In this paper, we present a novel local and parallel two-grid finite element scheme for solving the Stokes equations, and rigorously establish its a priori error estimates. The scheme admits simultaneously small scales of subproblems and distances between subdomains and its expansions, and hence can be expandable. Based on the a priori error estimates, we provide a corresponding iterative scheme with suitable iteration number. The resulting iterative scheme can reach the optimal convergence orders within specific two-grid iterations ($O(|\ln H|^2)$ in 2-D and $O(|\ln H|)$ in 3-D) if the coarse mesh size $H$ and the fine mesh size $h$ are properly chosen. Finally, some numerical tests including 2-D and 3-D cases are carried out to verify our theoretical results.
翻译:在本文中,我们提出了一个解决斯托克斯方程式的新颖的本地和平行的双格限制要素方案,并严格地确立其先验误差估计。 如果正确选择粗微网格大小的$H美元和精细网格大小的$h美元,那么这个方案同时承认子问题规模小,子领域及其扩展之间的距离,因此是可以扩大的。根据先验误差估计,我们提供了相应的迭接方案,并配有适当的迭代编号。由此产生的迭代方案可以在特定的两格迭代内达到最佳趋同顺序(O(O)-D)美元和3-D美元。最后,进行了一些数字测试,包括2-D和3-D案例,以核实我们的理论结果。