A conformal mapping algorithm for the Bernoulli free boundary value problem

Abstract

International audienceWe propose a new numerical method for the solution of Bernoulli's free boundary valueproblem for harmonic functions in a doubly connected domain $D$ in $\real^2$ where an unknown free boundary $\Gamma_0$ is determined by prescribed Cauchy data on $\Gamma_0$ in addition to a Dirichlet condition on the known boundary $\Gamma_1$.Our main idea is to involve the conformal mapping methodas proposed and analyzed by Akduman, Haddar and Kress~\cite{AkKr,HaKr05}for the solution of a related inverse boundary value problem. For this we interpret the free boundary $\Gamma_0$as the unknown boundary in the inverse problem to construct $\Gamma_0$ from the Dirichlet condition on $\Gamma_0$ and Cauchy data on the known boundary $\Gamma_1$. Our method for the Bernoulli problem iterates on the missing normal derivative on $\Gamma_1$by alternating between the application of the conformal mapping method for the inverse problemand solving a mixed Dirichlet--Neumann boundary value problem in $D$. We present the mathematicalfoundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach