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

Shashank K. Shukla, Gobinda Rakshit, Akshay S. Rane

Abstract

We consider the eigenvalue problem $K x = λx$. 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.

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

Abstract

We consider the eigenvalue problem . Our analysis focuses on the convergence rates of eigenvalue and spectral subspace approximations for compact linear integral operator with Green's kernels. By employing orthogonal and interpolatory projections at 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.
Paper Structure (11 sections, 12 theorems, 159 equations, 4 tables)

This paper contains 11 sections, 12 theorems, 159 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximations based on projections
  4. Orthogonal projection
  5. Interpolatory projection
  6. Estimates in the methods of approximation
  7. Convergence orders in the approximations
  8. Orthogonal projection based approximations
  9. Interpolatory projection based approximations
  10. Numerical Results
  11. Conclusion

Key Result

Theorem 2.1

For sufficiently large $n$, where $C$ denotes a generic constant independent of $n$ and $\left( \mathcal{K} - \mathcal{K}_n \right)\mathcal{K}|_{\mathcal{R}(E)}$ represents the restriction of $(\mathcal{K} - \mathcal{K}_n)\mathcal{K}$ to $\mathcal{R}(E)$.

Theorems & Definitions (17)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • Theorem 4.5
  • ...and 7 more