Nonuniformly weighted Schwarz smoothers for spectral element multigrid

TL;DR

This paper introduces a nonuniformly weighted Schwarz smoother for spectral element multigrid methods, significantly improving convergence rates and reducing computational costs for solving the Poisson equation.

Contribution

It extends existing Schwarz methods by incorporating nonuniform weights, achieving faster convergence and better robustness for spectral element solvers.

Findings

Logarithmic convergence rates between 1.2 and 1.9 achieved.

Iteration count reduced by a factor of 1.5 to 3.

Method is robust for various mesh sizes and polynomial degrees.

Abstract

A hybrid Schwarz/multigrid method for spectral element solvers to the Poisson equation in $\mathbb R^2$ is presented. It extends the additive Schwarz method studied by J. Lottes and P. Fischer (J. Sci. Comput. 24:45--78, 2005) by introducing nonuniform weight distributions based on the smoothed sign function. Using a V-cycle with only one pre-smoothing, the new method attains logarithmic convergence rates in the range from 1.2 to 1.9, which corresponds to residual reductions of almost two orders of magnitude. Compared to the original method, it reduces the iteration count by a factor of 1.5 to 3, leading to runtime savings of about 50 percent. In numerical experiments the method proved robust with respect to the mesh size and polynomial orders up to 32. Used as a preconditioner for the (inexact) CG method it is also suited for anisotropic meshes and easily extended to diffusion problems with variable coefficients.