Numerical computation of the conformal map onto lemniscatic 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.

Minor revision; simplified Example 6.1, and changed Example 6.2 to a set without symmetry

References in corpus (5)

• Fast solution of boundary integral equations with the generalized Neumann kernel
• Fast and accurate computation of the logarithmic capacity of compact sets
• Fast computation of the circular map
• Properties and examples of Faber--Walsh polynomials
• On conformal maps from multiply connected domains onto lemniscatic domains

Cited by in corpus (9)

• Fast and accurate computation of the logarithmic capacity of compact sets
• Properties and examples of Faber--Walsh polynomials
• On conformal maps from multiply connected domains onto lemniscatic domains
• Computing the logarithmic capacity of compact sets having (infinitely) many components with the Charge Simulation Method
• Walsh's conformal map onto lemniscatic domains for polynomial pre-images I
• Walsh's Conformal Map onto Lemniscatic Domains for Polynomial Pre-images II
• Walsh's Conformal Map onto Lemniscatic Domains for Several Intervals
• Minimal degree rational open up mappings and related questions
• The Complex Green's Function for Symmetric Sets