Novel mass-based multigrid relaxation schemes for the Stokes equations

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.