Trial methods for Bernoulli's free boundary problem

Abstract

Free boundary problems deal with solving partial differential equations in a domain, a part of whose boundary is unknown – the so-called free boundary. Beside the standard boundary conditions that are needed in order to solve the partial differential equation, an additional boundary condition is imposed at the free boundary. One aims thus to determine both, the free boundary and the solution of the partial differential equation. This thesis is dedicated to the solution of the generalized exterior Bernoulli free boundary problem which is an important model problem for developing algorithms in a broad band of applications such as optimal design, fluid dynamics, electromagnentic shaping etc. Due to its various advantages in the analysis and implementation, the trial method, which is a fixed-point type iteration method, has been chosen as numerical method. The iterative scheme starts with an initial guess of the free boundary. Given one boundary condition at the free boundary, the boundary element method is applied to compute an approximation of the violated boundary data. The free boundary is then updated such that the violated boundary condition is satisfied at the new boundary. Taylor’s expansion of the violated boundary data around the actual boundary yields the underlying equation, which is formulated as an optimization problem for the sought update function. When a target tolerance is achieved the iterative procedure stops and the approximate solution of the free boundary problem is detected. How efficient or quick the trial method is converging depends significantly on the update rule for the free boundary, and thus on the violated boundary condition. Firstly, the trial method with violated Dirichlet data is examined and updates based on the first and the second order Taylor expansion are performed. A thorough analysis of the convergence of the trial method in combination with results from shape sensitivity analysis motivates the development of higher order convergent versions of the trial method. Finally, the gained experience is exploited to draw very important conclusions about the trial method with violated Neumann data, which is until now poorly explored and has never been numerically implemented.