We consider a multiple iteration method based on defect correction for the refinement of approximate eigenelements of an integral operator T obtained by projection, that constructs a finite rank operator approximating T and reducing the integral eigenvalue problem to a matrix one. Focus is on the case where the discretizing matrices are sparse, due to the underlying physics problem. Numerical experiments with a radiative transfer equation in Astrophysics are presented and compared with state-of-the-art methods.