Asymptotic analysis for elliptic equations with small perturbations on domains in high-contrast medium

Abstract

We provide a comprehensive study on the asymptotic solutions of an interface problem corresponding to an elliptic partial differential equation with Dirichlet boundary condition and transmission condition, subject to the small geometric perturbation and/or the high contrast ratio of the conductivity. All asymptotic terms can be solved in the unperturbed reference domains, which significantly reduces computations in practice, especially for random perturbations. Our setting is quite general and allows two types of elliptic problems: the perturbation of the domain boundary where the Dirchlet condition is imposed and the perturbation of the interface where the transmission condition is imposed. As the perturbation size and the ratio of the conductivities tends to zero, the two-parameter asymptotic expansions on the reference domain are derived to any order after the single parameter expansions are solved beforehand. The results suggest the emergence of the Neumann or Robin boundary condition, depending on the relation of the two asymptotic parameters. Our method is the classic asymptotic analysis techniques but in a new unified approach to both problems.