Unfitted mesh formulations for interface problems generally adopt two distinct methodologies: (i) penalty-based approaches and (ii) explicit enrichment space techniques. While Stable Generalized Finite Element Method (SGFEM) has been rigorously established for one-dimensional and linear-element cases, the construction of optimal enrichment spaces preserving approximation-theoretic properties within isogeometric analysis (IGA) frameworks remains an open challenge. In this paper, we introduce a stable quadratic generalized isogeometric analysis (SGIGA2) for two-dimensional elliptic interface problems. The method is achieved through two key ideas: a new quasi-interpolation for the function with C0 continuous along interface and a new enrichment space with controlled condition number for the stiffness matrix. We mathematically prove that the present method has optimal convergence rates for elliptic interface problems and demonstrate its stability and robustness through numerical verification.

翻译：界面问题的非拟合网格方法通常采用两种不同的策略：(i) 基于罚函数的方法和(ii)显式增强空间技术。虽然稳定广义有限元法(SGFEM)在一维和线性单元情形下已得到严格建立，但在等几何分析(IGA)框架内构建保持近似理论性质的最优增强空间仍然是一个开放挑战。本文针对二维椭圆界面问题，提出了一种稳定的二次广义等几何分析方法(SGIGA2)。该方法通过两个关键思想实现：为沿界面C0连续的函数构造新的拟插值算子，以及建立具有受控条件数的刚度矩阵新增强空间。我们从数学上证明了本方法对椭圆界面问题具有最优收敛阶，并通过数值验证展示了其稳定性和鲁棒性。