Long time numerical stability and asymptotic analysis for the viscoelastic oldroyd flows

In this article, we study the long time numerical stability and as- ymptotic behavior for the viscoelastic Oldroyd fluid motion equations. Firstly, with the Euler semi-implicit scheme for the temporal discretization, we deduce the global H2-stability result for the fully discrete finite element solution. Sec- ondly, based on the uniform stability of the numerical solution, we investigate the discrete asymptotic behavior and claim that the viscoelastic Oldroyd prob- lem converges to the stationary Navier-Stokes flows if the body force f(x, t) approaches to a steady-state f1(x) as t ! 1. Finally, some numerical exper- iments are given to verify the theoretical predictions.