A pseudo-transient finite element method for the computation of two-dimensional viscoelastic flows using the method of characteristics is presented. The momentum and continuity equations are first solved as a saddle-point problem using the Uzawa algorithm with the viscoelastic stress treated as a fixed body force. The constitutive equation is then integrated with known kinematics using the method of characteristics. Numerical results involving the four-to-one abrupt circular contraction and the flow around a sphere in a tube are presented to assess the validity and the robustness of the proposed approach.