Use of extrapolation for improving the order of convergence of eigenelement approximations
Rekha P. Kulkarni · IMA Journal of Numerical Analysis · 1997
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 ≤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(h2r) in the iterated collocation method can be improved to O(h2r+2). Similar results hold for the Nyström method and for the iterated Galerkin method. We illustrate the improvement in the order of convergence by numerical experiments.