  In this article, a finite element Galerkin method is applied to the
Kelvin-Voigt viscoelastic fluid model, when its forcing function is in
$L^{\infty}(\bL^2)$. Some new {\it a priori} bounds for the velocity as well as
for the pressure are derived which are independent of inverse powers of the
retardation time $\kappa$. Optimal error estimates for the velocity in
$L^{\infty} (\bL^2)$ as well as in $L^{\infty}(\bH^1_0)$-norms and for the
pressure in $L^{\infty}(L^2)$-norm of the semidiscrete method are discussed
which hold uniformly with respect to $\kappa$ as $\kappa\rightarrow 0$ with the
initial condition only in $\bH^2\cap\bH_0^1$. Further, under uniqueness
condition, these estimates are shown to be uniformly in time as $t \mapsto
\infty$. For the complete discretization of the semidiscrete system, a
first-order accurate backward Euler method is applied and fully discrete
optimal error estimates are established. Finally, numerical experiments are
conducted to verify the theoretical results. The results derived in this
article are sharper than those derived earlier for finite element analysis of
the Kelvin-Voigt fluid model in the sense that the error estimates in this
article hold true uniformly even as $\kappa\rightarrow 0$.
