An Asymptotically Compatible Approach For Neumann-Type Boundary Condition On Nonlocal Problems

Understand

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter $\delta$ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models.

• While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as $\delta\rightarrow 0$, we present a new generalization of classical local Neumann conditions that recovers the local case as $O(\delta^2)$ in the $L^{\infty}(\Omega)$ norm.
• This convergence rate is optimal considering the $O(\delta^2)$ convergence of the nonlocal equation to its local limit away from the boundary.
• We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced.