Derivative recovery and a posteriori error estimate for extended finite elements

This paper is the first attempt at error estimation for extended finite elements. The goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method (XFEM). In each element, the local estimator is the L₂ norm of the difference between the raw XFEM strain field and an enhanced strain field computed by extended moving least squares (XMLS) derivative recovery obtained from the raw nodal XFEM displacements. The XMLS construction is tailored to the nature of the solution. The technique is applied to linear elastic fracture mechanics, in which near-tip asymptotic functions are added to the MLS basis. The XMLS shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity. The result is a very smooth enhanced strain solution including the singularity at the crack tip. Results are shown for two- and three-dimensional linear elastic fracture mechanics problems in mode I and mixed mode. The effectivity index of the estimator is close to 1 and improves upon mesh refinement for the studied near-tip problem. It is also shown that for the linear elastic fracture mechanics problems treated, the proposed estimator outperforms one of the superconvergent patch recovery technique of Zienkiewicz and Zhu, which is only C₀. Parametric studies of the general performance of the estimator are also carried out.