椭圆界面问题数值方法研究综述
A Review on Numerical Methods for Elliptic Interface Problems
摘要: 椭圆界面问题广泛存在于复合材料力学、多相流、生物流体、多孔介质传输等科学与工程领域，其核心特征是求解区域被界面分割为多个子区域，方程系数、解或其法向通量在界面处存在跳跃，导致解全局正则性较低。本文系统梳理了椭圆界面问题的主流数值求解方法，将其划分为经典数值方法与无网格机器学习方法两大类。其中，经典数值方法又可进一步分为适配网格方法与非适配网格方法。针对不同类别的方法，本文详细阐述了其核心思想、技术特点及代表性研究进展。最后，对当前领域的研究现状进行总结，并展望未来的发展趋势。
Abstract: Elliptic interface problems widely exist in scientific and engineering fields such as composite material mechanics, multiphase flow, biological fluid dynamics and porous medium transport. The essential characteristic is that the computational domain is divided into multiple subdomains by interfaces, where jumps occur in equation coefficients, solutions or their normal fluxes, resulting in low global regularity of solutions. We systematically review the mainstream numerical methods for solving elliptic interface problems, which are classified into two categories: traditional numerical methods and meshless machine learning methods. Among them, traditional numerical methods are further divided into fitted mesh methods and unfitted mesh methods. For each type of method, we elaborate on their core principles, technical characteristics and representative research progress. Finally, the current research status in this field is summarized, and future development trends are prospected.
文章引用：邓唐川. 椭圆界面问题数值方法研究综述[J]. 应用数学进展, 2026, 15(6): 196-203. https://doi.org/10.12677/aam.2026.156277

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