Overlapping Schwarz methods are not anisotropy-robust multigrid smoothers

TL;DR

This paper demonstrates that overlapping Schwarz methods as smoothers in multigrid algorithms are not robust to anisotropy in diffusion problems unless the block size scales appropriately with the anisotropy strength, highlighting the necessity of global smoothers.

Contribution

The paper provides a detailed analysis showing that local overlapping Schwarz smoothers require large blocks proportional to the inverse square root of anisotropy to be effective, challenging the assumption that local methods suffice.

Findings

Local Fourier analysis confirms deterioration of smoothing with increased anisotropy.

Smoothing effectiveness requires blocks of diameter proportional to (psilon^{-1/2}) for anisotropy ratio (psilon).

Global smoothers are necessary for anisotropy-robust multigrid performance.

Abstract

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 that the smoothing properties of point-wise smoothers, such as Gauss--Seidel, rapidly deteriorate as the strength of anisotropy increases. On the other hand, global smoothers based on line smoothing are known to generally provide good smoothing for diffusion problems, independent of the anisotropy strength. A natural question is whether global methods are really necessary to achieve good smoothing in such problems, or whether it can be obtained with locally overlapping block smoothers using sufficiently large blocks and overlap. Through local Fourier analysis and careful numerical experimentation, we show that global methods are indeed necessary to achieve anisotropy-robust smoothing. Specifically, for any fixed block size bounded sufficiently far away from the global domain size, we find that the smoothing properties of overlapping multiplicative Schwarz rapidly deteriorate with increasing anisotropy, irrespective of the amount of overlap between blocks. Moreover, our results indicate that anisotropy-robust smoothing requires blocks of diameter ${\cal O}(\epsilon^{-1/2})$ for anisotropy ratio $\epsilon \in (0,1]$.