Two-Color Fourier Analysis of the Multigrid Method with Red-Rlack Gauss-Seidel Smoothing*

A twocolor 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 ii& the low-frequency region. We show that the twocolor two-grid method asymptotically reduces to a one-color twogrid method whose physical mechanism is more transparent than for its original two-color form. The twocolor Fourier analysis is also used to design variants of the standard multigrid algorithm.