In this paper we present a new time marching scheme for the time dependent simulation of viscoelastic flows governed by constitutive equations of differential form. Based on the ideas of Carvalho and Scriven , a domain perturbation technique is introduced that can be applied to viscoelastic flows with fluid/ fluid interfaces or free surfaces. This work mainly focuses on the development and, consequently, benchmarking of Finite Element algorithms (FEM) that can efficiently handle the stability problems of complex viscoelastic flows. Since spurious or non-physical solutions are easily generated for this type of analysis using Finite Element techniques (FEM), both the new time stepping scheme and the domain perturbation technique are benchmarked in simple shear flows of Upper Convected Maxwell (UCM) fluids. Both single and two layer flows are considered for which the dominating mode and associated growth rate of a perturbation are solutions of the 1-dimensional Generalized Eigenvalue Problem (GEVP). We show that both the growth rate and the most dangerous eigenmode of the simple shear flows can be accurately captured by our transient algorithm.