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

Shashank K. Shukla

Abstract

This paper studies the eigenvalue problem $K ψ= λψ$ 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.

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

Abstract

This paper studies the eigenvalue problem associated with a Fredholm integral operator 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 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.

Paper Structure

This paper contains 8 sections, 12 theorems, 58 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Projection-based Approximations
  3. Interpolatory Projection
  4. Approximation Methods for Interpolatory Projection
  5. Spectral Projection and Error Estimation
  6. Convergence Rates in Approximation Methods
  7. Numerical Results
  8. Conclusion

Key Result

Lemma 3.1

For all sufficiently large $n$, the gap between the spectral subspaces satisfies where $C >0$ is a constant independent of $n$, which takes different values at different places, and $(\mathcal{K}-\mathcal{K}_n)\mathcal{K}|_{\mathcal{R}(E)}$ denotes the restriction of $(\mathcal{K}-\mathcal{K}_n)\mathcal{K}$ to the spectral subspace $\mathcal{R}(E)$.

Theorems & Definitions (12)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Lemma 3.4: Kulkarni RPK6
  • Theorem 3.5
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Lemma 4.4
  • Lemma 4.5
  • ...and 2 more