Consideration is given to a methodology for achieving highly elastic solutions of complex flows. Steady solutions are obtained through an unsteady finite element approach that employs a pressure-correction scheme. A simplified Phan-Thien-Tanner constitutive model is used with parameter values representative of polymer solutions to simulate viscoelastic flows for two planar problems, a 4:1 contraction flow and a mixing and separating flow. Highly elastic flows are studied for both problems. For mixing and separating flow, various bifurcations are investigated for Newtonian and viscoelastic fluids to reveal the effects of increasing inertia, elasticity and variation in geometric gap width. Good agreement with experimental observations is achieved.