Computing the logarithmic capacity of compact sets having (infinitely) many components with the Charge Simulation Method

TL;DR

This paper introduces an efficient Charge Simulation Method for computing the logarithmic capacity of complex compact sets with infinitely many components, leveraging single charge points per component and advanced numerical techniques.

Contribution

The paper presents a novel, more efficient CSM approach for calculating capacities of intricate sets, combining single charge points with fast iterative solvers and the Fast Multipole Method.

Findings

Method achieves high accuracy with fewer computational resources.

Efficient solution of linear systems via preconditioned iterative methods.

Successful application to Cantor sets and dust demonstrates versatility.

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, S\`ete, Nasser, 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.