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Mathematics of Computation

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Analysis of two-grid methods: The nonnormal case


Author: Yvan Notay
Journal: Math. Comp. 89 (2020), 807-827
MSC (2010): Primary 65F08, 65F10, 65F50, 65N22
DOI: https://doi.org/10.1090/mcom/3460
Published electronically: July 11, 2019
MathSciNet review: 4044451
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Abstract: Core results about the algebraic analysis of two-grid methods are extended in relations bounding the field of values (or numerical range) of the iteration matrix. On this basis, bounds are obtained on its norm and numerical radius, leading to rigorous convergence estimates. Numerical illustrations show that the theoretical results deliver qualitatively good predictions, allowing one to anticipate success or failure of the two-grid method. They also indicate that the field of values and the associated numerical radius are much more reliable convergence indicators than the eigenvalue distribution and the associated spectral radius. On this basis, some discussion is developed about the role of local Fourier or local mode analysis for nonsymmetric problems.


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Additional Information

Yvan Notay
Affiliation: Service de Métrologie Nucléaire, Université Libre de Bruxelles (C.P. 165/84), 50, Av. F.D. Roosevelt, B-1050 Brussels, Belgium
Email: ynotay@ulb.ac.be

DOI: https://doi.org/10.1090/mcom/3460
Keywords: Iterative methods, convergence analysis, linear systems, multigrid, two-grid, nonnormal matrices, AMG, preconditioning
Received by editor(s): April 3, 2018
Received by editor(s) in revised form: March 5, 2019, and April 12, 2019
Published electronically: July 11, 2019
Additional Notes: The author is Research Director of the Fonds de la Recherche Scientifique – FNRS
Article copyright: © Copyright 2019 American Mathematical Society