Preconditioning techniques of all-at-once systems for multi-term time-fractional diffusion equations

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.

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

$$\begin{aligned} \left\| B_1- \widetilde{A}^{-1} \right\| _{\infty } \le \varepsilon /2 + l\hat{c}h + l\check{c}\tau . \end{aligned}$$

Proof

Consider

$$\begin{aligned} \begin{aligned} \left\| B_1- \widetilde{A}^{-1} \right\| _{\infty }&= \max _{1\le i\le M,1\le j\le N} \left\| \left( e_i\otimes e_j \right) ^T\left( B_1 - \widetilde{A}^{-1} \right) \right\| _1\\&= \max _{1\le i\le M,1\le j\le N}\left\| \left( e_i\otimes e_j \right) ^T\left( K_{j,i}^{-1} - \widetilde{A}^{-1} \right) \right\| _1\\&= \max _{1\le i\le M,1\le j\le N}\left\| \left( e_i\otimes e_j \right) ^T \widetilde{A}^{-1}\left( \widetilde{A} - K_{j,i} \right) K_{j,i}^{-1} \right\| _1\\&= \max _{1\le i\le M,1\le j\le N}\left\| \left( e_i\otimes e_j \right) ^T \widetilde{A}^{-1}\left( \widetilde{D} - \phi _j^iI_{NM} \right) K_{j,i}^{-1} \right\| _1\\&\le \max _{1\le i\le M,1\le j\le N}\left\| \left( e_i\otimes e_j \right) ^T \widetilde{A}^{-1}\left( \widetilde{D} - \phi _j^iI_{NM} \right) \right\| _1 \left\| K_{j,i}^{-1} \right\| _{\infty }. \end{aligned} \end{aligned}$$

(A1)

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

$$\widetilde{A}^{-1}((i-1)M+j,(p-1)M+q) = \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}$$

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

$$\begin{aligned}{} & {} \left\| \left( e_i\otimes e_j \right) ^T \widetilde{A}^{-1}\left( \widetilde{D} - \phi _j^iI_{NM} \right) \right\| _1 = \sum _{p=1}^{M}\sum _{q=1}^{N}\left| \left( \phi _q^p-\phi _j^i \right) \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p} \right| = \sum _{p=1}^{i-l}\sum _{q=1}^{j-l}\left| \left( \phi _q^p-\phi _j^i \right) \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p} \right| \nonumber \\{} & {} + \sum _{p=1}^{i-l}\sum _{q=j-l+1}^{j+l-1}\left| \left( \phi _q^p-\phi _j^i \right) \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p} \right| + \sum _{p=1}^{i-l}\sum _{q=j+l}^{N}\left| \left( \phi _q^p-\phi _j^i \right) \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p} \right| \nonumber \\{} & {} + \sum _{p=i-l+1}^{i+l-1}\sum _{q=1}^{j-l}\left| \left( \phi _q^p-\phi _j^i \right) \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p} \right| + \sum _{p=i-l+1}^{i+l-1}\sum _{q=j-l+1}^{j+l-1}\left| \left( \phi _q^p-\phi _j^i \right) \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p} \right| \nonumber \\{} & {} + \sum _{p=i-l+1}^{i+l-1}\sum _{q=j+l}^{N}\left| \left( \phi _q^p-\phi _j^i \right) \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p} \right| + \sum _{p=i+l}^{M}\sum _{q=1}^{j-l}\left| \left( \phi _q^p-\phi _j^i \right) \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p} \right| \nonumber \\{} & {} + \sum _{p=i+l}^{M}\sum _{q=j-l+1}^{j+l-1}\left| \left( \phi _q^p-\phi _j^i \right) \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p} \right| +\sum _{p=i+l}^{M}\sum _{q=j+l}^{N}\left| \left( \phi _q^p-\phi _j^i \right) \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p} \right| \nonumber \\{} & {} \le \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=1}^{i-l}\sum _{q=1}^{j-l} \dfrac{c_p}{[1+(i-p)M+(j-q)]^{s}} + \max _{\begin{array}{c} 1\le p\le i-l \\ j+l-1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=1}^{i-l}\sum _{q=j-l+1}^{j+l-1} \left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| \nonumber \\{} & {} + \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=1}^{i-l}\sum _{q=j+l}^{N} \dfrac{c_p}{[1+(i-p)M+(q-j)]^{s}} + \max _{\begin{array}{c} i-l+1\le p\le i+l-1 \\ 1\le q\le j-l \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i-l+1}^{i+l-1}\sum _{q=1}^{j-l} \left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| \nonumber \\{} & {} +\!\! \max _{\begin{array}{c} i-l+1\le p\le i+l-1 \\ j-l+1\le q\le j+l-1 \end{array}}\left| \phi _q^p\!-\!\phi _j^i \right| \sum _{p=i-l+1}^{i+l-1}\sum _{q=j-l+1}^{j+l-1} \left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| + \max _{\begin{array}{c} i-l+1\le p\le i+l-1 \\ j+l\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i-l+1}^{i+l-1}\sum _{q=j+l}^{N} \!\left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\!\right| \nonumber \\{} & {} +\!\! \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i+l}^{M}\sum _{q=1}^{j-l} \dfrac{c_p}{[1+(p-i)M+(j-q)]^{s}} \!\!+\!\! \max _{\begin{array}{c} i+l\le p\le M \\ j-l+1\le q\le j+l-1 \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i+l}^{M}\sum _{q=j-l+1}^{j+l-1} \left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| \nonumber \\{} & {} + \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i+l}^{M}\sum _{q=j+l}^{N} \dfrac{c_p}{[1+(p-i)M+(q-j)]^{s}}. \end{aligned}$$

