Analysis and improvements of global-local enrichments for the Generalized Finite Element Method

Partition of unity methods like the GFEM, rely on closed-form analytical enrichment functions. However, only limited a priori knowledge about the solution is available for many problems of engineering relevance. Control of discretization errors in this class of problems requires mesh refinement which offsets some of the advantages associated with these methods. This has led to the development of the GFEM with global-local enrichments (GFEMgl), which uses ideas of the classical global-local finite element method to numerically build enrichment functions for the GFEM. It involves the solution of local boundary value problems using boundary conditions from the solution of the global problem discretized with a coarse mesh. These local solutions are in turn used to enrich the solution space of the global problem with the help of the partition of unity framework.This paper presents an a priori error estimate accounting for the effect of inexact boundary conditions applied to local problems. Two strategies to control the error of the GFEMgl solution due to these boundary conditions are also presented. The first one is based on the use of a buffer or over-sampling zone in the local problems. The second strategy investigated is based on multiple global-local iterations. Several numerical experiments performed on three-dimensional fracture mechanics problems illustrate the effectiveness of these strategies in controlling the error of the GFEMgl solution. The problems are selected such that the boundary conditions on the local problems range from smooth to singular functions.