Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift

David J. Knezevic; Endre Süli

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2009)

  • Volume: 43, Issue: 3, page 445-485
  • ISSN: 0764-583X

Abstract

top
This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to + ∞ along the boundary ∂ D of the computational domain D . Using a symmetrization of the differential operator based on the Maxwellian M corresponding to U , which vanishes along ∂ D , we remove the unbounded drift coefficient at the expense of introducing a degeneracy, through M , in the principal part of the operator. The general class of admissible potentials considered includes the FENE (finitely extendible nonlinear elastic) model. We show the existence of weak solutions to the initial-boundary-value problem, and develop a fully-discrete spectral Galerkin method for such degenerate Fokker-Planck equations that exhibits optimal-order convergence in the Maxwellian-weighted H 1 norm on D . In the case of the FENE model, we also discuss variants of these analytical results when the Fokker-Planck equation is subjected to an alternative class of transformations proposed by Chauvière and Lozinski; these map the original Fokker-Planck operator with an unbounded drift coefficient into Fokker-Planck operators with unbounded drift and reaction coefficients, that have improved coercivity properties in comparison with the original operator. The analytical results are illustrated by numerical experiments for the FENE model in two space dimensions.

How to cite

top

Knezevic, David J., and Süli, Endre. "Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 43.3 (2009): 445-485. <http://eudml.org/doc/244908>.

@article{Knezevic2009,
abstract = {This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential $U$ that is equal to $+\infty $ along the boundary $\partial D$ of the computational domain $D$. Using a symmetrization of the differential operator based on the Maxwellian $M$ corresponding to $U$, which vanishes along $\partial D$, we remove the unbounded drift coefficient at the expense of introducing a degeneracy, through $M$, in the principal part of the operator. The general class of admissible potentials considered includes the FENE (finitely extendible nonlinear elastic) model. We show the existence of weak solutions to the initial-boundary-value problem, and develop a fully-discrete spectral Galerkin method for such degenerate Fokker-Planck equations that exhibits optimal-order convergence in the Maxwellian-weighted $\mathrm \{H\}^1$ norm on $D$. In the case of the FENE model, we also discuss variants of these analytical results when the Fokker-Planck equation is subjected to an alternative class of transformations proposed by Chauvière and Lozinski; these map the original Fokker-Planck operator with an unbounded drift coefficient into Fokker-Planck operators with unbounded drift and reaction coefficients, that have improved coercivity properties in comparison with the original operator. The analytical results are illustrated by numerical experiments for the FENE model in two space dimensions.},
author = {Knezevic, David J., Süli, Endre},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {spectral methods; Fokker-Planck equations; transport-diffusion problems; FENE; finitely extendible nonlinear elastic model (FENE); unbounded coefficients; polymer chains},
language = {eng},
number = {3},
pages = {445-485},
publisher = {EDP-Sciences},
title = {Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift},
url = {http://eudml.org/doc/244908},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Knezevic, David J.
AU - Süli, Endre
TI - Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2009
PB - EDP-Sciences
VL - 43
IS - 3
SP - 445
EP - 485
AB - This paper is concerned with the analysis and implementation of spectral Galerkin methods for a class of Fokker-Planck equations that arises from the kinetic theory of dilute polymers. A relevant feature of the class of equations under consideration from the viewpoint of mathematical analysis and numerical approximation is the presence of an unbounded drift coefficient, involving a smooth convex potential $U$ that is equal to $+\infty $ along the boundary $\partial D$ of the computational domain $D$. Using a symmetrization of the differential operator based on the Maxwellian $M$ corresponding to $U$, which vanishes along $\partial D$, we remove the unbounded drift coefficient at the expense of introducing a degeneracy, through $M$, in the principal part of the operator. The general class of admissible potentials considered includes the FENE (finitely extendible nonlinear elastic) model. We show the existence of weak solutions to the initial-boundary-value problem, and develop a fully-discrete spectral Galerkin method for such degenerate Fokker-Planck equations that exhibits optimal-order convergence in the Maxwellian-weighted $\mathrm {H}^1$ norm on $D$. In the case of the FENE model, we also discuss variants of these analytical results when the Fokker-Planck equation is subjected to an alternative class of transformations proposed by Chauvière and Lozinski; these map the original Fokker-Planck operator with an unbounded drift coefficient into Fokker-Planck operators with unbounded drift and reaction coefficients, that have improved coercivity properties in comparison with the original operator. The analytical results are illustrated by numerical experiments for the FENE model in two space dimensions.
LA - eng
KW - spectral methods; Fokker-Planck equations; transport-diffusion problems; FENE; finitely extendible nonlinear elastic model (FENE); unbounded coefficients; polymer chains
UR - http://eudml.org/doc/244908
ER -

