We consider the eigenvalue problem \(\varvec{\mathcal {K} x = \lambda x}\) . Our analysis focuses on the convergence rates of eigenvalue and spectral subspace approximations for compact linear integral operator \(\varvec{\mathcal {K}}\) with Green’s kernels. By employing orthogonal and interpolatory projections at \(\varvec{2r+1}\) collocation points (which are not necessarily Gauss points) onto an approximating space of piecewise even degree polynomials, we establish the superconvergence of eigenfunctions under iteration. The modified projection methods achieve faster convergence rates compared to classical projection methods. The enhancement in convergence rate is verified by numerical examples.
