A conformal mapping algorithm for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of Bernoulli’s free boundary value problem for harmonic functions in a doubly connected domain D in R2 where an unknown free boundary Γ0 is determined by prescribed Cauchy data on Γ0 in addition to a Dirichlet condition on the known boundary Γ1. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar and Kress [2, 9] for the solution of a related inverse boundary value problem. For this we interpret the free boundary Γ0 as the unknown boundary in the inverse problem to construct Γ0 from the Dirichlet condition on Γ0 and Cauchy data on the known boundary Γ1. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ1 by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach.