Numerical experiments with the Generalized Finite Element Method based on a flat-top Partition of Unity

Abstract The Stable Generalized Finite Element Method (SGFEM) is essentially an improved version of the Generalized Finite Element Method (GFEM). Besides of retaining the good flexibility for constructing local enriched approximations, the SGFEM has the advantage of presenting much better conditioning than that of the conventional GFEM. Actually, bad conditioning is well known as one of the main drawbacks of the GFEM, while affecting severely the precision of the numerical scheme used for solving the linear system associated to the problem. Despite of its consistent mathematical basis, the numerical experiments so far conducted on using SGFEM are not yet clearly conclusive, especially regarding the robustness of the method. Therefore, the main purpose of the present paper is to give a contribution in this direction, through further investigating the SGFEM accuracy and stability. In particular, the so called Flat-Top SGFEM is a recent proposed version of the method hereby considered. As a flat-top Partition of Unit (PoU) is used for constructing the augmented approximation space with polynomial enrichments this version of the method is called SGFEM with flat-top PoU, or simply FT-SGFEM. Some computational aspects are briefly addressed, as the ones related to the implementation and integration of the flat-top for 2-D analysis. The numerical simulations consist essentially of linear analysis of panels presenting edge cracks and reentrant corners on its boundaries. Our findings from the numerical tests done are highly relevant regarding accuracy of the SGFEM versions, which present order of convergence similar to the conventional GFEM. Moreover, the measure of stability given by the scaled condition number presented in particular by the FT-SGFEM is comparable to the conventional FEM order.