Numerical computation of the conformal map onto lemniscatic domains

TL;DR

This paper introduces a numerical method for computing conformal maps from complex, multiply-connected domains onto lemniscatic domains, efficiently handling challenging geometries with high connectivity.

Contribution

The paper develops a boundary integral equation-based numerical approach for conformal mapping onto lemniscatic domains, applicable to complex geometries with high connectivity.

Findings

Method solves boundary integral equations with $O( ext{ell}^2 n ext{log} n)$ complexity.

Numerical examples demonstrate effectiveness for domains with close-to-touching, non-convex, and high connectivity boundaries.

Approach works for piecewise smooth boundaries and high connectivity domains.

Abstract

We present a numerical method for the computation of the conformal map from unbounded multiply-connected domains onto lemniscatic domains. For $\ell$-times connected domains the method requires solving $\ell$ boundary integral equations with the Neumann kernel. This can be done in $O(\ell^2 n \log n)$ operations, where $n$ is the number of nodes in the discretization of each boundary component of the multiply connected domain. As demonstrated by numerical examples, the method works for domains with close-to-touching boundaries, non-convex boundaries, piecewise smooth boundaries, and for domains of high connectivity.