We consider the time-harmonic problem of the diffraction of an incident propagative mode by a localized defect, in an infinite straight isotropic elastic waveguide. We propose several iterative algorithms to compute an approximate solution of the problem, using a classical finite element discretization in a small area around the perturbation, and a modal expansion in unbounded straight parts of the guide. Each algorithm can be related to a so-called domain decomposition method, with or without an overlap between the domains. Specific transmission conditions are used, so that only the sparse finite element matrix has to be inverted, the modal expansion being obtained by a simple projection, using the Fraser bi-orthogonality relation. The benefit of using an overlap between the finite element domain and the modal domain is emphasized, in particular for the extension to the anisotropic case. Numerical validations for two-and three-dimensional configurations for both isotropic and anisotropic media, are finally presented.