Efficient treatments of stationary free boundary problems

In the present paper we consider the efficient treatment of free boundary problems by shape optimization. We reformulate the free boundary problem as shape optimization problem. A second order shape calculus enables us to realize a Newton scheme to solve this problem. In particular, all evaluations of the underlying state function are required only on the boundary of the domain. We compute these data by boundary integral equations which are numerically solved by a fast wavelet Galerkin scheme. Numerical results prove that we succeeded in finding a fast and robust algorithm for solving the considered class of problems. Furthermore, the stability of the solutions is investigated by treating the second order sufficient optimality conditions of the underlying shape problem

Coupling of FEM and BEM in shape optimization

In the present paper we consider the numerical solution of shape optimization problems which arise from shape functionals of integral type over a compact region of the unknown domain, especially $L^2$-tracking type functionals. The underlying state equation is assumed to satisfy a Poisson equation with Dirichlet boundary conditions. We proof that the shape Hessian is not strictly $H^1/2$-coercive at the optimal domain which implies ill-posedness of the optimization problem under consideration. Since the adjoint state depends directly on the state, we propose a coupling of finite element methods (FEM) and boundary element methods (BEM) to realize an efficient first order shape optimization algorithm. FEM is applied in the compact region while the rest is treated by BEM. The coupling of FEM and BEM essentially retains all the structural and computational advantages of treating the free boundary by boundary integral equations

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