This paper deals with the aim of coupling multigrid generation of boundary-fitted grids with an effective full multigrid technique. The author presents a full-FAS multigrid algorithm for the generation of curvilinear coordinate systems on physical domains. He adopts standard central differences for the approximation of the nonlinear generating 2D and 3D systems, full approximation storage (FAS) algorithms to solve the resulting discrete equations and full multigrid computation to obtain a better initial guess for the already developed multigrid algorithms. The elliptic grid generation models, the multigrid computation and the full multigrid algorithm, based on the nested iteration technique are introduced. As a test, the double ellipsoid problem is considered to generate C-type grids for external flow investigations.