On some coupled local and nonlocal diffusion models

TL;DR

This paper investigates coupled local and nonlocal diffusion models, developing new energy formulations, deriving strong forms, and analyzing regularity and discretization, with numerical experiments demonstrating key features.

Contribution

It introduces two coupled energy models using weighted norms, derives their strong formulations, and studies regularity and finite element discretizations for the combined local-nonlocal diffusion problem.

Findings

Development of coupled energy formulations for local and nonlocal diffusion.

Derivation of strong formulations and analysis of nonlocal fluxes.

Numerical experiments illustrating model features and discretization performance.

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 $H^1$-semi norm to model the local part, while two different weighted $H^s$-semi norms, with $s \in (0,1)$, 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 diffusivity, 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 fluxes. These nonlocal fluxes may enter the problem as a source term, thereby changing its structure. Finally, we focus on a specific problem, that we consider most relevant, and study regularity of solutions and finite element discretizations. We provide numerical experiments to illustrate the most salient features of the models.