Spectral refinement based on superconvergent Nyström and degenerate kernel methods

Abstract

In this paper are studied two spectral refinement schemes, Elementary and Double Iteration, for the approximate solution of eigenvalue problems for an integral operator with a smooth kernel. We use new superconvergent Nyström and degenerate kernel approximating operators based on interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree⩽r−1. We show that for the eigenvalue iterates the Elementary Iteration scheme exhibits order (k+4)r while the Double Iteration scheme exhibits order (3k+4)r, where k=0,1,2,… denotes the iteration step. We show that these orders of convergence are preserved in the corresponding discrete methods obtained by replacing the integration by a numerical quadrature formula. Similar improvement is observed for eigenvector iterates. Our results are extended to the case of discontinuous kernels along the diagonal for which superconvergence results are also obtained. Finally, numerical validation is given.