A Posteriori Error Estimation and Adaptivity for Second-Order Optimally Convergent G/XFEM and FEM

The Generalized/eXtended Finite Element Method (G/XFEM) is known to efficiently and accurately solve problems that are challenging for standard methodologies. The method can deliver optimal convergence rates in the energy norm and global matrices with a scaled condition number that has the same order as in the Finite Element Method (FEM). This is achieved even for problems of Linear Elastic Fracture Mechanics (LEFM), which have solutions containing singularities and discontinuities. Despite delivering optimal convergence rates, it has been shown [1], however, that first-order G/XFEM are not competitive with second-order FEM that uses quarter-point elements, especially for three-dimensional (3-D) problems. Because of this, optimally convergent second-order G/XFEM, customized to solve LEFM problems, have been recently proposed [1, 2, 3]. The formulations presented in these works augment both standard lagrangian FEM approximation spaces [3] and p FEM approximation spaces [1, 2] in order to insert into the G/XFEM numerical approximation the discontinuous and singular behaviors of fractures. It is important to note that, in addition to using enrichment functions, G/XFEM still needs local mesh refinement around crack fronts in order to achieve optimal convergence. This must be considered especially for 3-D problems that violate the assumptions of the adopted singular enrichments. While this local mesh refinement can be easily performed for simple cases, the level of refinement