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Journal of Scientific Computing

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30-12-2016

Nonuniformly Weighted Schwarz Smoothers for Spectral Element Multigrid

Author: Jörg Stiller

Published in: Journal of Scientific Computing | Issue 1/2017

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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 Lottes and 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 1.5–3, leading to runtime savings of about 50%. 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.

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Title: Nonuniformly Weighted Schwarz Smoothers for Spectral Element Multigrid
Author: Jörg Stiller
Publication date: 30-12-2016
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2017
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-016-0345-z

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