Numerical conformal mapping with rational functions

TL;DR

This paper introduces new numerical algorithms for conformal mapping using rational functions and least-squares boundary fitting, especially effective for regions with corners and singularities.

Contribution

It presents novel algorithms combining rational approximations with lightning Laplace solver techniques for efficient conformal mapping of complex regions.

Findings

Effective mapping of polygonal regions with corners.

Improved handling of singularities near corners.

Enhanced computational efficiency for conformal mapping.

Abstract

New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the Dirichlet problem is solved by the "lightning Laplace solver" with poles exponentially clustered near each singularity. For polygons and circular polygons, further simplifications are possible.