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A finite element program has been developed in this project, which is able to predict the x and y directional velocities and normal and shear stresses in the case of a two dimensional Newtonian, power-law and viscoelastic flow through the converging channel of an extrusion die. The solution in the Newtonian case is fairly straightforward and a penalty function technique has been used with a variational principle. The results in this case follow the predicted trends. For a power-law fluid, since the viscosity is not a constant but dependent on the strain-rate, an iterative sequence is needed for the solution. As the nonlinear dynamic terms in the momentum balance equations have been neglected because of creeping flow, the dependence of velocities on the viscosity coefficient is rather weak and for all practical purposes, fairly good results have been obtained for the power-law case without a real iteration but by simply recalculating the stresses and pressures on the basis of the corrected viscosity. Maxwell's upper convected model has been used for viscoelastic analysis but the technique can easily be extended to other convected derivatives. Since, the major problem for viscoelastic fluid analysis is the inability to eliminate stress from dynamic equations, most often the velocities and the stresses need to be solved simultaneously. In this technique, the dynamic and the constitutive equations are solved separately by assuming the stresses to be constant in the first case and the velocities to be constant in the second case. The Newtonian or the power-law stress values have been used as the initial guesses. The dynamic equations are solved, by using a Galerkin-Penalty technique, for velocities and the constitutive equations are solved, by using a Petrov-Galerkin technique, for stresses. The entire sequence is repeated until the final stress values match the initial ones. Thus, the system of equations are essentially solved as a set of linear equations. The weighting functions for the Petrov-Galerkin method has been chosen in such a manner that when the elastic effect is zero the Petrov-Galerkin case degenerates into the Galerkin case. The predictions for velocities and stresses are reasonable and the elastic effect is understandable for low values of relaxation time. At higher values of the relaxation time the results seem to be anomalous. For a zero value of the relaxation parameter the solution degenerates into the Newtonian case.