In this work, we propose three novel block-structured multigrid relaxation schemes based on distributive relaxation, Braess-Sarazin relaxation, and Uzawa relaxation, for solving the Stokes equations discretized by the mark-and-cell scheme. In our earlier work \cite{he2018local}, we discussed these three types of relaxation schemes, where the weighted Jacobi iteration is used for inventing the Laplacian involved in the Stokes equations. In \cite{he2018local}, we show that the optimal smoothing factor is $\frac{3}{5}$ for distributive weighted-Jacobi relaxation and inexact Braess-Sarazin relaxation, and is $\sqrt{\frac{3}{5}}$ for $\sigma$-Uzawa relaxation. Here, we propose mass-based approximation inside of these three relaxations, where mass matrix $Q$ obtained from bilinear finite element method is directly used to approximate to the inverse of scalar Laplacian operator instead of using Jacobi iteration. Using local Fourier analysis, we theoretically derive the optimal smoothing factors for the resulting three relaxation schemes. Specifically, mass-based distributive relaxation, mass-based Braess-Sarazin relaxation, and mass-based $\sigma$-Uzawa relaxation have optimal smoothing factor $\frac{1}{3}$, $\frac{1}{3}$ and $\sqrt{\frac{1}{3}}$, respectively. Note that the mass-based relaxation schemes do not cost more than the original ones using Jacobi iteration. Another superiority is that there is no need to compute the inverse of a matrix. These new relaxation schemes are appealing.

翻译：在这项工作中,我们提出三个基于分配放松的新颖的组合结构多格放松计划,即布拉伊斯-沙拉津放松计划,以及乌泽放松计划,以解决由标记和细胞计划分解的斯托克斯方程式。在我们早先的工作\cite{he2018loal}中,我们讨论了这三种类型的放松计划,其中加权的雅各分流用于发明参与斯托克斯方程式的拉普拉西亚。在{cite{he2018loal}中,我们表明最佳的顺畅系数是用于分配加权Jacobi放松和不完全的布拉什-沙拉辛放松计划$3美元。在我们早期的工作中,我们讨论了这三种类型的放松计划,其中加权的雅各分层迭接率用于创建参与斯托克斯方程方程式的拉普拉卡。在基于双线固定要素方法中直接使用QQQQ$,以至于马拉加平价操作者反调价,而不是使用雅的降价3美元放松计划。