A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 6, p. 1117-1156

We examine a heterogeneous alternating-direction method for the approximate solution of the FENE Fokker-Planck equation from polymer fluid dynamics and we use this method to solve a coupled (macro-micro) Navier-Stokes-Fokker-Planck system for dilute polymeric fluids. In this context the Fokker-Planck equation is posed on a high-dimensional domain and is therefore challenging from a computational point of view. The heterogeneous alternating-direction scheme combines a spectral Galerkin method for the Fokker-Planck equation in configuration space with a finite element method in physical space to obtain a scheme for the high-dimensional Fokker-Planck equation. Alternating-direction methods have been considered previously in the literature for this problem ($e.g.$ in the work of Lozinski, Chauvière and collaborators [J. Non-newtonian Fluid Mech. 122 (2004) 201-214; Comput. Fluids 33 (2004) 687-696; CRM Proc. Lect. Notes 41 (2007) 73-89; Ph.D. Thesis (2003); J. Computat. Phys. 189 (2003) 607-625]), but this approach has not previously been subject to rigorous numerical analysis. The numerical methods we develop are fully-practical, and we present a range of numerical results demonstrating their accuracy and efficiency. We also examine an advantageous superconvergence property related to the polymeric extra-stress tensor. The heterogeneous alternating-direction method is well suited to implementation on a parallel computer, and we exploit this fact to make large-scale computations feasible.

DOI : https://doi.org/10.1051/m2an/2009034
Classification:  65M70,  65M12,  35K20,  82C31,  82D60
Keywords: multiscale modelling, kinetic models, dilute polymers, alternating-direction methods, spectral methods, finite element methods, high-performance computing
@article{M2AN_2009__43_6_1117_0,
author = {Knezevic, David J. and S\"uli, Endre},
title = {A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {43},
number = {6},
year = {2009},
pages = {1117-1156},
doi = {10.1051/m2an/2009034},
zbl = {pre05636849},
mrnumber = {2588435},
language = {en},
url = {http://www.numdam.org/item/M2AN_2009__43_6_1117_0}
}

Knezevic, David J.; Süli, Endre. A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 6, pp. 1117-1156. doi : 10.1051/m2an/2009034. http://www.numdam.org/item/M2AN_2009__43_6_1117_0/

[1] A. Ammar, B. Mokdad, F. Chinesta and R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J. Non-Newtonian Fluid Mech. 139 (2006) 153-176.

[2] A. Ammar, B. Mokdad, F. Chinesta and R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. Part II: Transient simulation using space-time separated representations. J. Non-Newtonian Fluid Mech. 144 (2007) 98-121.

[3] S. Balay, K. Buschelman, V. Eijkhout, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. Mcinnes, B.F. Smith and H. Zhang, PETSc users manual. Tech. Rep. ANL-95/11 - Revision 2.1.5, Argonne National Laboratory (2004).

[4] 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 2419205 | Zbl 1158.35070

[5] B. Bialecki and R. Fernandes, An orthogonal spline collocation alternating direction implicit Crank-Nicolson method for linear parabolic problems on rectangles. SIAM J. Numer. Anal. 36 (1999) 1414-1434. | MR 1706778 | Zbl 0955.65073

[6] P.B. Bochev, M.D. Gunzburger and J.N. Shadid, Stability of the SUPG finite element method for transient advection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 193 (2004) 2301-2323. | MR 2055248 | Zbl 1067.76563

[7] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Second Edn., Springer (2002). | MR 1894376 | Zbl 1012.65115

[8] M. Celia and G. Pinder, An analysis of alternating-direction methods for parabolic equations. Numer. Methods Part. Differ. Equ. 1 (1985) 57-70. | MR 868051 | Zbl 0634.65098

[9] M. Celia and G. Pinder, Generalized alternating-direction collocation methods for parabolic equations. I. Spatially varying coefficients. Numer. Methods Partial Differ. Equ. 3 (1990) 193-214. | MR 1062376 | Zbl 0705.65073

[10] C. Chauvière and A. Lozinski, Simulation of complex viscoelastic flows using Fokker-Planck equation: 3D FENE model. J. Non-Newtonian Fluid Mech. 122 (2004) 201-214. | Zbl 1131.76307

[11] C. Chauvière and A. Lozinski, Simulation of dilute polymer solutions using a Fokker-Planck equation. Comput. Fluids 33 (2004) 687-696. | Zbl 1100.76549

[12] P. Clément, Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Anal. Numér. 9 (1975) 77-84. | Numdam | MR 400739 | Zbl 0368.65008

