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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Funding by German Research Foundation (DFG) in frame of the Project STI 157/4-1 is gratefully acknowledged.
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Stiller, J. Nonuniformly Weighted Schwarz Smoothers for Spectral Element Multigrid. J Sci Comput 72, 81–96 (2017). https://doi.org/10.1007/s10915-016-0345-z
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DOI: https://doi.org/10.1007/s10915-016-0345-z