In this paper, a backward Euler method combined with finite element discretization in spatial direction is discussed 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. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal {\it a priori} error estimate in $\textbf{L}^2$-norm is derived for the discrete problem with non-smooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition. Finally, we present some numerical results to validate our theoretical results.
翻译：在本文中,对于2D$ Undroyd rode of compaticic 流体的2D$ Unical motion 模型产生的运动方程式,将讨论与空间方向的有限元素分解相结合的后向 Euler 方法,该方法与空间方向后向的 Euler 法相结合,该方法与2D$ Undroved roductive roils of procisions of one with the process unitive or in time or in the time. 或 $ $L ⁇ infty} $L ⁇ infty $(在时间上具有强迫性) 或时间上以 $L ⁇ infty}$(在时间上具有强迫性) 。 显示Drichlettlet 规范中离散解决方案的估计数在时间上是一致的。 。 在$\ textb{L ⁇ 2$- $norm 中得出最佳的偏差的误差估计数,用于非moot 初始数据的离散问题。该估计数。这一估计数在时间上显示在假定独特性条件下是统一的,在时间上是统一的。我们理论结果。我们提出一些数字结果。我们提出一些数字结果的理论结果。
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