Block Preconditioning for All-at-Once Variable-Coefficient Fractional Evolution Equations via the GLT Analysis
- 1Department of Science and High Technology, University of Insubria, 22100 Como, Italy
- 2Department of Information Technology, Uppsala University, 75105 Uppsala, Sweden
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.
Related
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BibTeX
@article{khan2026block,
title = {{Block Preconditioning for All-at-Once Variable-Coefficient Fractional Evolution Equations via the GLT Analysis}},
author = {Muhammad Faisal Khan and Stefano Serra-Capizzano},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213925},
url = {https://www.mdpi.com/2227-7390/14/21/3925},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Block Preconditioning for All-at-Once Variable-Coefficient Fractional Evolution Equations via the GLT Analysis
AU - Muhammad Faisal Khan
AU - Stefano Serra-Capizzano
PY - 2026
DA - 2026/10/27
JO - Mathematics
DO - 10.3390/math14213925
UR - https://www.mdpi.com/2227-7390/14/21/3925
AB - 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.
SN - 2227-7390
ER -