Strongly Stable Generalized Finite Element Method (SSGFEM) for a non-smooth interface problem II: A simplified algorithm

Abstract

The solution of interface problems with non-smooth interfaces, having corners connected by straight lines, exhibit weak singularity along the interface and may have singularities at the corners. In an earlier paper Zhang et al. (2019), a Strongly Stable Generalized Finite Element Method (SSGFEM) was proposed to approximate solution of such problems. The method required two separate enrichment functions at O(h−1) number of nodes (h is the mesh parameter) and a local orthogonalization technique, requiring additional operations, was used to ensure its stability and robustness. In this paper, we present a simplified SSGFEM that does not require two enrichment functions at any of the nodes. As a result, we obtain an optimally converging GFEM that is stable, i.e., its scaled condition number is of the same order that of a standard FEM, and robust, i.e., the conditioning is independent of the position of meshes relative to the interface; these properties are illuminated by well-designed experiments. The simplified SSGFEM does not require LOT and requires less degrees of freedom, and is easier to implement, in contrast to the SSGFEM proposed in (Zhang et al., 2019). We also proved the optimal convergence for the simplified SSGFEM in this paper that required a more involved analysis based on a smart observation on approximation of weak singularities near the corner.