weighted Sobolev spaces. Ric. Mat. 52 (2003) 55-71. | MR | Zbl

[2] A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck equations. Comm. PDE 26 (2001) 43-100. | MR | Zbl

[3] 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. | MR | Zbl

[4] 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. | MR | Zbl

[5] 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. | MR | Zbl

[6] J.W. Barrett and E. Süli, Numerical approximation of corotational dumbbell models for dilute polymers. IMA J. Numer. Anal. 29 (2009) 937-959. | MR | Zbl

[7] J.W. Barrett, C. Schwab and E. Süli, Existence of global weak solutions for some polymeric flow models. Math. Models Methods Appl. Sci. 15 (2005) 939-983. | MR | Zbl

[8] R. Bird, C. Curtiss, R. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theory. John Wiley and Sons, New York (1987).

[9] S. Bobkov and M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 (2000) 1028-1052. | MR | Zbl

[10] J. Brandts, S. Korotov, M. Křížek and J. Šolc, On nonobtuse simplicial partitions. SIAM Rev. 51 (2009) 317-335. | MR | Zbl

[11] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin (1991). | MR | Zbl

[12] S. Cerrai, Second-order PDEs in Finite and Infinite Dimension, Lecture Notes in Mathematics 1762. Springer-Verlag, Berlin (2001). | MR | Zbl

[13] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl

[14] P. Constantin, Nonlinear Fokker-Planck Navier-Stokes systems. Commun. Math. Sci. 3 (2005) 531-544. | MR | Zbl

[15] G. Da Prato and A. Lunardi, Elliptic operators with unbounded drift coefficients and Neumann boundary condition. J. Differ. Equ. 198 (2004) 35-52. | MR | Zbl

[16] L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Comm. Pure Appl. Math. 54 (2001) 1-42. | MR | Zbl

[17] Q. Du, C. Liu and P. Yu, FENE dumbbell models and its several linear and nonlinear closure approximations. Multiscale Model. Simul. 4 (2005) 709-731. | MR | Zbl

[18] W. E, T.J. Li and P.-W. Zhang, Well-posedness for the dumbbell model of polymeric fluids. Com. Math. Phys. 248 (2004) 409-427. | MR | Zbl

[19] A.W. El-Kareh and L.G. Leal, Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion. J. Non-Newton. Fluid Mech. 33 (1989) 257-287. | Zbl

[20] D. Eppstein, J.M. Sullivan and A. Üngör, Tiling space and slabs with acute tetrahedra. Comput. Geom. 27 (2004) 237-255. | MR | Zbl

[21] G. Grün and M. Rumpf, Nonnegativity preserving numerical schemes for the thin film equation. Numer. Math. 87 (2000) 113-152. | MR | Zbl

[22] J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I: Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982) 275-311. | MR | Zbl

[23] J.-I. Itoh and T. Zamfirescu, Acute triangulations of the regular dodecahedral surface. European J. Combin. 28 (2007) 1072-1086. | MR | Zbl

[24] B. Jourdain, T. Lelièvre and C. Le Bris, Existence of solution for a micro-macro model of polymeric fluid: the FENE model. J. Funct. Anal. 209 (2004) 162-193. | MR | Zbl

[25] B. Jourdain, T. Lelièvre, C. Le Bris and F. Otto, Long-time asymptotics of a multiscle model for polymeric fluid flows. Arch. Rat. Mech. Anal. 181 (2006) 97-148. | MR | Zbl

[26] D. Knezevic and E. Süli, Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift. ESAIM: M2AN 43 (2009) 445-485. | Numdam | MR | Zbl

[27] D. Knezevic and E. Süli, A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model. ESAIM: M2AN 43 (2009) 1117-1156. | Numdam | MR | Zbl

[28] S. Korotov and M. Křížek, Acute type refinements of tetrahedral partitions of polyhedral domains. SIAM J. Numer. Anal. 39 (2001) 724-733. | MR | Zbl

[29] S. Korotov and M. Křížek, Global and local refinement techniques yielding nonobtuse tetrahedral partitions. Comput. Math. Appl. 50 (2005) 1105-1113. | MR | Zbl

[30] A. Kufner, Weighted Sobolev Spaces. Teubner, Stuttgart (1980). | MR | Zbl

[31] T. Lelièvre, Modèles multi-échelles pour les fluides viscoélastiques. Ph.D. Thesis, École National des Ponts et Chaussées, Marne-la-Vallée, France (2004).

[32] T. Li and P.-W. Zhang, Mathematical analysis of multi-scale models of complex fluids. Commun. Math. Sci. 5 (2007) 1-51. | MR | Zbl

[33] T. Li, H. Zhang and P.-W. Zhang, Local existence for the dumbbell model of polymeric fuids. Comm. Partial Differ. Equ. 29 (2004) 903-923. | MR | Zbl

[34] F.-H. Lin, C. Liu and P. Zhang, On a micro-macro model for polymeric fluids near equilibrium. Comm. Pure Appl. Math. 60 (2007) 838-866. | MR | Zbl

[35] P.-L. Lions and N. Masmoudi, Global existence of weak solutions to some micro-macro models. C. R. Math. Acad. Sci. Paris 345 (2007) 15-20. | MR | Zbl

[36] L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups. Chapman & Hall/CRC, Boca Raton (2007). | MR | Zbl

[37] A. Lozinski, C. Chauvière, J. Fang and R.G. Owens, Fokker-Planck simulations of fast flows of melts and concentrated polymer solutions in complex geometries. J. Rheol. 47 (2003) 535-561.

[38] A. Lozinski, R.G. Owens and J. Fang, A Fokker-Planck-based numerical method for modelling non-homogeneous flows of dilute polymeric solutions. J. Non-Newton. Fluid Mech. 122 (2004) 273-286. | Zbl

[39] N. Masmoudi, Well posedness of the FENE dumbbell model of polymeric flows. Comm. Pure Appl. Math. 61 (2008) 1685-1714. | MR | Zbl

[40] F. Otto and A. Tzavaras, Continuity of velocity gradients in suspensions of rod-like molecules. Comm. Math. Phys. 277 (2008) 729-758. | MR | Zbl

[41] M. Renardy, An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal. 22 (1991) 1549-151. | MR | Zbl

[42] J.D. Schieber, Generalized Brownian configuration field for Fokker-Planck equations including center-of-mass diffusion. J. Non-Newton. Fluid Mech. 135 (2006) 179-181. | Zbl

[43] W.H.A. Schilders and E.J.W. ter Maten, Eds., Numerical Methods in Electromagnetics, Handbook of Numerical Analysis XIII. Amsterdam, North-Holland (2005). | MR | Zbl

[44] J. Simon, Compact sets in the space Lp(0,T;B). Ann. Math. Pur. Appl. 146 (1987) 65-96. | MR | Zbl

[45] R. Temam, Navier-Stokes Equations - Theory and Numerical Analysis, Studies in Mathematics and its Applications 2. Third Edition, Amsterdam, North-Holland (1984). | MR | Zbl

[46] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. Second Edition, Johann Ambrosius Barth Publ., Heidelberg/Leipzig (1995). | MR | Zbl

[47] P. Yu, Q. Du and C. Liu, From micro to macro dynamics via a new closure approximation to the FENE model of polymeric fluids. Multiscale Model. Simul. 3 (2005) 895-917. | MR | Zbl