Backward Euler method for the equations of motion arising in Oldroyd model of order one with nonsmooth initial data

Abstract In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed for the equations of motion arising two-dimensional Oldroyd model viscoelastic fluids order one forcing term independent time or $L^{\infty }$ time. It shown that estimates discrete solution Dirichlet norm bounded uniformly Optimal priori error estimate $\textbf {L}^2$-norm derived problem nonsmooth initial data. This to be uniform time, under assumption uniqueness condition. Finally, we present some numerical results validate our theoretical results.