  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.
