Structure-preserving preconditioning of discrete space-fractional diffusion equations with variable coefficient and θ-Method

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 $Ω\subset \mathbb{R}^d$, $d=1,2$. 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 $θ$-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.

Figures (13)

• Figure 1: Comparison between the symbol $f_{\alpha}(\xi_{1})$ (blue stars) and the eigenvalues of ${\overline{G}}_{\alpha,N}$ (red stars) for different values of $N$ - case $\alpha = 1.5$.
• Figure 2: Comparison between the symbol $a(x)f_{\alpha}(\xi_{1})$ (blue stars) and the eigenvalues of $D_{N}(a) \,\overline{G}_{\alpha,N}$ (red stars) for different values of $N$ - case $a(x) = x^2+1$, $\alpha = 1.5$.
• Figure 3: Comparison between the symbol ${f_{\alpha_1}(\xi_{1})+f_{\alpha_2}(\xi_{2})}$ (blue stars) and the eigenvalues of $\overline{G}_{\alpha_1,N_1}\otimes I_{N_2} + I_{N_1} \otimes \overline{G}_{\alpha_2,N_2}$ (red stars) for different values of $N_1=N_2$ - case $\alpha_1 = \alpha_2 = 1.5$, $\gamma_1=\gamma_2=1$.
• Figure 4: Comparison between the symbol $a(x_1,x_2)(f_{\alpha_1}(\xi_{1})+f_{\alpha_2}(\xi_{2}))$ (blue stars) and the eigenvalues of $D_{\mathbf{n}}(a)({\overline{G}_{\alpha_1,N_1}\otimes I_{N_2} + I_{N_1} \otimes \overline{G}_{\alpha_2,N_2}})$ (red stars) for different values of $N_1=N_2$ - case $a(x_1,x_2) = x_1^2+x_2^2+1$, $\alpha_1 = \alpha_2 = 1.5$, $\gamma_1=\gamma_2=1$. .
• Figure 5: Eigenvalues of $A_{\alpha,N,M}$ - $N=M=2^6$, $a(x)=1$.

Theorems & Definitions (29)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4
• Theorem 2.5
• Proposition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Proposition 3.4