High Order Topological Asymptotics: Reconciling Layer Potentials and Compound Asymptotic Expansions

A systematic two-step procedure is proposed for the derivation of full asymptotic expansions of the solution of elliptic partial differential equations set on a domain perforated with a small hole on which a Dirichlet boundary condition is applied. First, an integral representation of the solution is sought, which enables us to exploit the explicit dependence with respect to the small parameter to predict the correct form of a two-scale ansatz. Second, the terms of the ansatz are characterized by the method of compound asymptotic expansions as the solutions of a cascade of successive interior and exterior domains. This allows us to interpret them as high-order correctors, for which error bounds can be proved using variational estimates. The methodology is illustrated on two different problems: we start by revisiting the perforated Poisson problem with Dirichlet boundary conditions on both the hole and the outer boundary, where we highlight how the method enables us to retrieve very naturally the correct form of the ansatz in the most delicate two-dimensional setting. Then, we provide original and complete asymptotic expansions for a perforated cell problem featuring periodic conditions.