A systematic approach to reduced GLT

TL;DR

This paper introduces Reduced GLT sequences and symbols to analyze the spectral properties of matrix-sequences from PDE discretizations on generic domains, extending existing GLT theory beyond hypercubes.

Contribution

The paper develops the theory of Reduced GLT sequences and symbols, enabling spectral analysis on arbitrary domains, and demonstrates applications to finite difference and finite element methods.

Findings

Reduced GLT sequences generalize spectral analysis to generic domains.

The theory is applied to convection-diffusion-reaction PDE discretizations.

New insights into eigenvalue asymptotics for complex geometries.

Abstract

This paper concerns the spectral analysis of matrix-sequences that are generated by the discretization and numerical approximation of partial differential equations (PDEs), in case the domain is a generic Peano-Jordan measurable set. It is observed that such matrix-sequences often present a spectral symbol, that is a measurable function describing the asymptotic behaviour of the eigenvalues. When the domain is a hypercube, the analysis can be conducted using the theory of generalized locally Toeplitz (GLT) sequences, but in case of generic domain, a new kind of matrix-sequences and theory has to be formalized. We thus introduce the Reduced GLT sequences and symbols, developing in full detail its theory, and presenting some application to finite differences and finite elements discretization for convection-diffusion-reaction differential equations.