The Generalized Finite Element Method (GFEM) is an effective unfitted numerical method for handling interface problems. By augmenting the standard FEM space with an appropriate enrichment space, GFEM can accurately capture C^0 solutions across the interfaces. While numerous GFEMs for interface problems have been studied, establishing a stable high-order GFEM with optimal convergence rates and robust system conditioning remains a challenge. The highest known order of two was established by Zhang and Babu\v{s}ka (SGFEM2, Comput. Methods Appl. Mech. Engrg. 363 (2020), 112889). In this paper, we propose a unified enrichment space construction and establish arbitrary high-order stable GFEMs (HoSGFEM) for elliptic interface problems. The main idea distinguishes itself from Zhang and Babu\v{s}ka's SGFEM2 substantially and it is twofold: a) we construct dimensionality-reduced auxiliary locally supported piecewise polynomials that satisfy the partition of unity property for elements containing interfaces; b) we construct the enrichment scheme based on d{1,(x-x_c^e),...,(y-y_c^e)^{p-1}} (d is the distance function; (x_c^e, y_c^e) is the center of the element containing interface, thus element-based) for arbitrary p-th order elements instead of d, d{1,x,y} or d{1,x,y,x^2,xy,y^2} (global functions) for p=1,2 in the literature. This idea results in an enrichment space that has a large angle with the standard FEM space, leading to the stability of the method with system condition number growing in order O(h^{-2}). We establish optimal convergence rates for HoSGFEM solutions under the proposed construction. Various numerical experiments with both straight and curved interfaces demonstrate the optimal convergence, FEM-comparable system condition number with O(h^{-2}) growth, and robustness as element boundaries approach interfaces.

翻译：广义有限元法（GFEM）是处理界面问题的有效非拟合数值方法。通过在标准有限元空间中增加适当的增强空间，GFEM能够精确捕捉跨越界面的C^0解。尽管已有多种针对界面问题的GFEM被研究，但建立具有最优收敛速率和鲁棒系统条件数的高阶稳定GFEM仍具挑战性。目前已知的最高阶数为二阶，由Zhang和Babuška提出（SGFEM2，Comput. Methods Appl. Mech. Engrg. 363 (2020)，112889）。本文提出统一的增强空间构造方法，并为椭圆界面问题建立了任意高阶的稳定GFEM（HoSGFEM）。其核心思想与Zhang和Babuška的SGFEM2有本质区别，主要体现在两个方面：a）我们构造了满足单位分解性质的降维辅助局部支撑分段多项式，用于处理包含界面的单元；b）对于任意p阶单元，我们基于d{1,(x-x_c^e),...,(y-y_c^e)^{p-1}}（其中d为距离函数；(x_c^e, y_c^e)是包含界面的单元中心，因此具有单元局部性）构建增强方案，而非文献中针对p=1,2阶单元采用的d、d{1,x,y}或d{1,x,y,x^2,xy,y^2}等全局函数。该思想使得增强空间与标准有限元空间形成大夹角，从而保证方法稳定性，其系统条件数增长阶数为O(h^{-2})。我们在所提构造框架下证明了HoSGFEM解的最优收敛速率。通过包含直线与曲线界面的多种数值实验，验证了该方法具有最优收敛性、与有限元法相当的O(h^{-2})阶系统条件数增长特性，以及当单元边界逼近界面时的鲁棒性。