Spectral and norm estimates for matrix sequences arising from a finite difference approximation of elliptic operators

TL;DR

This paper analyzes the spectral and norm properties of matrices from finite difference approximations of elliptic operators, providing theoretical estimates validated by numerical experiments.

Contribution

It introduces new spectral and norm estimates for matrices from finite difference schemes for elliptic problems, utilizing Toeplitz and GLT matrix theory.

Findings

Spectral estimates align with numerical results.

Norm bounds are established for matrix sequences.

Numerical experiments confirm theoretical predictions.

Abstract

When approximating elliptic problems by using specialized approximation techniques, we obtain large structured matrices whose analysis provides information on the stability of the method. Here we provide spectral and norm estimates for matrix sequences arising from the approximation of the Laplacian via ad hoc finite differences. The analysis involves several tools from matrix theory and in particular from the setting of Toeplitz operators and Generalized Locally Toeplitz matrix sequences. Several numerical experiments are conducted, which confirm the correctness of the theoretical findings.