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Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method

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Abstract

We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.

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Authors and Affiliations

  1. Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, IL, 61801, USA

    James W. Lottes

  2. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 60439, USA

    Paul F. Fischer

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  1. James W. Lottes
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  2. Paul F. Fischer
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Correspondence to Paul F. Fischer.

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Lottes, J.W., Fischer, P.F. Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method. J Sci Comput 24, 45–78 (2005). https://doi.org/10.1007/s10915-004-4787-3

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  • DOI: https://doi.org/10.1007/s10915-004-4787-3

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