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In this paper, a backward Euler method is discussed for the equations ofmotion arising in the 2D Oldroyd model of viscoelastic fluids of order one withthe forcing term independent of time or in $L^{\infty}$ in time. It is shownthat the estimates of the discrete solution in Dirichlet norm is boundeduniformly in time. Optimal a priori error estimate in L2-norm is derived forthe discrete problem with non-smooth initial data. This estimate is shown to beuniform in time, under the assumption of uniqueness condition.

Backward Euler method for the Equations of Motion Arising in Oldroyd Fluids of Order One with Nonsmooth Initial Data