Use of extrapolation for improving the order of convergence of eigenelement approximations

We consider the approximation of the eigenelements of a compact integral operator defined on C[0, 1] with a smooth kernel. We use the iterated collocation method based on r Gauss points and piecewise polynomials of degree less than or equal to r - 1 on each subinterval of a nonuniform partition of [0,1]. We obtain asymptotic expansions for the arithmetic means of m eigenvalues and also for the associated spectral projections. Using Richardson extrapolation, we show that the order of convergence O(h(2r)) in the iterated collocation method can be improved to O(h(2r+2)). Similar results hold for the Nystrom method and for the iterated Galerkin method. We illustrate the improvement in the order of convergence by numerical experiments