Abstract:

A variational formulation of the Poisson equation with homogeneous boundary condition is considered as a state equation on a two-dimensional domain. A part of the boundary has to be found to minimize a smooth cost functional. The primal hybrid formulation of the state problem is used to obtain not only a solution of the original state equation but also its derivative with respect to the outward unit normal to the boundary of the domain. Simple approximative spaces are introduced and a convergence of approximate state solutions as well as approximate optimal domains are proved.