Superconvergence of Legendre projection methods for the eigenvalue problem of a compact integral operator

In this paper, we consider the Galerkin and collocation methods for the eigenvalue problem of a compact integral operator with a smooth kernel using the Legendre polynomials of degree ≤n≤n. We prove that the error bounds for eigenvalues are of the order O(n−2r)O(n−2r) and the gap between the spectral subspaces are of the orders O(n−r)O(n−r) in L2L2-norm and O(n1/2−r)O(n1/2−r) in the infinity norm, where rr denotes the smoothness of the kernel. By iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence O(n−2r)O(n−2r) in both L2L2-norm and infinity norm. We illustrate our results with numerical examples.