Multigrid Smoothing Factors for Red-Black Gauss–Seidel Relaxation Applied to a Class of Elliptic Operators

Analytic formulae are obtained for the smoothing factors yielded by Gauss–Seidel relaxation in two-color ordering for a class of scalar elliptic operators. Block and point relaxations, in conjunction with full or partial coarsening, are encompassed for operators with general (constant, positive) coefficients in general dimensions and for an arbitrary number of relaxation sweeps. It is found that there is no direct dependence of the smoothing factors on the dimension, and that the effect of the number of relaxation sweeps on the smoothing factor is usually independent of the operator coefficients and of the relaxation scheme. The results are compared with computed results of two-level analyses. Smoothing strategies implied by the formulae are discussed.