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 or , 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 for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian .
Mots-clés : finite element method, convergence analysis, existence of weak solutions, kinetic polymer models, FENE dumbbell, Navier-Stokes equations, Fokker-Planck equations
@article{M2AN_2012__46_4_949_0,
author = {Barrett, John W. and S\"uli, Endre},
title = {Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {949--978},
publisher = {EDP-Sciences},
volume = {46},
number = {4},
year = {2012},
doi = {10.1051/m2an/2011062},
mrnumber = {2891476},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2011062/}
}
TY - JOUR AU - Barrett, John W. AU - Süli, Endre TI - Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 949 EP - 978 VL - 46 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2011062/ DO - 10.1051/m2an/2011062 LA - en ID - M2AN_2012__46_4_949_0 ER -
%0 Journal Article %A Barrett, John W. %A Süli, Endre %T Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 949-978 %V 46 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2011062/ %R 10.1051/m2an/2011062 %G en %F M2AN_2012__46_4_949_0
Barrett, John W.; Süli, Endre. Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 4, pp. 949-978. doi : 10.1051/m2an/2011062. http://www.numdam.org/articles/10.1051/m2an/2011062/
[1] , Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158 (2004) 227-260. | MR | Zbl
[2] and , 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
[3] and , Existence of global weak solutions to some regularized kinetic models of dilute polymers. Multiscale Model. Simul. 6 (2007) 506-546. | MR | Zbl
[4] and , 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
[5] and , Numerical approximation of corotational dumbbell models for dilute polymers. IMA J. Numer. Anal. 29 (2009) 937-959. | MR | Zbl
[6] and , Existence and equilibration of global weak solutions to finitely extensible nonlinear bead-spring chain models for dilute polymers. Available as arXiv:1004.1432v2 [math.AP] from http://arxiv.org/abs/1004.1432 (2010). | Zbl
[7] and , Existence and equilibration of global weak solutions to kinetic models for dilute polymers I : Finitely extensible nonlinear bead-spring chains. Math. Models Methods Appl. Sci. 21 (2011) 1211-1289. | MR | Zbl
[8] and , Finite element approximation of kinetic dilute polymer models with microscopic cut-off. ESAIM : M2AN 45 (2011) 39-89. | Numdam | MR | Zbl
[9] and , Existence and equilibration of global weak solutions to kinetic models for dilute polymers II : Hookean bead-spring chains. Math. Models Methods Appl. Sci. 22 (2012), to appear. Extended version available as arXiv:1008.3052 [math.AP] from http://arxiv.org/abs/1008.3052. | Zbl
[10] , and , Kinetic theory and rheology of dilute, nonhomogeneous polymer solutions. J. Chem. Phys. 95 (1991) 2988-3000.
[11] , , and , On acute and nonobtuse simplicial partitions. Helsinki University of Technology, Institute of Mathematics, Research Reports, A503 (2006). | Zbl
[12] and , Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin (1991). | MR | Zbl
[13] , The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl
[14] and , Kinetic models for polymers with inertial effects. Netw. Heterog. Media 4 (2009) 625-647. | MR | Zbl
[15] and , Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989) 511-547. | MR | Zbl
[16] , and , Tiling space and slabs with acute tetrahedra. Comput. Geom. 27 (2004) 237-255. | MR | Zbl
[17] and , Nonnegativity preserving numerical schemes for the thin film equation. Numer. Math. 87 (2000) 113-152. | MR | Zbl
[18] and , Acute triangulations of the regular dodecahedral surface. Eur. J. Comb. 28 (2007) 1072-1086. | MR | Zbl
[19] and , A deterministic multiscale approach for simulating dilute polymeric fluids, in BAIL 2008 - boundary and interior layers. Lect. Notes Comput. Sci. Eng. 69 (2009) 23-38. | MR | Zbl
[20] and , A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model. ESAIM : M2AN 43 (2009) 1117-1156. | Numdam | MR | Zbl
[21] and , Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift. ESAIM : M2AN 43 (2009) 445-485. | Numdam | MR | Zbl
[22] and , Acute type refinements of tetrahedral partitions of polyhedral domains. SIAM J. Numer. Anal. 39 (2001) 724-733. | MR | Zbl
[23] and , Global and local refinement techniques yielding nonobtuse tetrahedral partitions. Comput. Math. Appl. 50 (2005) 1105-1113. | MR | Zbl
[24] and , Global existence of weak solutions to some micro-macro models. C. R. Math. Acad. Sci. Paris 345 (2007) 15-20. | MR | Zbl
[25] , Well posedness of the FENE dumbbell model of polymeric flows. Comm. Pure Appl. Math. 61 (2008) 1685-1714. | MR | Zbl
[26] , Global existence of weak solutions to the FENE dumbbell model of polymeric flows. Preprint (2010). | MR | Zbl
[27] , Finite element methods for parabolic free boundary problems, in Advances in Numerical Analysis I. Lancaster (1990); Oxford Sci. Publ., Oxford Univ. Press, New York (1991) 34-95. | MR | Zbl
[28] , Generalized Brownian configuration field for Fokker-Planck equations including center-of-mass diffusion. J. Non-Newtonian Fluid Mech. 135 (2006) 179-181. | Zbl
[29] W.H.A. Schilders and E.J.W. ter Maten, Eds., Numerical Methods in Electromagnetics, Handbook of Numerical Analysis XIII. North-Holland, Amsterdam (2005). | MR | Zbl
[30] , Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications 2. North-Holland, Amsterdam (1984). | MR | Zbl
Cité par Sources :
