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

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.