Quasinonlocal coupling of nonlocal diffusions

TL;DR

This paper introduces a new coupling strategy for nonlocal diffusion models that ensures stability, consistency, and adherence to physical principles, demonstrated through finite difference discretization and numerical examples.

Contribution

A novel self-adjoint, stable coupling method for nonlocal diffusions inspired by atomistic-to-continuum approaches, ensuring coercivity and maximum principle compliance.

Findings

Coupling model is coercive in energy and L^2 norms.

Finite difference scheme inherits continuous properties.

Coupled diffusion matches fully nonlocal diffusion results.

Abstract

We developed a new self-adjoint, consistent, and stable coupling strategy for nonlocal diffusion models, inspired by the quasinonlocal atomistic-to-continuum method for crystalline solids. The proposed coupling model is coercive with respect to the energy norms induced by the nonlocal diffusion kernels as well as the $L^2$ norm, and it satisfies the maximum principle. A finite difference approximation is used to discretize the coupled system, which inherits the property from the continuous formulation. Furthermore, we design a numerical example which shows the discrepancy between the fully nonlocal and fully local diffusions, whereas the result of the coupled diffusion agrees with that of the fully nonlocal diffusion.