Work

Block Preconditioning for All-at-Once Variable-Coefficient Fractional Evolution Equations via the GLT Analysis

Year: 2026

Type: article

Abstract: We study a class of nonlocal evolutionary partial differential equations with a weakly singular temporal kernel and a spatially variable diffusion coefficient. The model is posed on Ω⊂R, and involves a left-sided Riemann–Liouville fractional derivative in space multiplied by a variable coefficient a(x). The temporal derivative is approximated by an L1 type scheme, while the spatial operator is discretized by finite difference techniques, resulting in large-scale all-at-once linear systems with a two-level Toeplitz-like structure. We develop and analyze a block lower triangular strategy that mimics the structure of the coefficient matrix while simplifying its components for computational efficiency. The analysis is carried out at the level of matrix sequences by means of generalized locally Toeplitz (GLT) theory. Within this framework, we characterize the asymptotic spectral distribution of the discretized operators and use the associated GLT symbol to guide the construction of the structured approximation. Numerical experiments using the GMRES solver demonstrate that the proposed preconditioning strategy significantly improves convergence rates, robustness, and scalability for large-scale problems. Open problems and possible extensions are briefly discussed at the end of the present work.

Source: Mathematics

DOI: 10.3390/math14213925

Authors: Muhammad Faisal Khan, Stefano Serra-Capizzano

Institutions: University of Insubria, Uppsala University

Language: English

Topic: Matrix Theory and Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $3,137

Funder: Univerzita Karlova v Praze