An a Posteriori Error Estimator for the Generalized Finite Element Method With Global-Local Enrichments in Fracture Mechanics

Abstract

This work presents, for the first time, an a posteriori error estimator specifically developed for the Generalized Finite Element Method with Global-Local Enrichments (${\text{GFEM}}^{gl}$). The proposed estimator is built upon a recovery procedure originally formulated for the Generalized/eXtended Finite Element Method (G/XFEM), where a recovered stress field is computed using a block-diagonal system of equations. Designed for 2D Linear Elastic Fracture Mechanics problems, the estimator effectively evaluates errors in the energy norm at both the local and global scales of ${\text{GFEM}}^{gl}$, accounting for the particularities of its enrichment strategy. Additionally, this study incorporates the stable version of ${\text{GFEM}}^{gl}$ (${\text{SGFEM}}^{gl}$) to improve convergence and accuracy in the approximate solution. Numerical results, considering variations in the local domain size within a mixed-mode problem, validate the efficiency and reliability of the proposed error estimator. Effectivity, indices close to unity and precise error distributions confirm the robustness of the approach, even in scenarios with elevated error levels.