In this paper a modification of the asymptotic expansion (AE) given by Vishik and Lyustenik for the eigenelements of a spectral problem in elliptic-elliptic singular perturbation is described. We prove that the AE is valid in Cj(ω)j=O,…,n where n is the order of AE. A finite element method (FEM) is used to compute each term and an error estimate is proved in H1(ω) and H2(ω). We give some numerical results which prove that this method is better than a direct FEM when e is small enough.