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

TL;DR

This paper develops a local Fourier analysis for multigrid methods with coarsening by three for the Stokes equations, showing it is computationally advantageous and simplifies grid transfer operator construction.

Contribution

It extends local Fourier analysis to coarsening by three, deriving optimal smoothing factors and demonstrating computational benefits over standard coarsening.

Findings

Optimal smoothing factors nearly match standard coarsening.

Coarsening by three is computationally more efficient.

Creates a nested grid hierarchy simplifying transfer operators.

Abstract

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.