A Certified Reduced Basis Method for the Fokker--Planck Equation of Dilute Polymeric Fluids: FENE Dumbbells in Extensional Flow

In this paper we present a reduced basis method for the parametrized Fokker–Planck equation associated with evolution of Finitely Extensible Nonlinear Elastic (FENE) dumbbells in a Newtonian solvent for a (prescribed) extensional macroscale flow. There are two new important ingredients: a projection–based POD–Greedy sampling procedure for the stable identification of optimal reduced basis spaces; and a finite–time a posteriori bound for the error in the reduced basis prediction of the two outputs of interest — the optical anisotropy and the first normal stress difference. We present numerical results for stress–conformation hysteresis as a function of Weissenberg number and final time that demonstrate the rapid convergence of the reduced basis approximation and the effectiveness of the a posteriori error bounds.