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

- Volume: 45, Issue: 1, page 39-89
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topBarrett, John W., and Süli, Endre. "Finite element approximation of kinetic dilute polymer models with microscopic cut-off." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 45.1 (2011): 39-89. <http://eudml.org/doc/273175>.

@article{Barrett2011,

abstract = {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 ⊂ $\mathbb \{R\}^d$,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 $\beta ^L(\cdot ) :=\min (\cdot ,L)$ in the drag and convective terms, whereL ≫ 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.},

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

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {finite element method; polymeric flow models; convergence analysis; existence of weak solutions; Navier–Stokes equations; Fokker–Planck equations; FENE; polymeric flow model; incompressible Navier-Stokes equations; weak solutions},

language = {eng},

number = {1},

pages = {39-89},

publisher = {EDP-Sciences},

title = {Finite element approximation of kinetic dilute polymer models with microscopic cut-off},

url = {http://eudml.org/doc/273175},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Barrett, John W.

AU - Süli, Endre

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

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2011

PB - EDP-Sciences

VL - 45

IS - 1

SP - 39

EP - 89

AB - 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 ⊂ $\mathbb {R}^d$,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 $\beta ^L(\cdot ) :=\min (\cdot ,L)$ in the drag and convective terms, whereL ≫ 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.

LA - eng

KW - finite element method; polymeric flow models; convergence analysis; existence of weak solutions; Navier–Stokes equations; Fokker–Planck equations; FENE; polymeric flow model; incompressible Navier-Stokes equations; weak solutions

UR - http://eudml.org/doc/273175

ER -

## References

top- [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. Zbl1330.46029MR2091081
- [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. PDE26 (2001) 43–100. Zbl0982.35113MR1842428
- [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. Zbl1143.76473MR2046180
- [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. Zbl1228.76004MR2338493
- [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. Zbl1158.35070MR2419205
- [6] J.W. Barrett and E. Süli, Numerical approximation of corotational dumbbell models for dilute polymers. IMA J. Numer. Anal.29 (2009) 937–959. Zbl1180.82232MR2557051
- [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. Zbl1161.76453MR2149930
- [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. Zbl0969.26019MR1800062
- [10] J. Brandts, S. Korotov, M. Křížek and J. Šolc, On nonobtuse simplicial partitions. SIAM Rev.51 (2009) 317–335. Zbl1172.51012MR2505583
- [11] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin (1991). Zbl0788.73002MR1115205
- [12] S. Cerrai, Second-order PDEs in Finite and Infinite Dimension, Lecture Notes in Mathematics 1762. Springer-Verlag, Berlin (2001). Zbl0983.60004MR1840644
- [13] P. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). Zbl0511.65078MR520174
- [14] P. Constantin, Nonlinear Fokker–Planck Navier–Stokes systems. Commun. Math. Sci.3 (2005) 531–544. Zbl1110.35057MR2188682
- [15] G. Da Prato and A. Lunardi, Elliptic operators with unbounded drift coefficients and Neumann boundary condition. J. Differ. Equ.198 (2004) 35–52. Zbl1046.35025MR2037749
- [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. Zbl1029.82032MR1787105
- [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. Zbl1108.76006MR2203938
- [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. Zbl1060.35169MR2073140
- [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. Zbl0679.76004
- [20] D. Eppstein, J.M. Sullivan and A. Üngör, Tiling space and slabs with acute tetrahedra. Comput. Geom.27 (2004) 237–255. Zbl1054.65020MR2039173
- [21] G. Grün and M. Rumpf, Nonnegativity preserving numerical schemes for the thin film equation. Numer. Math.87 (2000) 113–152. Zbl0988.76056MR1800156
- [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. Zbl0487.76035MR650052
- [23] J.-I. Itoh and T. Zamfirescu, Acute triangulations of the regular dodecahedral surface. European J. Combin.28 (2007) 1072–1086. Zbl1115.52004MR2305575
- [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. Zbl1047.76004MR2039220
- [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. Zbl1089.76006MR2221204
- [26] D. Knezevic and E. Süli, Spectral Galerkin approximation of Fokker–Planck equations with unbounded drift. ESAIM: M2AN 43 (2009) 445–485. Zbl1180.82136MR2536245
- [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. Zbl1180.82137MR2588435
- [28] S. Korotov and M. Křížek, Acute type refinements of tetrahedral partitions of polyhedral domains. SIAM J. Numer. Anal.39 (2001) 724–733. Zbl1069.65017MR1860255
- [29] S. Korotov and M. Křížek, Global and local refinement techniques yielding nonobtuse tetrahedral partitions. Comput. Math. Appl. 50 (2005) 1105–1113. Zbl1086.65116MR2167747
- [30] A. Kufner, Weighted Sobolev Spaces. Teubner, Stuttgart (1980). Zbl0455.46034MR664599
- [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. Zbl1129.76006MR2310632
- [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. Zbl1058.76010MR2059152
- [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. Zbl1113.76017MR2306223
- [35] 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. Zbl1117.35312MR2340887
- [36] L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups. Chapman & Hall/CRC, Boca Raton (2007). Zbl1109.35005MR2313847
- [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. Zbl1143.76338
- [39] N. Masmoudi, Well posedness of the FENE dumbbell model of polymeric flows. Comm. Pure Appl. Math. 61 (2008) 1685–1714. Zbl1157.35088MR2456183
- [40] F. Otto and A. Tzavaras, Continuity of velocity gradients in suspensions of rod-like molecules. Comm. Math. Phys. 277 (2008) 729–758. Zbl1158.76051MR2365451
- [41] M. Renardy, An existence theorem for model equations resulting from kinetic theories of polymer solutions. SIAM J. Math. Anal. 22 (1991) 1549–151. Zbl0735.35101MR1084958
- [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. Zbl1195.76113
- [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). Zbl1064.65001MR2146791
- [44] J. Simon, Compact sets in the space Lp(0,T;B). Ann. Math. Pur. Appl. 146 (1987) 65–96. Zbl0629.46031MR916688
- [45] R. Temam, Navier–Stokes Equations – Theory and Numerical Analysis, Studies in Mathematics and its Applications 2. Third Edition, Amsterdam, North-Holland (1984). Zbl0568.35002MR769654
- [46] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. Second Edition, Johann Ambrosius Barth Publ., Heidelberg/Leipzig (1995). Zbl0830.46028MR1328645
- [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. Zbl1108.76007MR2164242

## NotesEmbed ?

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