Optimal smoothing factor with coarsening by a factor of three for the MAC scheme for the Stokes equations

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 [1], local Fourier analysis has been applied to a mass-based Braess-Sarazin relaxation, a mass-based σ-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 a factor of three for these relaxation schemes. We derive theoretically optimal smoothing factors for this coarsening strategy. The optimal smoothing factors of coarsening by a factor of three are nearly equal to those obtained from standard coarsening. Thus, coarsening by a factor of three is superior computationally. Moreover, coarsening by a factor of three generates a nested hierarchy of grids, which simplifies and unifies the construction of grid-transfer operators.