A stable generalized finite element method (SGFEM) of degree two for interface problems

Abstract

Generalized or Extended Finite Element Methods (GFEM/XFEM) of degree 1 for interface problems have been reported in the literature; they (i) yield optimal order of convergence, i.e., O(h) (h is a discretization parameter), (ii) are stable in a sense that conditioning is not worse than that of the standard FEM, and (iii) are robust in that the conditioning does not deteriorate as interface curves are close to boundaries of underlying elements. However, higher order GFEM/XFEM with the properties (i)–(iii) have not been successfully addressed yet. Various enrichment schemes for GFEM/XFEM based on D or DPk (D is a distance function or the absolute value of level set function, and Pk is the polynomial basis of degree k) have been reported recently to obtain higher order convergence, but they are not stable or robust in general; in fact, they even may not yield the optimal orders of convergence. In this paper, we propose a stable GFEM/XFEM of degree 2 (SGFEM2) for the interface problems, where we used the enrichment scheme based on D{1,x,y}, instead of D or D{1,x,y,x2,xy,y2} in the literature. We prove that the SGFEM2 yields the optimal order of convergence, i.e., O(h2), for the interface problems with curved (smooth) interfaces. A local principal component analysis technique has been proposed, which ensures that the SGFEM2 is stable and robust. Numerical experiments for straight and curved interfaces have been presented to illuminate these properties.