How to compute the eigenvalues and eigenfunctions of a compact Fredholm integral operator whose kernel is of Green's function type - smooth on each side of the diagonal but only continuous across it - by projecting onto piecewise polynomials of even degree at most 2r on a uniform mesh of width h, and how fast each scheme converges. The classical projection method replaces the operator K by pi_n K, where pi_n is either the L2-orthogonal projection (Galerkin) or interpolation at 2r+1 equally spaced points per subinterval (collocation); the modified projection method instead uses pi_n K + K pi_n - pi_n K pi_n. The central result is that iteration and modification each buy extra powers of h even though the kernel is non-smooth on the diagonal and the collocation nodes are equidistant rather than Gauss points: eigenvalue error improves from O(h^(2r+3)) to O(h^(2r+5)) in the Galerkin family and from O(h^(2r+2)) to O(h^(2r+3)) in the collocation family, with matching gains for the spectral subspace measured as the gap between ranges of spectral projections. The price is a matrix eigenvalue problem at most twice as large, since the modified operator's range sits in X_n + K(X_n).
Analyses how accurately projection methods compute the eigenvalues and eigenfunctions of a compact Fredholm integral operator (Kx)(s) = int_0^1 k(s,t) x(t) dt whose kernel is of Green's function type…