Closed-form multigrid smoothing factors for lexicographic Gauss-Seidel (Q2902192)

The authors study the computation of smoothing factors on uniform meshes in arbitrary dimensions. Applying the complex analysis results on the maximum modulus principle and properties of Möbius transformations they are able to derive closed-form expressions for the smoothing factors of the lexicographic pointwise and block Gauss-Seidel method. In the case of the pointwise method the effect of a relaxation parameter is analysed as well. The results presented in the paper are applicable to various model problems, such as the Poisson and anisotropic diffusion equations and the convection-diffusion equations in arbitraty space dimensions. It is shown that the lexicographic Gauss-Seidel smoothing is efficient for equations with strong asymmetry such as the convection-diffusion equations. For the constant coefficient convection-diffusion equation with equal mesh Reynolds numbers it is shown for the upwind discretizations that the lexicographic Gauss-Seidel smoothing has a smaller factor than the red-black Gauss-Seidel smoothing.