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 $\leq r-1$, 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.