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Journal Articles Mathematical Methods in the Applied Sciences Year : 2015

A conformal mapping algorithm for the Bernoulli free boundary value problem

Abstract

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 $\real^2$ 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 method as 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 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.
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Dates and versions

hal-01214261 , version 1 (13-10-2015)

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Houssem Haddar, Rainer Kress. A conformal mapping algorithm for the Bernoulli free boundary value problem. Mathematical Methods in the Applied Sciences, 2015, ⟨10.1002/mma.3708⟩. ⟨hal-01214261⟩
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