On Trefftz' integral equation for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of Bernoulli’s free boundary value problem for a harmonic function w in a doubly connected domain D in R2 where an unknown free boundary Γ0 is determined by prescribed Cauchy data of w on Γ0 in addition to a Dirichlet condition on the known boundary Γ1. Our method is based on a two-by-two system of boundary integral equations for the unknown boundary Γ0 and the unknown normal derivative g = ∂νw of w on Γ1. This system is nonlinear with respect to Γ0 and linear with respect to g and we suggest to solve it simultaneously for Γ0 and g by Newton iterations. We establish a local convergence result and exhibit the feasibility of the method by a few numerical examples.