AI 中文总结
本文针对带异质局域化和局部狄利克雷边界条件的一维非局部泊松问题，提出渐近相容协调有限元方法，推导误差估计并通过数值实验验证其收敛性与边界性能优势。
AI 中文摘要
本文研究一类一维非局部泊松问题的有限元近似，该问题带有异质局域化和齐次局部狄利克雷边界条件。这些局部边界条件会诱导出具有空间变化相互作用邻域的局域化核，这对奇异非局部刚度矩阵的组装带来了重大数值挑战。针对该变分形式，本文提出了一种渐近相容的协调有限元方法，同时给出了奇异刚度矩阵组装的精确几何分解。在局域化剖面的额外光滑性假设下，本文建立了二阶算子一致性，并推导了对应收敛阶的误差估计。数值实验验证了理论收敛行为，且证明了异质局域化模型的边界行为有所改善。
英文摘要
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.
