In this study a modified finite element :method (FEM) is utilized to simulate the 4:1 planar entry flow of an Oldroyd-B fluid. Several numerical techniques are employed, such as the pseudo-transient method, the decoupled approach, the penalty formulation, and streamline upwinding skill, to facilitate calculation of convergent numerical solutions when fluid elasticity is increased. This procedure yields more satisfactory results in that the upper limit of the Weissenberg number, where a convergent solution is obtainable, is much higher.