A Priori Error Estimates for Semidiscrete Finite Element Approximations to Equations of Motion Arising in Oldroyd Fluids of Order One

In this paper, a semidiscrete finite element Galerkin method for the equations of motion arising in the 2D Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty}$ in time, is analyzed. A step-by-step proof of the estimate in the Dirichlet norm for the velocity term which is uniform in time is derived for the non-smooth initial data. Further, new regularity results are obtained which reflect the behavior of solutions as $t\rightarrow 0$ and $t\rightarrow\infty.$ Optimal $L^\infty({\bf L}^2)$ error estimates for the velocity which is of order $O(t^{-1/2}h^2)$ and for the pressure term which is of order $O(t^{-1/2}h)$ are proved for the spatial discretization using conforming elements, when the initial data is divergence free and in $H_0^1.$ Moreover, compared to the results available in the literature even for the Navier-Stokes equations, the singular behavior of the pressure estimate as $t\rightarrow 0,$ is improved by an order $1/2,$ from $t^{-1}$ to $t^{-1/2},$ when conforming elements are used. Finally, under the uniqueness condition, error estimates are shown to be uniform in time