Communications in Mathematical Sciences

Volume 17 (2019)

Number 6

On uniform second order nonlocal approximations to linear two-point boundary value problems

Pages: 1737 – 1755

(Fast Communication)



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

Jiwei Zhang (School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science,Wuhan University, Wuhan, China)

Chunxiong Zheng (Department of Mathematical Sciences, Tsinghua University, Beijing, China; and College of Mathematics and Systems Science, Xinjiang University, Urumqi, China)


In this paper, nonlocal approximations are considered for linear two-point boundary value problems (BVPs) with Dirichlet and mixed boundary conditions, respectively. These nonlocal formulations are constructed from nonlocal variational problems that are analogous to local problems. The well-posedness and regularity of the resulting nonlocal problems are established, along with the convergence to local problem as the nonlocal horizon parameter $\delta$ tends to $0$. Uniform second order accuracy with respect to $\delta$ of the nonlocal approximation to the local solution, spatially in the pointwise sense, can be achieved under suitable conditions. Numerical simulations are carried out to examine the order of convergence rate, which also motivate further refined asymptotic estimates.


nonlocal two-point boundary value problems, nonlocal operator and maximum principle, nonlocal Dirichlet and Neumann-type problems with volume-constraints, local limit, the weak regularity of nonlocal solutions

2010 Mathematics Subject Classification

45A05, 46N20, 65M60, 65R20, 82C21

Qiang Du is supported in part by the U.S. NSF grants DMS-1719699, AFOSR MURI Center for material failure prediction through peridynamics, and the ARO MURI W911NF-15-1-0562 on Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications. Jiwei Zhang is partially supported by NSFC under Nos. 11771035, and the Natural Science Foundation of Hubei Province No. 2019CFA007, and Xiangtan University 2018ICIP01. Chunxiong Zheng is supported by Natural Science Foundation of Xinjiang Autonomous Region under No. 2019D01C026, and National Natural Science Foundation under No. 11771248.

Received 10 December 2018

Accepted 15 September 2019

Published 26 December 2019