First-order convergence in time and space is proved 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 grid-ratio conditions. The proof utilises 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:事实证明,时间和空间的第一阶趋同是二维导航-斯托克方程式的完全离散的半隐性限定要素方法,在不附加常规假设或网格-纬度条件的情况下,在convex多边形域的初始数据为$L2美元,没有额外的常规假设或网格-纬度条件。该证据利用了纳维-斯托克方程式的平滑特性、适当的双重性论点,以及离散值$L2(0美元)/t_m;H1美元标准中数字溶液的微小,如果$t_m美元小于某些恒定值,则使用数字示例支持理论分析。