Fully discrete finite element method for the viscoelastic fluid motion equations

In this article, a fully discrete finite element method isconsidered for the viscoelastic fluid motion equations arising inthe two-dimensional Oldroyd model. A finite element method isproposed for the spatial discretization and the time discretizationis based on the backward Euler scheme. Moreover, the stability andoptimal error estimates in the $L^2$- and $H^1$-norms for thevelocity and $L^2$-norm for the pressure are derived for all time$t>0.$ Finally, some numerical experiments are shown to verify thetheoretical predictions.