A Shape-Newton Method for Free-Boundary Problems Subject to the Bernoulli Boundary Condition

We develop a shape-Newton method for solving generic free-boundary problems where one of the free-boundary conditions is governed by the nonlinear Bernoulli equation. The method is a Newton-like scheme that employs shape derivatives of the governing equations. In particular, we derive the shape derivative of the Bernoulli equation, which turns out to depend on the curvature in a nontrivial manner. The resulting shape-Newton method allows one to update the position of the free boundary by solving a special linear boundary-value problem at each iteration. We prove solvability of the linearised problem under certain conditions of the data. We verify the effectiveness of the shape-Newton approach applied to free-surface flow over a submerged triangular obstacle using a finite element method on a deforming mesh. We observe superlinear convergence behaviour for our shape-Newton method as opposed to the unfavourable linear rate of traditional methods.