In this study, an approach for the prediction of inception and evolution of viscoelastic instabilities without a need for closed form constitutive equations is presented. This methodology combines Brownian dynamics simulations with time-dependent finite element techniques to allow investigation of stability of viscoelastic flows in complex geometries. In order to demonstrate the feasibility of this new approach, flow transitions in plane Couette flow of Hookean dumbbells have been studied. In turn, the results of Hookean dumbbell simulations (i.e. in terms of the most dangerous eigenvalues and their corresponding eigenfunctions) have been compared with the corresponding closed form constitutive equation, the Oldroyd-B model. The excellent comparison between the two results clearly demonstrates the viability of this new approach. Finally to demonstrate the versatility of our methodology, the effect of the finite extensibility of the entropic spring on the stability of plane Couette flow of finitely extensible non-linear elastic (FENE) dumbbells has been examined. Specifically, the simulation results demonstrate that the plane Couette flow of FENE dumbbells is stable and the corresponding eigenfunctions have a similar structure as the Hookean dumbbell eigenfunctions. Although, the plane Couette flow of Hookean and FENE dumbbells is linearly stable, selected number of non-linear stability analyses have been performed to demonstrate the capability of this new approach.