LAUSR

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 $$\mathbb {R}^2$$R2 where an unknown free boundary $$\varGamma _0$$Γ0 is determined by prescribed Cauchy data of w on $$\varGamma _0$$Γ0 in addition to a Dirichlet condition on the known boundary $$\varGamma _1$$Γ1. Our method is based on a two-by-two system of boundary integral equations for the unknown boundary $$\varGamma _0$$Γ0 and the unknown normal derivative $$g=\partial _\nu w$$g=∂νw of w on $$\varGamma _1$$Γ1. This system is nonlinear with respect to $$\varGamma _0$$Γ0 and linear with respect to g and we suggest to solve it simultaneously for $$\varGamma _0$$Γ0 and g by Newton iterations. We establish a local convergence result and exhibit the feasibility of the method by a few numerical examples.