Two-color fourier analysis of the multigrid method with red-black Gauss-Seidel smoothing

摘要

A two-color Fourier analytical approach is proposed to analyze the multigrid method which employs the red-black Gauss-Seidel smoothing iteration for solving the Poisson equation. In this approach, Fourier components in the high-frequency region are folded into the low-frequency region so that the coupling between the low and high Fourier components is transformed into a coupling between components of red and black computational waves in the low-frequency region. We show that the two-color two-grid method asymptotically reduces to a one-color two-grid method whose physical mechanism is more transparent than for its original two-color form. The two-color Fourier analysis is also used to design variants of the standard multigrid algorithm.