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/

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