A combination of the finite element method and the finite volume method has been developed for time-dependent viscoelastic flow problems. The governing system of equations is decoupled provided that a reasonably small time-increment is used, which is obtained from a Courant-Friedrichs-Lewy (CFL) condition taking into account a local vorticity wave speed. The momentum equation is solved explicitly with respect to time by a finite element method. The constitutive equation for an upper-convected Maxwell model is solved implicitly by finite volume methods. Case studies have been conducted for start-up of planar Poiseuille flow and planar four-to-one contraction flow for upper-convected Maxwell and Oldroyd-B fluids. Numerical results agree with analytical solutions for shock propagation in start-up of Poiseuille flow with no upper limit on the Weissenberg number We. In planar 4:1 contraction flow, the size of the corner vortex compares well with experimental measurements and other numerical predictions. An attempt to seek a critical value of We was stopped at We = 2 because of transient behaviour of the viscoelatic flow. There are new findings: vorticity separation from a re-entrant corner and a temporary lip vortex in transient flow induced by an instantaneous increase of We.