In　this　article,　we　study　the　long　time　numerical　stability　and　asymptotic　behavior　for　the　viscoelastic　Oldroyd　fluid　motion　equations.　Firstly,　with　the　Euler　semi-implicit　scheme　for　the　temporal　discretization,　we　deduce　the　global　H-2-stability　result　for　the　fully　discrete　finite　element　solution.　Secondly,　based　on　the　uniform　stability　of　the　numerical　solution,　we　investigate　the　discrete　asymptotic　behavior　and　claim　that　the　viscoelastic　Oldroyd　problem　converges　to　the　stationary　Navier-Stokes　flows　if　the　body　force　f(x,t)　approaches　to　a　steady-state　f(infinity)(x)　as　t　-＞　infinity.　Finally,　some　numerical　experiments　are　given　to　verify　the　theoretical　predictions.