Communications in Mathematical Sciences

Volume 20 (2022)

Number 7

Second-order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems

Pages: 1815 – 1837



Hwi Lee (Department of Applied Physics and Applied Mathematics, Columbia University, New York, N.Y., U.S.A.; and School of Mathematics, Georgia Institute of Technology, Atlanta, Ga., U.S.A.)

Qiang Du (Department of Applied Physics and Applied Mathematics, and Data Science Institute, Columbia University, New York, N.Y., U.S.A.)


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 operator 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.


nonlocal models, nonlocal diffusion, peridynamics, smoothed particle hydrodynamics, nonlocal gradient, nonlocal boundary condition, extrapolation, nonlocal interaction kernels

2010 Mathematics Subject Classification

34B10, 35A01, 35B40, 45A05, 65N12

This research is supported in part by the NSF DMS-2012562, DMS-1937254 and ARO MURI Grant W911NF-15-1-0562.

Received 16 August 2021

Received revised 4 December 2021

Accepted 24 January 2022

Published 21 October 2022