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Smoothing properties of two-color distributive relaxation for solving atwo-dimensional (2D) Stokes flow by multigrid method are theoreticallyinvestigated by using the local Fourier analysis (LFA) method. The governingequation of the 2D Stokes flow in consideration is discretized with thenon-staggered grid and an added pressure stabilization term with stabilizedparameters to be determined is introduced into the discretization system inorder to enhance the smoothing effectiveness in the analysis. So, an importantproblem caused by the added pressure stabilization term is how to determine asuitable zone of parameters in the added term. To that end, theoretically, atwo-color distributive relaxation, developed on the two-color Jacobi pointrelaxation, is established for the 2D Stokes flow. Firstly, a mathematicalconstitution based on the Fourier modes with various frequency components isconstructed as a base of the two-color smoothing analysis, in which the relatedFourier representation is presented by the form of two-color Jacobi pointrelaxation. Then, an optimal one-stage relaxation parameter and relatedsmoothing factor for the two-color distributive relaxation are applied to thediscretization system, and an analytical expression of the parameter zone onthe added pressure stabilization term is established by LFA. The obtainedanalytical results show that numerical schemes for solving 2D Stokes flow bymultigrid method on the two-color distributive relaxation have a specificconvergence zone on the parameters of the added pressure stabilization term,and the property of convergence is independent of mesh size, but depends on theparameters of the pressure stabilization term.

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