Second order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems

TL;DR

This paper introduces a second order accurate method for implementing Dirichlet boundary conditions in nonlocal diffusion models, using nonlocal gradient operators to improve upon traditional finite difference approaches.

Contribution

It proposes a novel boundary condition handling technique that incorporates nonlocal gradients, enhancing accuracy and generality for nonlocal diffusion problems.

Findings

Proves well-posedness of the new nonlocal boundary value problems.

Shows asymptotic convergence to local diffusion models as nonlocality diminishes.

Establishes second order localization rate as optimal in boundary approximation.

Abstract

We present an approach to handle Dirichlet type nonlocal boundary conditions for nonlocal diffusion models with a finite range of nonlocal interactions. Our approach utilizes a linear extrapolation of prescribed boundary data. A novelty is, instead of using local gradients of the boundary data that are not available a priori, we incorporate nonlocal gradient operators into the formulation to generalize the finite differences-based methods which are pervasive in literature; our particular choice of the nonlocal gradient operators is based on the interplay between a constant kernel function and the geometry of nonlocal interaction neighborhoods. Such an approach can be potentially useful to address similar issues in peridynamics, smoothed particle hydrodynamics and other nonlocal models. We first show the well-posedness of the newly formulated nonlocal problems and then analyze their asymptotic convergence to the local limit as the nonlocality parameter shrinks to zero. We justify the second order localization rate, which is the optimal order attainable in the absence of physical boundaries.