# Analysis of the spectral symbol function for spectral approximation of a differential operator

## TL;DR

This paper investigates the spectral symbol function for discretized differential operators, establishing conditions under which it accurately approximates the spectrum and challenging the effectiveness of uniform sampling methods.

## Contribution

It proves that the spectral symbol provides a necessary condition for spectral approximation and shows that uniform sampling of the symbol is generally insufficient for accurate spectrum approximation.

## Key findings

- Spectral symbol is necessary for spectral approximation.
- Uniform sampling of spectral symbol often fails to approximate spectra accurately.
- Proper discretization and grid refinement improve spectral approximation.

## Abstract

Given a differential operator $\mathcal{L}$ along with its own eigenvalue problem $\mathcal{L}u = \lambda u$ and an associated algebraic equation $\mathcal{L}^{(n)} \mathbf{u}_n = \lambda\mathbf{u}_n$ obtained by means of a discretization scheme (like Finite Differences, Finite Elements, Galerkin Isogeometric Analysis, etc.), the theory of Generalized Locally Toeplitz (GLT) sequences serves the purpose to compute the spectral symbol function $\omega$ associated to the discrete operator $\mathcal{L}^{(n)}$ We prove that the spectral symbol $\omega$ provides a necessary condition for a discretization scheme in order to uniformly approximate the spectrum of the original differential operator $\mathcal{L}$. The condition measures how far the method is from a uniform relative approximation of the spectrum of $\mathcal{L}$. Moreover, the condition seems to become sufficient if the discretization method is paired with a suitable (non-uniform) grid and an increasing refinement of the order of approximation of the method. On the other hand, despite the numerical experiments in many recent literature, we disprove that in general a uniform sampling of the spectral symbol $\omega$ can provide an accurate relative approximation of the spectrum, neither of $\mathcal{L}$ nor of the discrete operator $\mathcal{L}^{(n)}$.