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

Abstract

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. We apply a proper orthogonal decomposition (POD)–greedy sampling procedure for the stable identification of optimal reduced basis spaces, and we develop a rigorous 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.