On convergence in elliptic shape optimization

This paper is aimed at analyzing the existence and convergence of approximate solutions in shape optimization. Two questions arise when one applies a Ritz-Galerkin discretization to solve the necessary condition: does there exists an approximate solution and how good does it approximate the solution of the original infinite dimensional problem? We motivate a general setting by some illustrative examples, taking into account the so-called two norm discrepancy. Provided that the infinite dimensional shape problem admits a stable second order optimizer, we are able to prove the existence of approximate solutions and compute the rate of convergence. Finally, we verify the predicted rate of convergence by numerical results