A second order convergent trial method for a free boundary problem in three dimensions

The present article is concerned with the solution of a generalized Bernoulli free boundary problem in three spatial dimensions. We parametrize the free boundary under consideration over the sphere and apply a trial method which updates the free boundary into the normal direction. At the free boundary, we prescribe the Neumann boundary condition and update the free boundary with the help of the remaining Dirichlet boundary condition. An inexact Newton update is employed which leads to a novel second order convergent trial method. Numerical examples show the feasibility of the present approach. In particular, a parametric representation is utilized which imposes no restriction to the free boundary under consideration except for its genus.