On the convergence of Fourier spectral methods involving non-compact operators
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909177678397440 |
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| author | Trogdon, Thomas |
| author_facet | Trogdon, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_14319 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the convergence of Fourier spectral methods involving non-compact operators Trogdon, Thomas Numerical Analysis Functional Analysis Spectral Theory 65N35, 34L16, 45E10 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. |
| title | On the convergence of Fourier spectral methods involving non-compact operators |
| topic | Numerical Analysis Functional Analysis Spectral Theory 65N35, 34L16, 45E10 |
| url | https://arxiv.org/abs/2305.14319 |