This paper presents a coupling technique for integrating the element free Galerkin method (EFGM) with the fractal finite element method (FFEM) to analyze unbounded problems in the half space. FFEM is adopted to model the far field of an unbounded domain and EFGM is used in the near field. In the transition region interface elements are employed. The shape functions of interface elements which comprise both the element free Galerkin and the finite element shape functions, satisfy the consistency condition thus ensuring convergence of the proposed coupled EFGM FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no structured mesh or special enriched basis functions are necessary. The numerical results show that the proposed method performs extremely well converging rapidly to the analytical solution. Also a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the weight function, the scaling parameter and the number of transformation terms, on the quality of the numerical solutions.