On nonlocal problems with Neumann boundary conditions: scaling and convergence for nonlocal operators and solutions

Michael L. Parks; Petronela Radu

doi:10.1186/s13662-025-03923-x

On nonlocal problems with Neumann boundary conditions: scaling and convergence for nonlocal operators and solutions

Michael L. Parks, Petronela Radu

Computer Science and Mathematics Div

Research output: Contribution to journal › Article › peer-review

Abstract

Formulations of Neumann-type boundary conditions for boundary value problems in the nonlocal framework are beset with difficulties, some related to the choice of a proper scaling. Here we identify a space-dependent scaling for a nonlocal Neumann operator, for which we prove linear in δ (δ being the radius for the support for the kernel) convergence of the Neumann operator and O(δ2) convergence of solutions to their classical counterparts. The pointwise-like convergence of the nonlocal normal operator is cast as a new type of two-scale operator-point convergence, which we call condensated convergence. The results hold for general integrable kernels, a setting which is favored in numerical simulations. We support this analysis with numerical convergence studies using a piecewise linear discontinuous Galerkin discretization and show an O(δ2) rate of convergence of solutions, also exhibiting an O(h2) convergence, where h is the mesh size.

Original language	English
Article number	66
Journal	Advances in Continuous and Discrete Models
Volume	2025
Issue number	1
DOIs	https://doi.org/10.1186/s13662-025-03923-x
State	Published - Dec 2025

Funding

This manuscript has been authored by UT-Battelle LLC under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the paper for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan ( http://energy.gov/downloads/doe-public-access-plan ). MLP\u2019s work was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics Program under the Physics-Informed Learning Machines for Multiscale and Multiphysics Problems (PhILMs) project. PR\u2019s work was supported by the awards NSF \u2013 DMS 1716790, NSF \u2013 DMS 2109149, and the Leland J. and Dorothy H. Olson Professorship at University of Nebraska-Lincoln. The authors acknowledge helpful conversations with Robert Lipton, Qiang Du, and Xiaochuan Tian.

Keywords

Condensated convergence
Neumann-type boundary condition
Nonlocal boundary
Nonlocal flux
Nonlocal model
Peridynamics
Space-dependent scaling

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10.1186/s13662-025-03923-x

Cite this

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abstract = "Formulations of Neumann-type boundary conditions for boundary value problems in the nonlocal framework are beset with difficulties, some related to the choice of a proper scaling. Here we identify a space-dependent scaling for a nonlocal Neumann operator, for which we prove linear in δ (δ being the radius for the support for the kernel) convergence of the Neumann operator and O(δ2) convergence of solutions to their classical counterparts. The pointwise-like convergence of the nonlocal normal operator is cast as a new type of two-scale operator-point convergence, which we call condensated convergence. The results hold for general integrable kernels, a setting which is favored in numerical simulations. We support this analysis with numerical convergence studies using a piecewise linear discontinuous Galerkin discretization and show an O(δ2) rate of convergence of solutions, also exhibiting an O(h2) convergence, where h is the mesh size.",

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On nonlocal problems with Neumann boundary conditions: scaling and convergence for nonlocal operators and solutions

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