The Finite Element Method (FEM) has consolidated itself in recent decades as one of the most widely used numerical techniques to solve a variety of problems in different areas of engineering such as structural analysis, thermal analysis, fluids analysis, manufacturing processes, etc. One application where the method is most interesting is the analysis of problems within the area of Fracture Mechanics, facilitating the study and evaluation of the structural integrity of mechanical components, reliability, and detection and control of cracks.

Recently, the development of new techniques as the Extended Finite Element Method (XFEM) has increased even more the potential of FEM. These techniques improve the description
of features like singularities, discontinuities, etc.,  by adding special functions that enrich the space of the standard finite element approximation. 
 
However, whenever one approximates a problem using numerical techniques, the solution presents differences with respect to the system it represents. In techniques based on a discrete representation of the domain through finite elements (FEM, XFEM, ...) it is a matter of utmost concern the control of the discretization error. In the literature, many references can be found to techniques that allow quantification of the error in standard finite element formulations. Nevertheless, being the XFEM a relatively recent method, there are not yet sufficiently developed techniques for estimating the error for enriched finite element approximations.

The objective of this Thesis is to quantify the discretization error when XFEM approximations are used to represent problems in the field of  Linear Elastic Fracture Mechanics (LEFM), as is the case of modelling a crack. In this sense, it is proposed to develop an a posteriori  error estimator based on the recovery of the finite element solution, which has been specially adapted to XFEM approximations. A recovery technique which can be considered as an extension of the Superconvergent Patch Recovery (SPR) technique to enriched approximations has been used for the evaluation of the recovered field.

Moreover, the error estimators can underestimate or overestimate the exact error in energy norm. In practice, it is of greater interest to ensure that the solution has reached a certain level of accuracy, bounding the error so that a safety criterion to accept the FE solution can be defined. For this reason, this Thesis proposes a technique for evaluating upper bounds of the error well suited to XFEM approximations, which is based on the recovery of the solution, and the evaluation of the defects introduced into the equilibrium by forcing the continuity of the recovered field.

The error estimation and error bounding techniques have been verified through numerical examples with known reference solution. The results indicate a high accuracy of the error estimate and the upper bound of the error.