Convergence of Newton's method in shape optimisation via approximate normal functions

In this paper we propose a Newton method for shape functions defined on an image set generated by the (Micheletti) metric group. We review basic properties of the metric group and a quotient associated with the metric group and a fixed domain. Taking into account the special structure of the second shape derivative and its symmetric part allows us to distinguish between two Hessians, the domain shape Hessian and the boundary shape Hessian. Using the domain Hessian we define a Newton method on the metric group by discretising the tangent space of the quotient via approximate normal functions using reproducing kernels. Under suitable assumptions we are able to show superlinear convergences of the Newton iterations and additionally convergence of the shapes in the metric group. Finally we verify our findings in a number of numerical experiments including a thorough numerical study of the impact of the discretisation on the convergence speed.