Trial methods for Bernoulli's free boundary problem

Free boundary problems deal with solving partial differential equations in a domain, \na part of whose boundary is unknown – the so-called free boundary. Beside \nthe standard boundary conditions that are needed in order to solve the partial \ndifferential equation, an additional boundary condition is imposed at the free \nboundary. One aims thus to determine both, the free boundary and the solution \nof the partial differential equation. \nThis thesis is dedicated to the solution of the generalized exterior Bernoulli \nfree boundary problem which is an important model problem for developing algorithms \nin a broad band of applications such as optimal design, fluid dynamics, \nelectromagnentic shaping etc. Due to its various advantages in the analysis and \nimplementation, the trial method, which is a fixed-point type iteration method, \nhas been chosen as numerical method. \nThe iterative scheme starts with an initial guess of the free boundary. Given \none boundary condition at the free boundary, the boundary element method is \napplied to compute an approximation of the violated boundary data. The free \nboundary is then updated such that the violated boundary condition is satisfied \nat the new boundary. Taylor’s expansion of the violated boundary data around \nthe actual boundary yields the underlying equation, which is formulated as an \noptimization problem for the sought update function. When a target tolerance is \nachieved the iterative procedure stops and the approximate solution of the free \nboundary problem is detected. \nHow efficient or quick the trial method is converging depends significantly \non the update rule for the free boundary, and thus on the violated boundary \ncondition. Firstly, the trial method with violated Dirichlet data is examined and \nupdates based on the first and the second order Taylor expansion are performed. \nA thorough analysis of the convergence of the trial method in combination with \nresults from shape sensitivity analysis motivates the development of higher order \nconvergent versions of the trial method. Finally, the gained experience is \nexploited to draw very important conclusions about the trial method with violated \nNeumann data, which is until now poorly explored and has never been \nnumerically implemented.