Numerical solution of a spectral problem for an ode with a small parameter using an asymptotic expansion and a finite element method

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 H 1 (ω) and H 2 (ω). We give some numerical results which prove that this method is better than a direct FEM when e is small enough.