Smoothing analysis of two-color distributive relaxation for solving 2D Stokes flow by multigrid method

TL;DR

This paper theoretically analyzes the smoothing properties of a two-color distributive relaxation method for 2D Stokes flow, using local Fourier analysis to optimize parameters and improve multigrid convergence.

Contribution

It develops a mathematical framework for the two-color distributive relaxation method, deriving optimal parameters and establishing a convergence zone for the pressure stabilization term.

Findings

The smoothing effectiveness depends on the pressure stabilization parameters.

A specific parameter zone ensures convergence independent of mesh size.

Analytical expressions guide optimal parameter selection.

Abstract

Smoothing properties of two-color distributive relaxation for solving a two-dimensional (2D) Stokes flow by multigrid method are theoretically investigated by using the local Fourier analysis (LFA) method. The governing equation of the 2D Stokes flow in consideration is discretized with the non-staggered grid and an added pressure stabilization term with stabilized parameters to be determined is introduced into the discretization system in order to enhance the smoothing effectiveness in the analysis. So, an important problem caused by the added pressure stabilization term is how to determine a suitable zone of parameters in the added term. To that end, theoretically, a two-color distributive relaxation, developed on the two-color Jacobi point relaxation, is established for the 2D Stokes flow. Firstly, a mathematical constitution based on the Fourier modes with various frequency components is constructed as a base of the two-color smoothing analysis, in which the related Fourier representation is presented by the form of two-color Jacobi point relaxation. Then, an optimal one-stage relaxation parameter and related smoothing factor for the two-color distributive relaxation are applied to the discretization system, and an analytical expression of the parameter zone on the added pressure stabilization term is established by LFA. The obtained analytical results show that numerical schemes for solving 2D Stokes flow by multigrid method on the two-color distributive relaxation have a specific convergence zone on the parameters of the added pressure stabilization term, and the property of convergence is independent of mesh size, but depends on the parameters of the pressure stabilization term.