How Many Numerical Eigenvalues can We Trust?

TL;DR

This paper investigates the reliability of numerically approximated eigenvalues in high-order elliptic problems, revealing that the proportion of trustworthy eigenvalues diminishes as the system size grows, despite an increase in their total number.

Contribution

It provides new insights into the diminishing reliability percentage of eigenvalues in finite element and finite difference methods for high-order elliptic problems.

Findings

The percentage of reliable eigenvalues decreases as degrees of freedom increase.

The total number of reliable eigenvalues still grows with system size.

Previous assumptions about eigenvalue trustworthiness are overly optimistic.

Abstract

When using finite element and finite difference methods to approximate eigenvalues of $2m^{th}$-order elliptic problems, the number of reliable numerical eigenvalues can be estimated in terms of the total degrees of freedom $N$ in resulting discrete systems. The truth is worse than what we used to believe in that the percentage of reliable eigenvalues decreases with an increased $N$, even though the number of reliable eigenvalues increases with $N$.