General enrichments of stable GFEM for interface problems: Theory and extreme learning machine construction

Generalized finite element methods (GFEMs), when applied to interface problems (IPs), need to be enriched with special functions to enhance approximation accuracy. These functions include distance functions, one-side distance functions, level set functions, and exponential forms of level set function. For the IP with geometrically complex interface curves, computation of the distance function or level set function could be challenging, and algorithms of computational geometry are usually involved. Moreover, theoretical analysis on optimal convergence of the GFEM enriched by these functions has not been fully investigated. In this study we propose a general enrichment scheme, based on which all the aforementioned enrichments can be viewed as special examples. We prove that a stable GFEM (SGFEM) with such a new enrichment scheme reaches the optimal convergence rate. Most importantly, the new scheme provides an instruction to construct machine learning (ML) based enrichments, which advances the ability of GFEM to handle geometrically complex interfaces. Two ML methods, deep neural network (DNN) and extreme learning machine (ELM), are studied. Among them, the ELM is highly suggested because it exhibits high accuracy for the interface curve with complex geometries. The learning dimension for the ML is one dimension less than that of the domain so that the proposed ML algorithm can be implemented efficiently. The numerical experiments demonstrate that the SGFEM with the ELM enrichment achieves the optimal convergence rates for the IP, as predicted theoretically.