Fast and accurate computation of the logarithmic capacity of compact sets

TL;DR

This paper introduces a fast, accurate numerical method for computing the logarithmic capacity of complex, multiply connected compact sets in the plane using conformal maps and boundary integral equations.

Contribution

It develops a new computational approach leveraging conformal mapping and boundary integral equations to efficiently estimate logarithmic capacities of complex sets.

Findings

Method is fast and accurate based on numerical examples.

Successfully applied to Cantor middle third set and its generalizations.

Handles sets with multiple components and no symmetry.

Abstract

We present a numerical method for computing the logarithmic capacity of compact subsets of $\mathbb{C}$, which are bounded by Jordan curves and have finitely connected complement. The subsets may have several components and need not have any special symmetry. The method relies on the conformal map onto lemniscatic domains and, computationally, on the solution of a boundary integral equation with the Neumann kernel. Our numerical examples indicate that the method is fast and accurate. We apply it to give an estimate of the logarithmic capacity of the Cantor middle third set and generalizations of it.