An innovative approach for numerically solving viscoelastic fluid problems, recently introduced to solve a time-dependent one-dimensional problem, is extended to two-dimensional space for a steady-state problem. Referred to as CONNFFESSIT (Calculation of Non-Newtonian Flow : Finite Elements and Stochastic Simulation Technique), this new approach uses standard finite element techniques for solving the momentum and continuity equations but replaces the integral or differential constitutive equation, traditionally used to compute polymer stresses, with stochastic simulations of polymer dynamics. In CONNFFESSIT, the polymer contribution to the stress tenser is determined from a Stochastic simulation of an ensemble of model polymer molecules from whose configurations the polymer stress can be computed as an ensemble average. The problem considered for testing the CONNFFESSIT approach in two dimensions is the steady now of an Oldroyd-B fluid in an abrupt 4:1 axisymmetric contraction, assuming incompressibility and isothermal conditions. After details of the implementation of the stochastic simulation technique are presented, a comparison is made between the numerical results obtained using the CONNFFESSIT approach and those obtained using the closed-form integral constitutive equation of the Oldroyd-B model. We find excellent agreement in most cases and explain reasons for the sometimes large differences.