The Method of Difference Potentials (DPM) has proven an efficient tool for the solution of boundary value problems (BVPs) in various fields of research including acoustics and fluid mechanics. The method converts the solution of problems of complex geometry to the multiple solutions of a simple, well defined auxiliary problem, on which efficient solvers can be used, and which also avoids the numerical computation of stiffness matrices. So far, most problems solved by the method have been considered for regular domains. Here the method is considered for the solution of Linear Elastic Fracture Mechanics (LEFM) problems. These problems contain a crack which introduces irregularities into the solution space in the form of a discontinuity across the crack boundary and a strain singularity at the crack tip. The relative ease with which the DPM can solve problems of complex geometry makes it particularly attractive for LEFM problems due to the often complex geometries of cracks and the possibility of multiple cracks. The DPM is developed here for the solution of crack problems with the aim of demonstrating the method's potential within this field.As part of this development, for the first time the DPM is combined with the Finite Element Method (FEM). In particular the Extended Finite Element Method (XFEM) is used in order to deal with the irregularities at the crack. Using a geometrical enrichment scheme for the XFEM, near-optimal convergence rates are achieved. The computation time is then significantly reduced by introducing a system of basis functions along the physical boundary of the problem. Applying the DPM with the XFEM, the discontinuity and singularity are dealt with entirely within the XFEM space, therefore avoiding the need to approximate the singularity along the physical crack boundary. With the intention of further reducing the computation time, a Fast Fourier Transform (FFT) algorithm is provided for the solution of the enriched auxiliary problem. The algorithm utilises the regular grid of the auxiliary problem to provide a potentially fast solution method. The above research was applied using Matlab. A Matlab script was written formulating the DPM and XFEM along with various interpolation functions required for the utilisation of the system of boundary basis functions. These included local spline functions and Lagrange polynomials. The FFT algorithm was also applied within Matlab. A Python script was also written for the application of a simple DPM algorithm within Code_Aster, EDF's open source finite element code for thermo-mechanical analyses.These developments are documented in two academic journal papers submitted during the course of the PhD and included in the appendix of this thesis. The Python script for the application of the method within Code_Aster is also included in the appendix.