On some coupled local and nonlocal diffusion models.

Abstract

We study problems in which a local model is coupled with a nonlocal one. We propose two energies: both of them are based on the same classical weighted H1-semi norm to model the local part, while two dierent weighted Hs-semi norms, with s ∈ (0; 1), are used to model the nonlocal part. The corresponding strong formulations are derived. In doing so, one needs to develop some technical tools, such as suitable integration by parts formulas for operators with variable diusivity, and one also needs to study the mapping properties of the Neumann operators that arise. In contrast to problems coupling purely local models, in which one requires transmission conditions on the interface between the subdomains, the presence of a nonlocal operator may give rise to nonlocal uxes. These nonlocal uxes may enter the problem as a source term, thereby changing its structure. Finally, we focus on a specic problem, that we consider most relevant, and study regularity of solutions and nite element discretizations. We provide numerical experiments to illustrate the most salient features of the models.