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arXiv · 2608.16079

Finite Element Approximation of Nonlocal Problems with Heterogeneous Localization and Local Boundary Conditions

Yuyan Chang·Hui Liang·Zhonghua Qiao
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Abstract

This paper studies the finite element approximation of a one-dimensional nonlocal Poisson problem with heterogeneous localization and homogeneous local Dirichlet boundary conditions. These local boundary conditions induce localization kernels with spatially varying interaction neighborhoods, which lead to substantial numerical challenges for the assembly of the singular nonlocal stiffness matrix. An asymptotically compatible conforming finite element method is developed for the variational formulation, together with an exact geometric decomposition for the singular stiffness matrix assembly. Under additional smoothness assumptions on the localization profile, second-order operator consistency is established and error estimates are derived with the corresponding convergence orders. Numerical experiments confirm the theoretical convergence behavior and demonstrate the improved boundary behavior of the heterogeneous localization model.

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Yuyan Chang, Hui Liang, Zhonghua Qiao. 2026-08-17. Finite Element Approximation of Nonlocal Problems with Heterogeneous Localization and Local Boundary Conditions. https://arxiv.org/abs/2608.16079

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