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

2023/05/23 by Trogdon, Thomas
#34L16 #45E10 #65N35 #FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2305.14319

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.

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