In this paper, we consider the use of H(div) elements in the velocity-pressure formulation to discretize Stokes equations in two dimensions. We address the error estimate of the element pair RT0-P0, which is known to be suboptimal, and render the error estimate optimal by the symmetry of the grids and by the superconvergence result of Lagrange inter-polant. By enlarging RT0 such that it becomes a modified BDM-type element, we develop a new discretization [Formula: see text]. We, therefore, generalize the classical MAC scheme on rectangular grids to triangular grids and retain all the desirable properties of the MAC scheme: exact divergence-free, solver-friendly, and local conservation of physical quantities. Further, we prove that the proposed discretization [Formula: see text] achieves the optimal convergence rate for both velocity and pressure on general quasi-uniform grids, and one and half order convergence rate for the vorticity and a recovered pressure. We demonstrate the validity of theories developed here by numerical experiments.

中文翻译：

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二维Stokes方程的三角形MAC方案的收敛性分析。

在本文中，我们考虑在速度-压力公式中使用H（div）元素来离散二维Stokes方程。我们解决了已知为次优的元素对RT0-P0的误差估计，并通过网格的对称性和Lagrange Inter-polant的超收敛结果使误差估计最佳。通过扩大RT0使其成为经过修改的BDM类型元素，我们开发了一种新的离散化[公式：参见文本]。因此，我们将矩形网格上的经典MAC方案推广到三角形网格，并保留了MAC方案的所有理想特性：精确无散度，对求解器友好以及对物理量的局部守恒。此外，我们证明了建议的离散化[公式：参见文本]在一般准均匀网格上实现了速度和压力的最佳收敛速度，涡度和恢复压力的收敛速度为一阶和半阶。我们通过数值实验证明了这里开发的理论的有效性。