Computing the logarithmic capacity of compact sets having (infinitely) many components with the charge simulation method

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Authors
Liesen, Jörg
Nasser, Mohamed M. S.
Sète, Olivier
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Issue Date
2023-02-28
Type
Article
Keywords
Logarithmic capacity , Charge simulation method , Cantor set , Cantor dust , Fast multipole method , GMRES method
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Liesen, J., Nasser, M.M.S. & Sète, O. Computing the logarithmic capacity of compact sets having (infinitely) many components with the charge simulation method. Numer Algor (2023). https://doi.org/10.1007/s11075-022-01428-2
Abstract

We apply the charge simulation method (CSM) in order to compute the logarithmic capacity of compact sets consisting of (infinitely) many “small” components. This application allows to use just a single charge point for each component. The resulting method therefore is significantly more efficient than methods based on discretizations of the boundaries (for example, our own method presented in Liesen et al. (Comput. Methods Funct. Theory 17, 689–713, 2017)), while maintaining a very high level of accuracy. We study properties of the linear algebraic systems that arise in the CSM, and show how these systems can be solved efficiently using preconditioned iterative methods, where the matrix-vector products are computed using the fast multipole method. We illustrate the use of the method on generalized Cantor sets and the Cantor dust.

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Springer Link
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Series
Numerical Algorithms
2023
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ISSN
1572-9265
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