| Summary: | 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 <em>U</em> that is equal to +∞ along the boundary ∂<em>D</em> of the computational domain <em>D</em>. Using a symmetrization of the differential operator based on the Maxwellian <em>M</em>corresponding to <em>U</em>, which vanishes along ∂<em>D</em>, we remove the unbounded drift coefficient at the expense of introducing a degeneracy, through <em>M</em>, 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 <strong>H</strong><sup>1</sup> norm on <em>D</em>. The theoretical results are illustrated by numerical experiments for the FENE model in two space dimensions.
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