# Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift

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

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topKnezevic, 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] 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] 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] 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] 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] 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] 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] 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] C. Bernardi, Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 26 (1989) 1212–1240. Zbl0678.65003MR1014883
- [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] O.V. Besov and A. Kufner, The density of smooth functions in weight spaces. Czechoslova. Math. J. 18 (1968) 178–188. Zbl0193.41502MR223877
- [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] 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] 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] 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] C. Canuto, A. Quarteroni, M.Y. Hussaini and T.A. Zang, Spectral Methods: Fundamentals in Single Domains. Springer (2006). Zbl1093.76002MR2223552
- [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] 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] C. Chauvière and A. Lozinski, Simulation of dilute polymer solutions using a Fokker-Planck equation. Comput. Fluids 33 (2004) 687–696. Zbl1100.76549
- [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] 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] H. Eisen, W. Heinrichs and K. Witsch, Spectral collocation methods and polar coordinate singularities. J. Comput. Phys. 96 (1991) 241–257. Zbl0731.65095MR1128222
- [22] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Springer (1973). Zbl0294.58004MR341518
- [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] A.N. Kolmogorov, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104 (1931). Zbl0001.14902
- [25] A. Kufner, Weighted Sobolev Spaces, Teubner-Texte zur Mathematik. Teubner (1980). Zbl0455.46034MR664599
- [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] 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] M. Marcus, V.J. Mizel and Y. Pinchover, On the best constant for Hardy’s inequality in ${\mathbb{R}}^{n}$. Trans. Amer. Math. Soc. 350 (1998) 3237–3255. Zbl0917.26016MR1458330
- [29] T. Matsushima and P.S. Marcus, A spectral method for polar coordinates. J. Comput. Phys. 120 (1995) 365–374. Zbl0842.65051MR1349468
- [30] H.C. Öttinger, Stochastic Processes in Polymeric Fluids. Springer (1996). Zbl0995.60098MR1383323
- [31] J. Shen, Efficient spectral Galerkin methods III: Polar and cylindrical geometries. SIAM J. Sci. Comput. 18 (1997) 1583–1604. Zbl0890.65117MR1480626
- [32] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis. Third edition, North-Holland, Amsterdam (1984). Zbl0568.35002MR769654
- [33] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. Second edition, Johan Ambrosius Barth, Heidelberg (1995). Zbl0830.46028MR1328645
- [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

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