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On the convergence of Fourier spectral methods involving non-compact operators

Thomas Trogdon

TL;DR

This paper addresses the problem of proving convergence and estimating rates for Fourier-based spectral methods applied to operator equations $\mathcal L u=f$ when the operator involves non-compact perturbations. Its core idea is a left-Fredholm regulator $\mathcal N$ with a finite-dimensional approximation, yielding a general convergence theorem that does not require using the regulator in the numerical method itself. The authors apply the framework to periodic differential operators via finite-section and collocation projections and to Riemann–Hilbert problems on the unit circle, obtaining spectral convergence for eigenvalues and $H^s$ convergence in the RH setting, with rates tied to smoothness. They also derive improved eigenvalue convergence results in the self-adjoint case and demonstrate the theory with numerical experiments that confirm the predicted rates, underscoring the practical impact for spectral discretizations of non-compact problems.

Abstract

Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to Fourier finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann--Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.

On the convergence of Fourier spectral methods involving non-compact operators

TL;DR

This paper addresses the problem of proving convergence and estimating rates for Fourier-based spectral methods applied to operator equations when the operator involves non-compact perturbations. Its core idea is a left-Fredholm regulator with a finite-dimensional approximation, yielding a general convergence theorem that does not require using the regulator in the numerical method itself. The authors apply the framework to periodic differential operators via finite-section and collocation projections and to Riemann–Hilbert problems on the unit circle, obtaining spectral convergence for eigenvalues and convergence in the RH setting, with rates tied to smoothness. They also derive improved eigenvalue convergence results in the self-adjoint case and demonstrate the theory with numerical experiments that confirm the predicted rates, underscoring the practical impact for spectral discretizations of non-compact problems.

Abstract

Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations . The framework posits the existence of a left-Fredholm regulator for and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to Fourier finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann--Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.
Paper Structure (21 sections, 22 theorems, 220 equations, 4 figures)

This paper contains 21 sections, 22 theorems, 220 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Overview of main results
  4. Relation to previous work
  5. The classical framework
  6. The classical framework
  7. Extending the classical framework -- left-Fredholm operators
  8. Spaces of periodic functions
  9. Projections
  10. Periodic differential operators
  11. An operator equation
  12. Spectrum approximation
  13. Improvements for self-adjoint operators
  14. Riemann--Hilbert problems on the circle
  15. Numerical demonstrations
  16. ...and 6 more sections

Key Result

Lemma 2.1

Suppose that for any $u \in \mathbb W$. Then if $\mathcal{K} : \mathbb V \to \mathbb W$ is compact

Figures (4)

  • Figure 1: The measured $H^s(\mathbb U)$ error in solving the euquation \ref{['eq:3rdop']} The diamonds (blue) give the computed error by comparing against a solution computed using $N = 2001.$ The rate of convergence closely matches $N^{s-t + 3}$ reflecting the fact that that Theorem \ref{['t:diff-conv']} is optimal.
  • Figure 2: The errors $d_j^{(N)}$ (left panel) and the rescaled errors $r_j^{(N)}$ plotted versus $1 + |\lambda_j|$ for the finite $N$ approximation of \ref{['eq:2']}. The rescaled errors verify the estimates in Theorem \ref{['t:spectrum-main']}.
  • Figure 3: The errors $d_j^{(N)}$ (left panel) and the rescaled errors $r_j^{(N)}$ plotted versus $1 + |\lambda_j|$ for the finite $N$ approximation of \ref{['eq:3op']}. The rescaled errors verify the estimates in Theorem \ref{['t:spectrum-main']}.
  • Figure 4: The measured $H^s(\mathbb U)$ error in solving the Riemann--Hilbert problem on the circle with jump condition given by \ref{['eq:gg']}. The diamonds (blue) give the computed error by comparing against a solution computed using $N = 2000.$ The rate of convergence closely matches $N^{s-t}$ indicating that Theorem \ref{['t:rhp-1']} is optimal with respect to the rate it predicts.

Theorems & Definitions (47)

  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Remark 3.6
  • ...and 37 more