Tracking Neumann Data for Stationary Free Boundary Problems

The present paper is dedicated to the numerical solution of stationary free boundary problem by means of shape optimization. We assume that the state satisfies the Dirichlet problem and track the Neumann data at the free boundary. The gradient and Hessian of the shape functional under consideration are computed. By analysing the shape Hessian in case of matching data a sufficient criterion for its coercivity is derived. Coercivity implies existence and convergence of approximate shapes. The derived coercivity criterion is exactly the same as in [13] even though the present shape functional is completely different to the Dirichlet energy functional considered there. Numerical experiments are carried out in three spatial dimensions.