References

top
  1. [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. Zbl1195.76337
  2. [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. Zbl1196.76047
  3. [3] F.G. Avkhadiev and K.-J. Wirths, Unified Poincaré and Hardy inequalities with sharp constants for convex domains. ZAMM Z. Angew. Math. Mech. 87 (2007) 632–642. Zbl1145.26005MR2354734
  4. [4] J.W. Barrett and E. Süli, Existence of global weak solutions to kinetic models of dilute polymers. Multiscale Model. Simul. 6 (2007) 506–546. Zbl1228.76004MR2338493
  5. [5] J.W. Barrett and E. Süli, Existence of global weak solutions to dumbbell models for dilute polymers with microscopic cut-off. Math. Mod. Meth. Appl. Sci. 18 (2008) 935–971. Zbl1158.35070MR2419205
  6. [6] J.W. Barrett and E. Süli, Numerical approximation of corotational dumbbell models for dilute polymers. IMA J. Numer. Anal. (2008) online. Available at http://imajna.oxfordjournals.org/cgi/content/abstract/drn022. Zbl1180.82232MR2557051
  7. [7] J.W. Barrett, C. Schwab and E. Süli, Existence of global weak solutions for some polymeric flow models. Math. Mod. Meth. Appl. Sci. 15 (2005) 939–983. Zbl1161.76453MR2149930
  8. [8] C. Bernardi, Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 26 (1989) 1212–1240. Zbl0678.65003MR1014883
  9. [9] C. Bernardi and Y. Maday, Spectral methods, in Handbook of Numerical Analysis V, P. Ciarlet and J. Lions Eds., Elsevier (1997). Zbl0991.65124MR1470226
  10. [10] O.V. Besov and A. Kufner, The density of smooth functions in weight spaces. Czechoslova. Math. J. 18 (1968) 178–188. Zbl0193.41502MR223877
  11. [11] O.V. Besov, J. Kadlec and A. Kufner, Certain properties of weight classes. Dokl. Akad. Nauk SSSR 171 (1966) 514–516. Zbl0168.11103MR204842
  12. [12] R.B. Bird, C.F. Curtiss, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 1, Fluid Mechanics. Second edition, John Wiley and Sons (1987). 
  13. [13] R.B. Bird, C.F. Curtiss, R.C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, Vol. 2, Kinetic Theory. Second edition, John Wiley and Sons (1987). 
  14. [14] S. Bobkov and M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 (2000) 1028–1052. Zbl0969.26019MR1800062
  15. [15] C. Canuto, A. Quarteroni, M.Y. Hussaini and T.A. Zang, Spectral Methods: Fundamentals in Single Domains. Springer (2006). Zbl1093.76002MR2223552
  16. [16] S. Cerrai, Second Order PDE’s in Finite and Infinite Dimensions, A Probabilistic Approach, Lecture Notes in Mathematics 1762. Springer (2001). Zbl0983.60004MR1840644
  17. [17] 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. Zbl1131.76307
  18. [18] C. Chauvière and A. Lozinski, Simulation of dilute polymer solutions using a Fokker-Planck equation. Comput. Fluids 33 (2004) 687–696. Zbl1100.76549
  19. [19] G. Da Prato and A. Lunardi, On a class of elliptic operators with unbounded coefficients in convex domains. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 15 (2004) 315–326. Zbl1162.35345MR2148888
  20. [20] Q. Du, C. Liu and P. Yu, FENE dumbbell model and its several linear and nonlinear closure approximations. Multiscale Model. Simul. 4 (2005) 709–731. Zbl1108.76006MR2203938
  21. [21] H. Eisen, W. Heinrichs and K. Witsch, Spectral collocation methods and polar coordinate singularities. J. Comput. Phys. 96 (1991) 241–257. Zbl0731.65095MR1128222
  22. [22] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Springer (1973). Zbl0294.58004MR341518
  23. [23] B. Jourdain, T. Lelièvre and C. Le Bris, Numerical analysis of micro-macro simulations of polymeric fluid flows: A simple case. Math. Mod. Meth. Appl. Sci. 12 (2002) 1205–1243. Zbl1041.76003MR1927023
  24. [24] A.N. Kolmogorov, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104 (1931). Zbl0001.14902
  25. [25] A. Kufner, Weighted Sobolev Spaces, Teubner-Texte zur Mathematik. Teubner (1980). Zbl0455.46034MR664599
  26. [26] T. Li and P.-W. Zhang, Mathematical analysis of multi-scale models of complex fluids. Commun. Math. Sci. 5 (2007) 1–51. Zbl1129.76006MR2310632
  27. [27] A. Lozinski and C. Chauvière, A fast solver for Fokker-Planck equation applied to viscoelastic flows calculation: 2D FENE model. J. Comput. Phys. 189 (2003) 607–625. Zbl1060.82525MR1996059
  28. [28] M. Marcus, V.J. Mizel and Y. Pinchover, On the best constant for Hardy’s inequality in ℝ n . Trans. Amer. Math. Soc. 350 (1998) 3237–3255. Zbl0917.26016MR1458330
  29. [29] T. Matsushima and P.S. Marcus, A spectral method for polar coordinates. J. Comput. Phys. 120 (1995) 365–374. Zbl0842.65051MR1349468
  30. [30] H.C. Öttinger, Stochastic Processes in Polymeric Fluids. Springer (1996). Zbl0995.60098MR1383323
  31. [31] J. Shen, Efficient spectral Galerkin methods III: Polar and cylindrical geometries. SIAM J. Sci. Comput. 18 (1997) 1583–1604. Zbl0890.65117MR1480626
  32. [32] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis. Third edition, North-Holland, Amsterdam (1984). Zbl0568.35002MR769654
  33. [33] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. Second edition, Johan Ambrosius Barth, Heidelberg (1995). Zbl0830.46028MR1328645
  34. [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. Zbl0889.76071MR1468626

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.