Two-dimensional time-dependent calculations for a molecular model of finite extensibility in the journal-bearing geometry are presented. The flow is considered to be incompressible and isothermal. The momentum conservation equation is integrated using a time-marching procedure in which local ensembles of dumbbells act as stress calculators. The calculations are based on the Calculation of non-Newtonian flows: finite elements and stochastic simulation technique (CONNFFESSIT) and combine deterministic (finite elements) and stochastic techniques to advance the velocity and stress fields in time. The ability of CONNFFESSIT to treat models for which no closed-form constitutive equation can be derived is illustrated by performing calculations using FENE dumbbells. Significant differences in the stress field between the true FENE and the linearized FENE-P are found, especially during the inception period. Steady-state kinematics are, however, identical within error bars for both FENE and FENE-P and for the Newtonian fluid. The essential algorithm of 2-D CONNFFESSIT is detailed, as well as experience gathered from its parallel and vector versions.