In many applications, which are modeled by partial differential equations, there is a small number of spatially distributed materials or parameters distinguished by interfaces. In order to identify these parameters, it is often more favorable to treat the shape of the interfaces as a variable instead of the parameter itself. Since the involved materials may form complex contours, high resolutions are required in the underlying finite element discretizations. The challenge here is to combine methods from PDE constraint shape optimization with HPC techniques and prepare algorithms for supercomputing.

We examine the interaction of multigrid methods and shape optimization in appropriate shape spaces. Our aim is a scalable algorithm for application on supercomputers, which can only be achieved by mesh-independent convergence. The impact of discrete approximations of geometrical quantities, like the mean curvature, on a multigrid shape optimization algorithm with quasi-Newton updates is investigated. For the purpose of illustration, we consider a complex model for the identification of cellular structures in biology with minimal compliance in terms of elasticity and diffusion equations.