Stable generalized finite element for two-dimensional and three-dimensional non-homogeneous interface problems

In this paper, we propose a Stable Generalized Finite Element Method (SGFEM) to address non-homogeneous elliptic interface problems with discontinuous coefficients. Our approach utilizes the homogenization method to transform non-homogeneous interface conditions into homogeneous ones, thereby facilitating the application of the SGFEM. Specifically, we construct functions that satisfy the jump conditions, streamlining the problem-solving process. In the SGFEM, enrichment functions are used to efficiently construct these specialized functions, reducing the computational demands of the homogenization method. Notably, this method involves only calculations of additional right-hand terms near the interface, which require significantly less computation compared to the calculation of the stiffness matrix, thus avoiding any alterations to the stiffness matrix and preserving the stability and robustness of the SGFEM. We apply our method in both two-dimensional and three-dimensional scenarios, employing distinct strategies for each. In the 2D examples, the displacement homogenization method simplifies the implementation by addressing only the displacement jumps, thereby reducing the computational load. In the 3D examples, the synchronous homogenization method further simplifies numerical implementation by eliminating surface integrals on the interface. Through error analysis and numerical experiments, we demonstrate the efficiency and optimal convergence of our proposed method.