In the framework of finite element analysis we numerically analyze both the steady and transient 4:1 contraction creeping viscoelastic flow. In the analysis of steady solutions, there exists upper limit of available numerical solutions in contraction flow of the Leonov fluid, and it is free from the frustrating mesh dependence when we incorporate the tensor-logarithmic formulation (Fattal and Kupferman, 2004). With the time dependent flow modeling with pressure difference imposed slightly below the steady limit, the $1^{st}$ and $2^{nd}$ order conventional approximation schemes have demonstrated fluctuating solution without approaching the steady state. From the result, we conclude that the existence of upper limit for convergent steady solution may imply flow transition to highly elastic time-fluctuating field without steady asymptotic. However definite conclusion certainly requires further investigation and devising some methodology for its proof.