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

TL;DR

This paper introduces an asymptotically compatible unfitted finite element method for 1D nonlocal elliptic interface problems, achieving optimal error estimates and second-order convergence without body-fitted meshes.

Contribution

The paper develops a novel AC unfitted finite element method with a Nitsche-type formulation and rigorous convergence analysis for nonlocal elliptic interface problems.

Findings

Achieves second-order convergence in maximum norm.

Confirms optimal convergence rates in energy and L2 norms.

Demonstrates robustness and efficiency through numerical experiments.

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.