Adaptive multilevel-based shape optimization for stationary Stokes flow by path-following primal-dual interior point methods

We consider shape optimization problems for channel-like domains occupied by stationary Stokes flow. The aim is to design the lateral walls of the domain such that a tracking type functional in terms of the velocity and the pressure is minimized. The design variables are the Bezier control points of a composite Bezier curve representation of the lateral walls. After reformulating the minimization problem within a primal-dual interior point framework and considering the associated optimality conditions, one is lead to a parameter-dependent nonlinear system which gives rise to the so-called central path. We present a predictor-corrector based path-following method with an adaptive choice of the continuation steplength originating from the affine covariant convergence theory for Newton-like methods. This scheme can be extended to a multigrid predictor-corrector method with nested iterations in the continuation step and a Newton-type corrector featuring a multigrid solver for the Stokes problems with respect to the primal and dual variables.;Numerical examples show the robustness of the adaptive path-following scheme and underline the efficiency of the multigrid version of the optimization algorithm.