Convergence Analysis of BCF Method for Hookean Dumbbell Model with Finite Difference Scheme

Abstract

A convergence analysis of the Brownian configuration fields (BCF) method [M. A. Hulsen, A. P. G. van Heel, and B. H. A. A. van den Brule, J. Non-Newtonian Fluid Mech., 70 (1997), pp. 79-101] for the Hookean dumbbell model with finite difference scheme in dimension 2 or 3 is given in this paper under the assumption that the continuous solution is smooth enough. An explicit solution of the Hookean dumbbell model is obtained via deformation tensor. A large deviation-type estimate for the error of polymeric stress $\mathbb{E}(\bfmath{QQ})$ is given, which is a key step in the proof. It is shown that if the number of configuration fields N, the space stepsize h, and the time stepsize $\delta t$ are chosen appropriately, the convergence of second order in space and first order in time may be proved after excluding a set of small probability. Simultaneous discretization of Monte Carlo and space and the inverse inequality trick are essential for the proof.