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arXiv · 2512.18939

An asymptotically compatible unfitted finite element methods for nonlocal elliptic Interfaces: local limits and sharp error estimates

Haixia Dong·Ziqing Xie·Jiwei Zhang
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Abstract

This paper presents the development and analysis of an asymptotically compatible (AC) unfitted finite element method for one-dimensional nonlocal elliptic interface problems. The proposed method achieves optimal error estimates through three principal contributions: (i) an extended maximum principle, coupled with an asymptotic consistency analysis of the flux operator, which establishes second-order convergence of nonlocal solutions to their local counterparts in the maximum norm; (ii) a Nitsche-type formulation that directly incorporates nonlocal jump conditions into the weak form, enabling high accuracy without body-fitted meshes; and (iii) a rigorous proof of optimal convergence rates in both the energy and L2 norms via the nonlocal maximum principle, flux consistency, and a newly derived nonlocal Poincare inequality. Numerical experiments confirm the theoretical findings and demonstrate the robustness and efficiency of the proposed approach, thereby providing a foundation for extensions to higher dimensions.

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Haixia Dong, Ziqing Xie, Jiwei Zhang. 2025-12-22. An asymptotically compatible unfitted finite element methods for nonlocal elliptic Interfaces: local limits and sharp error estimates. https://arxiv.org/abs/2512.18939

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