The Generalized Finite Element Method (GFEM) is developed from the Partition of the Unity Method (PUM), which expands the standard finite element space by using non-polynomial function spaces called the enrichment spaces. GFEM has been successfully applied to various problems, but it still has some drawbacks. It lacks robustness in adjusting meshes when solving interface problems, and the condition number of the stiffness matrix will increase dramatically when the interface is close to the mesh boundary. This phenomenon can lead to ill-conditioned linear equations. A stable GFEM called SGFEM is proposed for the Stokes interface problem in this paper, which modifies the enrichment space. The SGFEM space of the velocity is divided into a basic part and an enrichment part . The discretization of space uses element or the Taylor-Hood element for the study. uses different interpolation functions. Numerical studies show that SGFEM has the optimal convergence order of the error and robustness. The growth rate of the scaled condition number of the stiffness matrix is the same as that of a standard FEM.

中文翻译：

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Stokes接口问题的稳定广义有限元法

广义有限元法 (GFEM) 是从统一划分法 (PUM) 发展而来的，它通过使用称为富集空间的非多项式函数空间来扩展标准有限元空间。 GFEM 已成功应用于各种问题，但它仍然存在一些缺点。解决界面问题时调整网格缺乏鲁棒性，当界面接近网格边界时，刚度矩阵的条件数会急剧增加。这种现象可能导致病态线性方程。本文针对 Stokes 界面问题提出了一种称为 SGFEM 的稳定 GFEM，它修改了富集空间。速度的SGFEM空间分为基本部分和富集部分。空间的离散化采用元或Taylor-Hood元进行研究。使用不同的插值函数。数值研究表明SGFEM具有最优的误差收敛阶数和鲁棒性。刚度矩阵的缩放条件数的增长率与标准 FEM 的增长率相同。