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 . We prove that the error bounds for eigenvalues are of the order O ( n − 2 r ) and the gap between the spectral subspaces are of the orders O ( n − r ) in L 2 -norm and O ( n 1 / 2 − r ) in the infinity norm, where r denotes the smoothness of the kernel. By iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence O ( n − 2 r ) in both L 2 -norm and infinity norm. We illustrate our results with numerical examples.