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Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift

Published online by Cambridge University Press:  18 December 2008

David J. Knezevic  and
Endre Süli
David J. Knezevic
Affiliation:
OUCL, University of Oxford, Parks Road, Oxford, OX1 3QD, UK. david.knezevic@balliol.ox.ac.uk; davek@comlab.ox.ac.uk; endre.suli@comlab.ox.ac.uk
Endre Süli
Affiliation:
OUCL, University of Oxford, Parks Road, Oxford, OX1 3QD, UK. david.knezevic@balliol.ox.ac.uk; davek@comlab.ox.ac.uk; endre.suli@comlab.ox.ac.uk
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Abstract

This paper is concerned with the analysis and implementation of spectral Galerkinmethods for a class of Fokker-Planck equations that arisesfrom the kinetic theory of dilute polymers. A relevant feature of the class of equationsunder consideration from the viewpoint of mathematical analysis and numerical approximation isthe presence of an unbounded drift coefficient, involving a smooth convex potential U that is equal to +∞ alongthe 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 expenseof introducing a degeneracy, through M, in the principal part of the operator.The general class of admissible potentials consideredincludes the FENE (finitely extendible nonlinear elastic) model.We show the existence of weak solutions to the initial-boundary-value problem, and develop a fully-discretespectral Galerkin method for such degenerate Fokker-Planck equationsthat exhibits optimal-order convergence in the Maxwellian-weighted H1 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 theoriginal operator. The analytical results are illustrated by numerical experiments for the FENE model in two space dimensions.

Keywords

Spectral methodsFokker-Planck equationstransport-diffusion problemsFENE.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 3 , May 2009 , pp. 445 - 485
Copyright
© EDP Sciences, SMAI, 2009

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