Anisotropy-Robustness Limits of Overlapping Schwarz Multigrid Smoothers
A local-Fourier-analysis and numerical study proving that overlapping multiplicative Schwarz methods cannot serve as anisotropy-robust smoothers in geometric multigrid for two-dimensional anisotropic diffusion. For any fixed block size bounded away from the global domain size, the smoothing factor deteriorates rapidly as the anisotropy ratio grows, no matter how much the blocks overlap; anisotropy-robust smoothing instead requires blocks of diameter of order epsilon^(-1/2), establishing that global (line-type) smoothers are genuinely necessary.
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We analyze overlapping multiplicative Schwarz methods as smoothers in the geometric multigrid solution of two-dimensional anisotropic diffusion problems. For diffusion equations, it is well known tha…