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

ESAIM - Mathematical Modelling and Numerical Analysis

Volumn 43, Issue 6, 2009, Pages 1117-1156

  Knezevic, David J ^a   Süli, Endre ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Alternating direction methods; Dilute polymers; Finite element methods; High performance computing; Kinetic models; Multiscale modelling; Spectral methods

Indexed keywords

EID: 76249085920     PISSN: 0764583X     EISSN: 12903841     Source Type: Journal    
DOI: 10.1051/m2an/2009034     Document Type: Article

References (37)

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.
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.
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.
7. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Second Edn., Springer (2002).
8. M. Celia and G. Pinder, An analysis of alternating-direction methods for parabolic equations. Numer. Methods Part. Differ. Equ. 1 (1985) 57-70.
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.
10. C. Chauvìere and A. Lozinski, Simulation of complex viscoelastic flows using Fokker-Planck equation: 3D FENE model. J. Non-Newtonian Fluid Mech. 122 (2004) 201-214.
11. C. Chauvìere and A. Lozinski, Simulation of dilute polymer solutions using a Fokker-Planck equation. Comput. Fluids 33 (2004) 687-696.
12. P. Cĺement, Approximation by finite element functions using local regularization. Rev. Fraņcaise Automat. Informat. Recherche Oṕerationnelle Śer. RAIRO Anal. Nuḿer. 9 (1975) 77-84.
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.
14. J. Douglas and T. Dupont, Alternating-direction Galerkin methods on rectangles. Numer. Solution Partial Differ. Equ. II (SYNSPADE 1970) (1971) 133-214.
15. H. Eisen, W. Heinrichs and K. Witsch, Spectral collocation methods and polar coordinate singularities. J. Comput. Phys. 96 (1991) 241-257.
16. H. Elman, D. Silvester and A. Wathen, Finite elements and fast iterative solvers. Oxford Science Publications, UK (2005).
17. C. Helzel and F. Otto, Multiscale simulations of suspensions of rod-like molecules. J. Comp. Phys. 216 (2006) 52-75.
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.).
19. B. Jourdain, T. Lelìevre 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.
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.
23. A.N. Kolmogorov, über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104 (1931).
24. T. Li and P. Zhang, Mathematical analysis of multi-scale models of complex fluids. Commun. Math. Sci. 5 (2007) 1-51.
25. C. Liu and H. Liu, Boundary conditions for the microscopic FENE models. SIAM J. Appl. Math. 68 (2008) 1304-1315.
26. A. Lozinski, Spectral methods for kinetic theory models of viscoelastic fluids. Ph.D. Thesis, ́Ecole Polytechnique F́ed́erale de Lausanne, Suisse (2003).
27. A. Lozinski and C. Chauvìere, A fast solver for Fokker-Planck equation applied to viscoelastic flows calculation: 2D FENE model. J. Computat. Phys. 189 (2003) 607-625.
28. J.N. Lyness and D. Jespersen, Moderate degree symmetric quadrature rules for the triangle. J. Inst. Math. Appl. 15 (1975) 19-32.
29. T. Matsushima and P.S. Marcus, A spectral method for polar coordinates. J. Comput. Phys. 120 (1995) 365-374.
30. H.C. öttinger, Stochastic Processes in Polymeric Fluids. Springer (1996).
31. R.G. Owens and T.N. Phillips, Computational Rheology. Imperial College Press (2002).
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.
33. L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493.
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.
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.