Projection-based approximations for eigenvalue problems of Fredholm integral operators with Green's kernels

TL;DR

This paper introduces projection-based methods for efficiently approximating eigenvalues and eigenfunctions of compact integral operators with Green's kernels, demonstrating superconvergence and improved rates over classical approaches.

Contribution

It develops modified projection techniques that achieve faster convergence for eigenvalue problems of Fredholm integral operators with Green's kernels, with theoretical and numerical validation.

Findings

Superconvergence of eigenfunctions under iteration

Modified projections outperform classical methods

Numerical examples confirm enhanced convergence rates

Abstract

We consider the eigenvalue problem $K x = \lambda x$. Our analysis focuses on the convergence rates of eigenvalue and spectral subspace approximations for compact linear integral operator $K$ with Green's kernels. By employing orthogonal and interpolatory projections at $2r+1$ collocation points (which are not necessarily Gauss points) onto an approximating space of piecewise even degree polynomials, we establish the superconvergence of eigenfunctions under iteration. The modified projection methods achieve a faster convergence rates compared to classical projection methods. The enhancement in convergence rate is verified by numerical examples.