A conformal mapping algorithm for the Bernoulli free boundary value problem

摘要

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in R2 where an unknown free boundary (0) is determined by prescribed Cauchy data on (0) in addition to a Dirichlet condition on the known boundary (1). Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary (0) as the unknown boundary in the inverse problem to construct (0) from the Dirichlet condition on (0) and Cauchy data on the known boundary (1). Our method for the Bernoulli problem iterates on the missing normal derivative on (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. Copyright (c) 2015 John Wiley & Sons, Ltd.