Robustness in stable generalized finite element methods (SGFEM) applied to Poisson problems with crack singularities

In this paper, we study the performance of the Generalized Finite Element Method (GFEM) applied to the Poisson problem with crack singularities. Recently, a GFEM with modified Heaviside enrichments was proposed in Gupta et al. (2013) to approximate the solution of a 2D elasticity problem with a crack. It was shown that the GFEM is indeed a Stable GFEM (SGFEM), i.e., it yields the optimal order of convergence and its conditioning is not worse than that of the standard finite element method (FEM). However, the robustness of the conditioning of the GFEM with respect to the position of the mesh relative to the crack was not addressed in Gupta et al. (2013). In this paper, we observed that the conditioning of the GFEM with the enrichments used in Gupta et al. (2013) is not robust when applied to approximate the solution of the Poisson problem. Moreover, the order of convergence may not be optimal when the position of the mesh is changed with respect to the crack interfaces. We proposed using additional singular enrichments at the nodes close to the crack near the crack-tip. We proved that the GFEM, with enrichments proposed in this paper, yields optimal order of convergence irrespective of the position of the mesh. Moreover, with a local orthogonalization procedure, we have shown through numerical experiments that the conditioning of this GFEM is not worse than that of the standard FEM and the conditioning is robust with respect to the position of the mesh. Thus the GFEM, with the enrichments suggested in this paper, is indeed an SGFEM when applied to a Poisson problem with the crack singularities.