In the finite element framework, we employ decoupled time integration scheme for viscoelastic fluid (the Leonov model) flow and then investigate highly nonlinear behavior in 2D creeping contraction flow. In the analysis of steady solutions as a preliminary study, the results are shown to be free from frustrating mesh dependence when we incorporate the tensor-logarithmic formulation [Fattal and Kupferman, J. Non-Newtonian Fluid Mech. 123, 281–285 (2004)]. Two kinds of elastic fluid have been chosen, that is, highly shear thinning and Boger-type liquids. According to each liquid property, the transient computational modeling has revealed qualitatively distinct dynamics of instability. With pressure difference imposed slightly below the steady convergence limit, the numerical scheme demonstrates fluctuating solution without approaching steady state for the shear thinning fluid. When the pressure fairly higher than the limit is enforced, severe fluctuation of flowrate, oscillation of corner vortices, and also asymmetric irregular stress wave propagation along the downstream channel wall are expressed. In addition, flow dynamics seems quite stochastic with scanty temporal correlation. For the Boger-type fluid, under the traction higher than steady limit, the flowrate and corner vortices exhibit periodic variation with asymmetry added to the dynamics. These express elastic flow instability in this inertialess flow approximation.