Abstract
In this paper, we consider solutions for discrete systems arising from multi-term time-fractional diffusion equations. Using discrete sine transform techniques, we find that all-at-once systems of such equations have a structure similar to that of diagonal-plus-Toeplitz matrices. We establish a generalized circulant approximate inverse preconditioner for the all-at-once systems. Through a detailed analysis of the preconditioned matrices, we show that the spectrum of the obtained preconditioned matrices is clustered around one. We give some numerical examples to demonstrate the effectiveness of the proposed preconditioner.
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The author wants to thank to anonymous reviewers for their valuable suggestions.
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This work was supported by the National Natural Science Foundation of China (No. 11771193), the Fundamental Research Funds for the Central Universities (No. lzujbky-2018-31), and the Natural Science Foundation of Gansu Province (No. 23JRRA1104).
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DG and G-FZ wrote the main manuscript text and Z-ZL prepared Tables 1-8 and Figure 1. All authors reviewed the manuscript.
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Appendix. Detailed proof of Lemma 4.3
Appendix. Detailed proof of Lemma 4.3
Lemma A.1
Let \(B_1\) and \(\widetilde{A}\) be defined by (21) and (16), respectively. Assume that \(\phi (x,y) \in C^1\) is a smooth binary function on \([a,b]\times [0,T]\). Then, for a given \(\varepsilon >0\), there exist two constants \(\hat{c}\), \(\check{c}\) and an integer l such that
Proof
Consider
From the analysis in subsection 4.1, we know that both \(\left\| K_{j,i}^{-1} \right\| _{\infty }\) and \(\left\| \widetilde{A}^{-1}\right\| _{\infty }\) are bounded and we let \(\left\| K_{j,i}^{-1} \right\| _{\infty } \le c_4\) and \(\left\| \widetilde{A}^{-1}\right\| _{\infty } \le c_5\), respectively. Let
denote the \(((i-1)M+j)\)th row and \(((p-1)M+q)\)th column entry of matrix \(\widetilde{A}^{-1}\). It follows from (34) that
Using the inequality \(\sum _{a = b+1}^{\infty }\frac{1}{a^{c+1}}\le \frac{1}{cb^c}\) [29], we have
We will separate the terms on the right-hand side of (A3) into two parts for individual analysis. The first part is as follows:
The second part is as follows:
Consider the first term at the right end of (A5).
Similarly, we can obtain the other parts of (A5) with
and
It can be seen from (A3)–(A9) that when M tends to infinity, we have
From the mean value theorem for binary functions [42], we have
For any given \(\varepsilon \ge 0\), let l be the integer satisfying
Then,
Let \(\hat{c} = \max \limits _{\begin{array}{c} x \in \left[ a,b\right] \\ y \in \left[ 0,T\right] \end{array}}\left| \phi _x(x,y) \right| c_4c_5\) and \(\check{c} = \max \limits _{\begin{array}{c} x \in \left[ a,b\right] \\ y \in \left[ 0,T\right] \end{array}}\left| \phi _y(x,y) \right| c_4c_5\), we naturally get the conclusion of Lemma A.1. \(\square \)
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Gan, D., Zhang, GF. & Liang, ZZ. Preconditioning techniques of all-at-once systems for multi-term time-fractional diffusion equations. Numer Algor 96, 1499–1531 (2024). https://doi.org/10.1007/s11075-023-01675-x
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DOI: https://doi.org/10.1007/s11075-023-01675-x