An Asymptotically Compatible Unfitted Finite Element Method for Nonlocal Elliptic Interfaces: Local Limits and Sharp Error Estimates

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 \(L^2\) norms via the nonlocal maximum principle, flux consistency, and a newly derived nonlocal Poincaré inequality. Numerical experiments confirm the theoretical findings and demonstrate the robustness and efficiency of the proposed approach, thereby providing a theoretical and numerical foundation for extensions to higher-dimensional problems.