  In this article, we discuss a couple of nonlinear Galerkin method (NLG) in
finite element set up for viscoelastic fluid flow, mainly equations of motion
arising in the flow of 2D Oldroyd model. We obtain improved error estimate in
$L^{\infty}(\bL^2)$ norm, which is optimal in nature, for linear finite element
approximation, in view of the error estimate available in literature, in
$L^2(\bH^1)$ norm.
