Legendre multi-projection methods for solving eigenvalue problems for a compact integral operator

In this paper, we consider the M-Galerkin and M-collocation methods for solving the eigenvalue problem for a compact integral operator with smooth kernels, using Legendre polynomial bases. We obtain error bounds for the eigenvalues and the gap between the spectral subspaces in both Legendre M-Galerkin and Legendre M-collocation methods. We also obtain superconvergence results for the eigenvalues and iterated eigenvectors in both L2 and infinity norm. We illustrate our results with numerical examples.