Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift. (English) Zbl 1180.82136
The paper is concerned with the development of a rigorous foundation for the numerical approximation of a class of Fokker-Planck equations arising in the kinetic theory of dilute polymers. A special feature of these equations is the appearance of an unbounded drift coefficient, determined from a smooth potential going to infinity on the boundary of a finite domain, reflecting the finite stretching of polymer chains. The admissible potentials include the finitely extendible nonlinear elastic model (FENE). In order to tackle the numerical study of the equation the authors remove the unbounded drift coefficient using a symmetrization of the differential operator based on the Maxwellian obtained from the potential. Analytical results are obtained, together with numerical ones for the case of the FENE model in two dimensions.
Reviewer: Bassano Vacchini (Milano)
MSC:
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 82D60 | Statistical mechanics of polymers |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35Q84 | Fokker-Planck equations |
Keywords:
spectral methods; Fokker-Planck equations; transport-diffusion problems; finitely extendible nonlinear elastic model (FENE); unbounded coefficients; polymer chainsReferences:
| [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. · Zbl 1195.76337 · doi:10.1016/j.jnnfm.2006.07.007 |
| [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 · Zbl 1196.76047 · doi:10.1016/j.jnnfm.2007.03.009 |
| [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. · Zbl 1145.26005 · doi:10.1002/zamm.200710342 |
| [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. Zbl1228.76004 · Zbl 1228.76004 · doi:10.1137/060666810 |
| [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. · Zbl 1158.35070 · doi:10.1142/S0218202508002917 |
| [6] | J.W. Barrett and E. Süli, Numerical approximation of corotational dumbbell models for dilute polymers. IMA J. Numer. Anal. (2008) online. Available at . URIhttp://imajna.oxfordjournals.org/cgi/content/abstract/drn022 |
| [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. · Zbl 1161.76453 · doi:10.1142/S0218202505000625 |
| [8] | C. Bernardi, Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal.26 (1989) 1212-1240. · Zbl 0678.65003 · doi:10.1137/0726068 |
| [9] | C. Bernardi and Y. Maday, Spectral methods, in Handbook of Numerical AnalysisV, P. Ciarlet and J. Lions Eds., Elsevier (1997). · Zbl 0991.65124 |
| [10] | O.V. Besov and A. Kufner, The density of smooth functions in weight spaces. Czechoslova. Math. J.18 (1968) 178-188. · Zbl 0193.41502 |
| [11] | O.V. Besov, J. Kadlec and A. Kufner, Certain properties of weight classes. Dokl. Akad. Nauk SSSR171 (1966) 514-516. · Zbl 0168.11103 |
| [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. · Zbl 0969.26019 · doi:10.1007/PL00001645 |
| [15] | C. Canuto, A. Quarteroni, M.Y. Hussaini and T.A. Zang, Spectral Methods: Fundamentals in Single Domains. Springer (2006). · Zbl 1093.76002 |
| [16] | S. Cerrai, Second Order PDE’s in Finite and Infinite Dimensions, A Probabilistic Approach, Lecture Notes in Mathematics1762. Springer (2001). |
| [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. · Zbl 1131.76307 · doi:10.1016/j.jnnfm.2003.12.011 |
| [18] | C. Chauvière and A. Lozinski, Simulation of dilute polymer solutions using a Fokker-Planck equation. Comput. Fluids33 (2004) 687-696. · Zbl 1100.76549 · doi:10.1016/j.compfluid.2003.02.002 |
| [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. · Zbl 1162.35345 |
| [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. · Zbl 1108.76006 · doi:10.1137/040612038 |
| [21] | H. Eisen, W. Heinrichs and K. Witsch, Spectral collocation methods and polar coordinate singularities. J. Comput. Phys.96 (1991) 241-257. · Zbl 0731.65095 · doi:10.1016/0021-9991(91)90235-D |
| [22] | M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Springer (1973). · Zbl 0294.58004 |
| [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. · Zbl 1041.76003 · doi:10.1142/S0218202502002100 |
| [24] | A.N. Kolmogorov, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann.104 (1931). · Zbl 0001.14902 · doi:10.1007/BF01457949 |
| [25] | A. Kufner, Weighted Sobolev Spaces, Teubner-Texte zur Mathematik. Teubner (1980). · Zbl 0455.46034 |
| [26] | T. Li and P.-W. Zhang, Mathematical analysis of multi-scale models of complex fluids. Commun. Math. Sci.5 (2007) 1-51. · Zbl 1129.76006 · doi:10.4310/CMS.2007.v5.n1.a1 |
| [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. · Zbl 1060.82525 · doi:10.1016/S0021-9991(03)00248-1 |
| [28] | M. Marcus, V.J. Mizel and Y. Pinchover, On the best constant for Hardy’s inequality in Rn. Trans. Amer. Math. Soc.350 (1998) 3237-3255. · Zbl 0917.26016 · doi:10.1090/S0002-9947-98-02122-9 |
| [29] | T. Matsushima and P.S. Marcus, A spectral method for polar coordinates. J. Comput. Phys.120 (1995) 365-374. Zbl0842.65051 · Zbl 0842.65051 · doi:10.1006/jcph.1995.1171 |
| [30] | H.C. Öttinger, Stochastic Processes in Polymeric Fluids. Springer (1996). |
| [31] | J. Shen, Efficient spectral Galerkin methods III: Polar and cylindrical geometries. SIAM J. Sci. Comput.18 (1997) 1583-1604. · Zbl 0890.65117 · doi:10.1137/S1064827595295301 |
| [32] | R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis. Third edition, North-Holland, Amsterdam (1984). · Zbl 0568.35002 |
| [33] | H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. Second edition, Johan Ambrosius Barth, Heidelberg (1995). · Zbl 0830.46028 |
| [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. · Zbl 0889.76071 · doi:10.1006/jcph.1997.5747 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
