Structure-preserving preconditioning of discrete space-fractional diffusion equations with variable coefficient and {\theta}-Method

TL;DR

This paper develops a structure-preserving preconditioning method for large linear systems from discretized space-fractional diffusion equations with variable coefficients, using GLT theory to analyze spectral properties and improve computational efficiency.

Contribution

It introduces a novel preconditioning strategy based on matrix structure and GLT theory that effectively handles variable coefficients in space-fractional diffusion equations.

Findings

Preconditioning improves convergence of iterative solvers.

Spectral analysis confirms the effectiveness of the preconditioner.

Numerical experiments validate theoretical predictions.

Abstract

This paper studies the spectral properties of large matrices and the preconditioning of linear systems, arising from the finite difference discretization of a time-dependent space-fractional diffusion equation with a variable coefficient $a(x)$ defined on $\Omega \subset \mathbb{R}^d$, $d=1,2$. The model involves a one-sided Riemann-Liouville fractional derivative multiplied by the function $a(x)$, discretized by the shifted Gr"unwald formula in space and the $\theta$-method in time. The resulting all-at-once linear systems exhibit a $(d+1)$-level Toeplitz-like matrix structure, with $d=1,2$ denoting the space dimension, while the additional level is due to the time variable. A preconditioning strategy is developed based on the structural properties of the discretized operator. Using the generalized locally Toeplitz (GLT) theory, we analyze the spectral distribution of the unpreconditioned and preconditioned matrix sequences. The main novelty is that the analysis fully covers the case where the variable coefficient $a$ is nonconstant. Numerical results are provided to support the GLT based theoretical findings, and some possible extensions are briefly discussed.