Communications in Mathematical Sciences
Volume 17 (2019)
On uniform second order nonlocal approximations to linear two-point boundary value problems
Pages: 1737 – 1755
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