Existing generalized or extended finite element methods for modeling cracks in three-dimensions require the use of a sufficiently refined mesh around the crack front. This offsets some of the advantages of these methods specially in the case of propagating threedimensional cracks. In this paper, a strategy to overcome this limitation is investigated. The approach involves the development of enrichment functions that are computed using a new global-local approach. This strategy allows the use of a fixed global mesh around the crack front and is specially appealing for non-linear or time dependent problems since it avoids mapping of solutions between meshes. The resulting technique enjoys the same flexibility of the so-called meshfree methods for this class of problem while being more computationally efficient. The proposed generalized FEM with global-local functions, by numerically constructing the enrichment functions, brings the benefits of existing generalized FEMto a broader class of problems. The procedure is applied to the solution of three-dimensional linear elastic fracture mechanics problems. Numerical experiments demonstrating the computational efficiency and accuracy of the method are presented.