  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.

Numerical Analysis;Numerical Analysis