Notes on the nonlinear dependence of a multiscale coupled system with respect to the interface

This work studies the dependence of the solution with respect to interface geometric perturbations in a multiscaled coupled Darcy flow system in direct variational formulation. A set of admissible perturbation functions and a sense of convergence are presented, as well as sufficient conditions on the forcing terms, in order to conclude strong convergence statements. For the rate of convergence of the solutions we start solving completely the one dimensional case using orthogonal decompositions on appropriate subspaces. Finally, the rate of convergence question is analyzed in a simple multiple dimensional setting, studying the nonlinear operators introduced by the geometric perturbations.