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 ℓ-times connected domains, the method requires solving ℓ boundary integral equations with the Neumann kernel. This can be done in O(ℓ2nlog 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.