Overlapping Schwarz methods are not anisotropy-robust multigrid smoothers
Oliver A. Krzysik, Ben S. Southworth, Bobby Philip
TL;DR
The paper investigates whether local overlapping block smoothers in geometric multigrid can deliver anisotropy-robust smoothing for 2D anisotropic diffusion. Using local Fourier analysis (LFA) on maximally overlapping Schwarz smoothers with $2\times 2$ and $\ell\times 1$ subdomains, it shows that smoothing degrades as the anisotropy ratio $\epsilon$ increases, for any fixed block size; robust smoothing only arises when subdomains grow like ${\cal O}(\epsilon^{-1/2})$, which becomes impractical for large anisotropy. Grid-aligned and rotated anisotropies are considered, with FD and FE discretizations; while line/plane smoothing remains robust, fixed-block overlapping Schwarz is not, except in regimes where blocks scale with $\epsilon^{-1/2}$, effectively approaching global lines. Numerical results corroborate the theoretical predictions and emphasize the practical preference for global line smoothing or specialized anisotropy-aware methods. These results guide multigrid solver design for highly anisotropic diffusion, suggesting limits on the effectiveness of local overlap and highlighting promising directions like semi-coarsening or algebraic multigrid adaptations.
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 ${\cal O}(ε^{-1/2})$ for anisotropy ratio $ε\in (0,1]$.