  In this work, we propose a local Fourier analysis for multigrid methods with
coarsening by a factor of three for the staggered finite-difference method
applied to the Stokes equations. In [21], local Fourier analysis has been
applied to a mass-based Braess-Sarazin relaxation, a mass-based $\sigma$-Uzawa
relaxation, and a mass-based distributive relaxation, with standard coarsening
on staggered grids for the Stokes equations. Here, we consider multigrid
methods with coarsening by three for these relaxation schemes. We derive
theoretically optimal smoothing factors for this coarsening strategy. The
optimal smoothing factors of coarsening by three are nearly equal to those
obtained from standard coarsening. Thus, coarsening by three is superior
computationally. Moreover, coarsening by three generates a nested hierarchy of
grids, which simplifies and unifies the construction of grid-transfer
operators.
