A note on superconvergence in projection-based numerical approximations of eigenvalue problems for Fredholm integral operators

TL;DR

This paper investigates the convergence properties of projection-based numerical methods for eigenvalue problems of Fredholm integral operators, demonstrating superconvergence phenomena and providing explicit rates.

Contribution

It introduces a modified collocation method that achieves faster eigenvalue and eigenfunction convergence, including superconvergence, for Fredholm integral operators.

Findings

Modified collocation method improves convergence rates.

Superconvergence of eigenfunction approximations is demonstrated.

Numerical experiments confirm theoretical results.

Abstract

This paper studies the eigenvalue problem $K \psi = \lambda \psi$ associated with a Fredholm integral operator $K$ defined by a smooth kernel. The focus is on analyzing the convergence behaviour of numerical approximations to eigenvalues and their corresponding spectral subspaces. The interpolatory projection methods are employed on spaces of piecewise polynomials of even degree, using $2r+1$ collocation points that are not restricted to Gauss nodes. Explicit convergence rates are established, and the modified collocation method attains faster convergence of approximation of eigenvalues and associated eigenfunctions than the classical collocation scheme. Moreover, it is shown that the iteration yields superconvergent approximations of eigenfunctions. Numerical experiments are presented to validate the theoretical findings.