Novel mass-based multigrid relaxation schemes for the Stokes equations

Abstract

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 marker-and-cell scheme. The mass matrix obtained from the bilinear finite element method is directly used to approximate the inverse of scalar Laplacian operator in the relaxation schemes. Using local Fourier analysis, we theoretically derive optimal smoothing factors for the resulting three relaxation schemes. Specifically, mass-based distributive relaxation, mass-based Braess–Sarazin relaxation, and mass-based σ-Uzawa relaxation have optimal smoothing factor 13, 13 and 13, respectively. These smoothing factors are smaller than those in our earlier work [1], where weighted Jacobi iteration is used for inventing Laplacian involved in the Stokes equations. The mass-based relaxation schemes do not cost more than the original ones using the Jacobi iteration. Another superiority of mass-based relaxation is that there is no need to compute the inverse of a matrix. These new relaxation schemes are appealing.