Shape optimization for free boundary problems : analysis and numerics

In this paper the solution of a Bernoulli type free boundary problem by means of shape optimization is considered. Four different formulations are com- pared from an analytical and numerical point of view. By analyzing the shape Hessian in case of matching data it is distinguished between well-posed and ill-posed formulations. A nonlinear Ritz-Galerkin method is applied for discretizing the shape optimization problem. In case of well-posedness existence and convergence of the approximate shapes is proven. In combination with a fast boundary element method efficient first and second order shape optimization algorithms are obtained.