A shape calculus based method for a transmission problem with random interface

Abstract

The present work is devoted to approximation of the statistical moments of the unknown solution of a class of elliptic transmission problems in $\mathbb R^3$ with randomly perturbed interfaces. Within this model, the diffusion coefficient has a jump discontinuity across the random transmission interface which models linear diffusion in two different media separated by an uncertain surface. We apply the shape calculus approach to approximate solution's perturbation by the so-called shape derivative, correspondingly statistical moments of the solution's perturbation are approximated by the moments of the shape derivative. We characterize the shape derivative as a solution of a related homogeneous transmission problem with nonzero jump conditions which can be solved with the aid of boundary integral equations. We develop a rigorous theoretical framework for this method, particularly i) extending the method to the case of unbounded domains and ii) closing the gaps and clarifying and adapting results in the existing literature. The theoretical findings are supported by and illustrated in two particular examples.