In this paper, we discuss the backward Euler method along with its linearized version
for the Kelvin-Voigt viscoelastic fluid flow model with non zero forcing function, which is either
independent of time or in L^∞(L²). After deriving some bounds for the semidiscrete scheme,
a priori estimates in Dirichlet norm for the fully discrete scheme are obtained, which are valid
uniformly in time using a combination of discrete Gronwall's lemma and Stolz-Cesaro's classical
result for sequences. Moreover, an existence of a discrete global attractor for the discrete problem
is established. Further, optimal a priori error estimates are derived, whose bounds may depend
exponentially in time. Under uniqueness condition, these estimates are shown to be uniform
in time. Even when f = 0, the present result improves upon earlier result of Bajpai et al.
(IJNAM,10 (2013), pp.481-507) in the sense that error bounds in this article depend on 1 ⁄ \sqrt{K} as
against 1 ⁄ K^r, r ≥ 1. Finally, numerical experiments are conducted which confirm our theoretical
findings.