A note on eigenvalues and singular values of variable Toeplitz matrices and matrix-sequences, with application to variable two-step BDF approximations to parabolic equations

TL;DR

This paper analyzes the eigenvalues and singular values of variable Toeplitz matrix-sequences using GLT theory, with applications to BDF methods for parabolic equations on non-uniform grids, including spectral analysis and numerical illustrations.

Contribution

It extends GLT theory to a broader class of matrix-sequences and applies it to analyze BDF approximation matrices for parabolic PDEs on non-uniform grids.

Findings

Eigenvalues and singular values characterized via GLT symbols.

Asymptotic spectral behavior of BDF matrices established.

Numerical examples illustrate theoretical results.

Abstract

Here, we consider a more general class of matrix-sequences and we prove that they belong to the maximal $*$-algebra of generalized locally Toeplitz (GLT) matrix-sequences. Then, we identify the associated GLT symbols and GLT momentary symbols in the general setting and in the specific case, by providing in both cases a spectral and singular value analysis. More specifically, we use the GLT tools in order to study the asymptotic behaviour of the eigenvalues and singular values of the considered BDF matrix-sequences, in connection with the given non-uniform grids. Numerical examples, visualizations, and open problems end the present work.