  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.
