In　this　paper,　the　asymptotic　analysis　of　the　two-dimensional　viscoelastic　Oldroyd　flows　is　presented.　With　the　physical　constant　rho/delta　approaches　zero,　where　rho　is　the　viscoelastic　coefficient　and　1/delta　the　relaxation　time,　the　viscoelastic　Oldroyd　fluid　motion　equations　converge　to　the　viscous　model　known　as　the　famous　Navier-Stokes　equations.　Both　the　continuous　and　discrete　uniform-in-time　asymptotic　errors　are　provided.　Finally,　the　theoretical　predictions　are　confirmed　by　some　numerical　experiments.