Trial methods for Bernoulli's free boundary problem

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.