A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model

Abstract

Abstract.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. , 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. Mathematics Subject Classification.

On the Approximation of the Fokker–Planck Equation of the Finitely Extensible Nonlinear Elastic Dumbbell Model I: A New Weighted Formulation and an Optimal Spectral-Galerkin Algorithm in Two Dimensions

“…We note that in the case of s = b/2, Knezevic and Süli [14] considered a similar semi-implicit scheme and proved a similar stability result.…”

“…While for given K and U (1.7) is linear, the unbounded FENE potential term ∇ · (f ∇U ) introduces difficulties in both mathematical analysis and numerical approximation. Recently, there have been many mathematical and numerical studies of the homogeneous system (1.7); see, for instance, [20,7,8,19,29,15,14] and the references therein. In particular, some delicate numerical algorithms based on spectral methods were developed recently for the Fokker-Planck equation of the FENE model in [20,7,15,14,6].…”

“…Later, Fan [10] improved Warner's method and used the results of the Fokker-Planck equation to verify the approximation of the FENE-P model. Our work is inspired by the recent work of Knezevic and Süli [15,14] and aimed at developing optimal spectral algorithms with minimal assumptions on the initial condition. More precisely, our approach will be based on a new weighted weak formulation which allows initial conditions in a much larger functional space, and we construct unconditionally stable semi-implicit schemes in time and spectral-Galerkin algorithms in space with optimal computational complexity while properly handing the pole conditions and unbounded drift.…”

Finite element approximation of finitely extensible nonlinear elastic dumbbell models for dilute polymers

“…A possible practical approach to its implementation might be based on operator-splitting, following the computational strategy advocated in the papers [19,20], for example; see also [21]. We note however that the schemes that were analyzed in [19][20][21] were shown to converge for the Fokker-Planck equation considered in isolation, i.e.…”

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

“…For a review of numerical algorithms for the approximation of kinetic models of dilute polymers see, for example, Section 4 of the survey article of Li and Zhang [32]; for recent progress on deterministic algorithms for the approximation of Fokker-Planck and coupled Navier-Stokes-Fokker-Planck systems, see, for example, Lozinski et al [37,38], and Knezevic and Süli [26,27].…”