A new superconvergent collocation method for eigenvalue problems

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, we show that the proposed method exhibits an error of the order of 4r for eigenvalue approximation and of the order of 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 4r can be obtained. This improves upon the order 2r for eigenvalue approximation in the collocation/iterated collocation method and the orders r and 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.