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

*David J. Knezevic and Endre Süli*

### Abstract

The paper is concerned with the analysis and implementation of a spectral Galerkin method 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 class of admissible potentials 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 approximation of such degenerate Fokker-Planck equations that exhibits optimal-order convergence in the Maxwellian-weighted **H**^{1} norm on *D*. The theoretical results are illustrated by numerical experiments for the FENE model in two space dimensions.