A seamless local-nonlocal coupling diffusion model with $H^1$ vanishing nonlocality convergence

TL;DR

This paper introduces a seamless coupling model combining local and nonlocal diffusion processes, ensuring stability and convergence to classical elliptic models with first-order accuracy in the H^1 norm.

Contribution

The paper develops a novel local-nonlocal coupling diffusion model with a new transmission condition and proves its well-posedness and convergence properties.

Findings

Model achieves stable coupling of local and nonlocal diffusion.

Convergence to classical elliptic models with first-order accuracy.

The approach ensures stability and well-posedness of the coupled system.

Abstract

Based on the development in dealing with nonlocal boundary conditions, we propose a seamless local-nonlocal coupling diffusion model in this paper. In our model, a finite constant interaction horizon is equipped in the nonlocal part and transmission conditions are imposed on a co-dimension one interface. To achieve a seamless coupling, we introduce an auxiliary function to merge the nonlocal model with the local part and design a proper coupling transmission condition to ensure the stability and convergence. In addition, by introducing bilinear form, well-posedness of the proposed model can be proved and convergence to a standard elliptic transmission model with first order in $H^1$ norm can be derived.