Analysis of transmission problems on Lipschitz boundaries in stronger norms

Abstract

We concentrate on a model diffusion equation on a Lipschitz simply connected bounded domain with a small diffusion coefficient in a Lipschitz simply connected subdomain located strictly inside of the original domain. We study asymptotic properties of the solution with respect to the small diffusion coefficient vanishing. It is known that the solution asymptotically turns into a solution of a corresponding diffusion equation with Neumann boundary conditions on a part of the boundary. One typical proof technique of this fact utilizes a reduction of the problem to the interface of the subdomain, using a transmission condition. An analogous approach appears in studying domain decomposition methods without overlap, reducing the investigation to the surface that separates the subdomains and in theoretical foundation of a fictitious domain, also called embedding, method, e.g., to prove a classical estimate that guaranties convergence of the solution of the fictitious domain problem to the solution of the original Neumann boundary value problem. On a continuous level, this analysis is usually performed in an H 1/2 norm for second order elliptic equations. This norm appears naturally for Poincaré-Steklov operators, which are convenient to employ to formulate the transmission condition. Using recent advances in regularity theory of Poincaré-Steklov operators for Lipschitz domains, we provide, in the present paper, a similar analysis in an H 1/2+α norm with α > 0, for a simple model problem. This result leads to a convergence theory of the fictitious domain method for a second order elliptic PDE in an H 1+α norm, while the classical result is in an H 1 norm. Here, α < 1/2 for the case of Lipschitz domains we consider.