Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift

Knezevic, David J.; Süli, Endre

ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 3, pp. 445-485.

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.

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