Superconvergent Nyström and degenerate kernel methods for eigenvalue problems

Abstract

The Nyström and degenerate kernel methods, based on projections at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ⩽ r - 1 , for the approximate solution of eigenvalue problems for an integral operator with a smooth kernel, exhibit order 2 r . We propose new superconvergent Nyström and degenerate kernel methods that improve this convergence order to 4 r for eigenvalue approximation and to 3 r for spectral subspace approximation in the case where the kernel is sufficiently smooth. Moreover for a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of order 4 r can be obtained. The methods introduced here are similar to that studied by Kulkarni in [10] and exhibit the same convergence orders, so a comparison with these methods is worked out in detail. Also, the error terms are analyzed and the obtained methods are numerically tested. Finally, these methods are extended to the case of discontinuous kernel along the diagonal and superconvergence results are also obtained.