Domain Truncation Method

In this chapter the domain decomposition technique combined with the Steklov–Poincare pseudo-differential boundary operator is presented. The main idea is introduced in the context of coupled elliptic boundary value problems. Then, the same framework is used for deriving the topological asymptotic expansion of a tracking-type shape functional with respect to singular domain perturbations produced by the nucleation of small circular holes endowed with homogeneous Neumann boundary condition. The resulting asymptotic method allows for obtaining sharp a priori estimates for the remainders, so that it can be seen as a rigorous mathematical justification for the derivations presented in the former chapters.