Trial methods for a nonlinear Bernoulli problem

In this work we consider the problem of determining the free surface for a stationary flow of inviscid fluid. We introduce several different formulations for the problem and study the numerical methods suggested by different formulations. The formulations to be studied include 'shape variational principles' where free boundary is sought after as a critical point of a 'shape' energy functional, as well as shape optimization formulations where one of the free boundary conditions is imposed as a cost functional. Analysis of the Hessians of the above functionals reveals that the conditioning of the minimization problem can depend on the formulation of the problem. Moreover we can characterise the situations where the 'variational' principles lead to practical numerical methods. Different numerical methods are implemented and tested. 1 Introduction Presence of free surfaces is characteristic in many flow situations. For example, the flow problems in ship industry involve typically a free w..