A new superconvergent collocation method for eigenvalue problems

Abstract

Here we propose a new method based on projections for the approximate solution of eigenvalue problems. For an integral operator with a smooth kernel, using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomials of degree ≤ r − 1 \leq r-1 , we show that the proposed method exhibits an error of the order of 4 r 4r for eigenvalue approximation and of the order of 3 r 3r for spectral subspace approximation. In the case of a simple eigenvalue, we show that by using an iteration technique, an eigenvector approximation of the order 4 r 4r can be obtained. This improves upon the order 2 r 2r for eigenvalue approximation in the collocation/iterated collocation method and the orders r r and 2 r 2r for spectral subspace approximation in the collocation method and the iterated collocation method, respectively. We illustrate this improvement in the order of convergence by numerical examples.