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

Parks, Michael L.; Radu, Petronela

doi:10.1186/s13662-025-03923-x

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

Journal Article · Fri Mar 14 00:00:00 UTC 2025 · Advances in Continuous and Discrete Models (Online)

DOI:https://doi.org/10.1186/s13662-025-03923-x· OSTI ID:2530975

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Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Univ. of Nebraska, Lincoln, NE (United States)

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 $$\mathcal{O}$$(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 $$\mathcal{O}$$(δ²) rate of convergence of solutions, also exhibiting an $$\mathcal{O}$$(h²) convergence, where h is the mesh size.

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Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: National Science Foundation (NSF); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 2530975

Alternate ID(s):: OSTI ID: 2538313

Journal Information:: Advances in Continuous and Discrete Models (Online), Journal Name: Advances in Continuous and Discrete Models (Online) Journal Issue: 1 Vol. 2025; ISSN 2731-4235

Publisher:: Springer NatureCopyright Statement

Country of Publication:: United States

Language:: English

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97 MATHEMATICS AND COMPUTING
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On nonlocal problems with Neumann boundary conditions: scaling and convergence for nonlocal operators and solutions

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