[13] P. Delaunay, A. Lozinski and R.G. Owens, Sparse tensor-product Fokker-Planck-based methods for nonlinear bead-spring chain models of dilute polymer solutions. CRM Proc. Lect. Notes 41 (2007) 73-89. | MR 2359669 | Zbl 1128.82017

[14] J. Douglas and T. Dupont, Alternating-direction Galerkin methods on rectangles. Numer. Solution Partial Differ. Equ. II (SYNSPADE 1970) (1971) 133-214. | MR 273830 | Zbl 0239.65088

[15] H. Eisen, W. Heinrichs and K. Witsch, Spectral collocation methods and polar coordinate singularities. J. Comput. Phys. 96 (1991) 241-257. | MR 1128222 | Zbl 0731.65095

[16] H. Elman, D. Silvester and A. Wathen, Finite elements and fast iterative solvers. Oxford Science Publications, UK (2005). | Zbl 1083.76001

[17] C. Helzel and F. Otto, Multiscale simulations of suspensions of rod-like molecules. J. Comp. Phys. 216 (2006) 52-75. | MR 2223436 | Zbl 1107.82040

[18] W. Huang and B. Guo, Fully discrete Jacobi-spherical harmonic spectral method for Navier-Stokes equations. Appl. Math. Mech. 29 (2008) 453-476 (English Ed.). | MR 2405135 | Zbl pre05318375

[19] 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 2039220 | Zbl 1047.76004

[20] B.S. Kirk, J.W. Peterson, R.M. Stogner and G.F. Carey, libMesh: A C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 23 (2006) 237-254.

[21] D.J. Knezevic, Analysis and implementation of numerical methods for simulating dilute polymeric fluids. Ph.D. Thesis, University of Oxford, UK (2008), http://www.comlab.ox.ac.uk/people/David.Knezevic.

[22] D.J. Knezevic and E. Süli, Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift. ESAIM: M2AN 43 (2009) 445-485. | Numdam | MR 2536245 | Zbl pre05574327

[23] A.N. Kolmogorov, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104 (1931). | Zbl 0001.14902

[24] T. Li and P. Zhang, Mathematical analysis of multi-scale models of complex fluids. Commun. Math. Sci. 5 (2007) 1-51. | MR 2310632 | Zbl 1129.76006

[25] C. Liu and H. Liu, Boundary conditions for the microscopic FENE models. SIAM J. Appl. Math. 68 (2008) 1304-1315. | MR 2407125 | Zbl 1147.76003

[26] A. Lozinski, Spectral methods for kinetic theory models of viscoelastic fluids. Ph.D. Thesis, École Polytechnique Fédérale de Lausanne, Suisse (2003).

[27] A. Lozinski and C. Chauvière, A fast solver for Fokker-Planck equation applied to viscoelastic flows calculation: 2D FENE model. J. Computat. Phys. 189 (2003) 607-625. | MR 1996059 | Zbl 1060.82525

[28] J.N. Lyness and D. Jespersen, Moderate degree symmetric quadrature rules for the triangle. J. Inst. Math. Appl. 15 (1975) 19-32. | MR 378368 | Zbl 0297.65018

[29] T. Matsushima and P.S. Marcus, A spectral method for polar coordinates. J. Comput. Phys. 120 (1995) 365-374. | MR 1349468 | Zbl 0842.65051

[30] H.C. Öttinger, Stochastic Processes in Polymeric Fluids. Springer (1996). | MR 1383323 | Zbl 0995.60098

[31] R.G. Owens and T.N. Phillips, Computational Rheology. Imperial College Press (2002). | MR 1906885 | Zbl 1015.76002

[32] C. Schwab, E. Süli and R.A. Todor, Sparse finite element approximation of high-dimensional transport-dominated diffusion problems. ESAIM: M2AN 42 (2008) 777-820. | Numdam | MR 2454623 | Zbl 1159.65094

[33] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | MR 1011446 | Zbl 0696.65007

[34] W.T.M. Verkley, A spectral model for two-dimensional incompressible fluid flow in a circular basin. I. Mathematical formulation. J. Comput. Phys. 136 (1997) 100-114. | MR 1468626 | Zbl 0889.76071

[35] N.J. Walkington, Quadrature on simplices of arbitrary dimension. http://www.math.cmu.edu/~nw0z/publications/00-CNA-023/023abs/.

[36] H.R. Warner, Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells. Ind. Eng. Chem. Fundamentals 11 (1972) 379-387.

[37] H. Zhang and P. Zhang, Local existence for the FENE-dumbbell model of polymeric fluids. Arch. Ration. Mech. Anal. 181 (2006) 373-400. | MR 2221211 | Zbl 1095.76004