Shape Optimization for Stokes flows: a finite-element convergence analysis

In this paper we analyze a two-dimensional shape optimization problem, governed by Stokes equations that are defined on a domain with a part of the boundary that is described as the graph of the control function. The state problem formulation is mapped onto a reference domain, which is independent of the control function, and the analysis is mainly led on such domain. The existence of an optimal control function is proved, and optimality conditions are derived. After the analytical inspection of the problem, finite element discretization is considered for both the control function and the state variables, and a priori convergence error estimates are derived. Numerical experiments assess the validity of the theoretical results.