The element free Galerkin method has been used to solve linear elastic fracture mechanics problems using least square approximations, where the dependent variable at any point is obtained by minimizing a weighted discrete -error norm involving the nodal variable within a small domain surrounding the point. Special techniques are presented for enriching the EFG approximations near the crack tip such as extrinsic and intrinsic enrichment. Extrinsic enrichment involves the addition of solution form to the trial function, whereas the EFG basis is expanded to include few terms from crack tip solution in intrinsic enrichment. Apart from enrichment techniques, four basic techniques of smoothing meshless approximations near nonconvex boundaries are also presented such as diffraction, transparency, see through and wedge model. These techniques are then used for the analysis of an edge crack problem under mode-1 loading. Out of these used techniques, the extrinsic PU enrichment technique was found to be more accurate as compared to other used approaches. Extrinsic PU enrichment technique has also been used for the analysis of a shear edge crack problem.