An error estimate is presented for a fully discrete semi-implicit finite element method for the two-dimensional Navier--Stokes equations with $L^2$ initial data in convex polygonal domains, without extra regularity assumptions or CFL conditions, by utilizing the smoothing properties of the Navier--Stokes equations, an appropriate duality argument, and the smallness of the numerical solution in the discrete $L^2(0,t_m;H^1)$ norm when $t_m$ is smaller than some constant. Numerical examples are provided to support the theoretical analysis.

翻译：对二维导航-斯托克斯方程式采用完全离散的半隐含的限定要素方法,在不附加常规假设或CFL条件的情况下,通过利用Navier-Stokes方程式的平滑性、适当的双重性参数和离散的$L2(0.t_m);H1-1美元标准中数值小于某些恒定值时的数值解算法,提出一个误差估计,该方法的初始数据为2美元,没有额外的常规假设或CFL条件。提供了支持理论分析的数字示例。