Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data

Abstract

aWe apply the trial method for the solution of Bernoulli's free boundary problem when the Dirichlet boundary condition is imposed for the solution of the underlying Laplace equation and the free boundary is updated according to the Neumann boundary condition. The Dirichlet boundary value problem for the Laplacian is solved by an exponentially convergent boundary element method. The update rule for the free boundary is derived from the linearization of the Neumann data around the actual free boundary. With the help of shape sensitivity analysis and Banach's fixed-point theorem, we shed light on the convergence of the respective trial method. Especially, we derive a stabilized version of this trial method. Numerical examples validate the theoretical findings.

A conformal mapping algorithm for the Bernoulli free boundary value problem

“…Then, firstly, the boundary value problem with one of the conditions on the free boundary omitted is solved, and, secondly, the remaining boundary condition is used to update the free boundary. These two steps are iterated until both free boundary conditions are satisfied up to some specified accuracy (see and further references therein).…”