(A2)

Using the inequality $\sum _{a = b+1}^{\infty }\frac{1}{a^{c+1}}\le \frac{1}{cb^c}$ [29], we have

$$\begin{aligned} \begin{aligned}&\left\| \left( e_i\otimes e_j \right) ^T \widetilde{A}^{-1}\left( \widetilde{D} - \phi _j^iI_{NM} \right) \right\| _1 \le \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{k=1+lM+l}^{\infty } \dfrac{c_p}{k^{s}} \\&+ \max _{\begin{array}{c} 1\le p\le i-l \\ j-l+1\le q\le j+l-1 \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=1}^{i-l}\sum _{q=j-l+1}^{j+l-1} \left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| + \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{k=1+lM+l}^{\infty } \dfrac{c_p}{k^{s}}\\&+ \max _{\begin{array}{c} i-l+1\le p\le i+l-1 \\ 1\le q\le j-l \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i-l+1}^{i+l-1}\sum _{q=1}^{j-l}\left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| + \max _{\begin{array}{c} i-l+1\le p\le i+l-1 \\ j-l+1\le q\le j+l-1 \end{array}}\left| \phi _q^p-\phi _j^i \right| \left\| \widetilde{A}^{-1}\right\| _{\infty }\\&+ \max _{\begin{array}{c} i-l+1\le p\le i+l-1 \\ j+l\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i-l+1}^{i+l-1}\sum _{q=j+l}^{N}\left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| + \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{k=1+lM+l}^{\infty } \dfrac{c_p}{k^{s}}\\&+ \max _{\begin{array}{c} i+l\le p\le M \\ j-l+1\le q\le j+l-1 \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i+l}^{M}\sum _{q=j-l+1}^{j+l-1}\left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| + \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{k=1+lM+l}^{\infty } \dfrac{c_p}{k^{s}}. \end{aligned} \end{aligned}$$

(A3)

We will separate the terms on the right-hand side of (A3) into two parts for individual analysis. The first part is as follows:

$$\begin{aligned}{} & {} \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p\!\!-\!\!\phi _j^i \right| \sum _{k=1+lM+l}^{\infty } \dfrac{c_p}{k^{s}} + \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{k=1+lM+l}^{\infty } \dfrac{c_p}{k^{s}} \!\!+\!\! \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{k=1+lM+l}^{\infty } \dfrac{c_p}{k^{s}}\nonumber \\{} & {} + \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{k=1+lM+l}^{\infty } \dfrac{c_p}{k^{s}} \le \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \dfrac{4c_p}{\alpha _1(M+1)^{\alpha _1}l^{\alpha _1}}. \end{aligned}$$

(A4)

The second part is as follows:

$$\begin{aligned}{} & {} \max _{\begin{array}{c} 1\le p\le i-l \\ j+l-1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=1}^{i-l}\sum _{q=j-l+1}^{j+l-1} \left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| \nonumber \\{} & {} + \max _{\begin{array}{c} i-l+1\le p\le i+l-1 \\ 1\le q\le j-l \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i-l+1}^{i+l-1}\sum _{q=1}^{j-l} \left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| \nonumber \\{} & {} + \max _{\begin{array}{c} i-l+1\le p\le i+l-1 \\ j+l\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i-l+1}^{i+l-1}\sum _{q=j+l}^{N} \left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| \nonumber \\{} & {} + \max _{\begin{array}{c} i+l\le p\le M \\ j-l+1\le q\le j+l-1 \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i+l}^{M}\sum _{q=j-l+1}^{j+l-1} \left| \left( \widetilde{A}^{-1}\right) _{j,i}^{q,p}\right| \nonumber \\{} & {} \le \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=1}^{i-l}\sum _{q=j-l+1}^{j+l-1} \dfrac{c_p}{\left[ 1+(i-p)M+\left| j-q \right| \right] ^s}\nonumber \\{} & {} + \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i-l+1}^{i+l-1}\sum _{q=1}^{j-l} \dfrac{c_p}{\left[ 1+\left| i-p\right| M +j-q\right] ^s}\nonumber \\{} & {} + \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i-l+1}^{i+l-1}\sum _{q=j+l}^{N} \dfrac{c_p}{\left[ 1+\left| i-p\right| M +q-j\right] ^s}\nonumber \\{} & {} + \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \sum _{p=i+l}^{M}\sum _{q=j-l+1}^{j+l-1} \dfrac{c_p}{\left[ 1+(p-i)M +\left| j-q \right| \right] ^s} \end{aligned}$$

(A5)

Consider the first term at the right end of (A5).

$$\begin{aligned} \begin{aligned}&\sum _{p=1}^{i-l}\sum _{q=j-l+1}^{j+l-1} \dfrac{c_p}{\left[ 1+(i-p)M+\left| j-q \right| \right] ^s} = \sum _{p=1}^{i-l}\sum _{q=j-l+1}^{j-1}\dfrac{c_p}{\left[ 1+(i-p)M+j-q \right] ^s}\\&+ \sum _{p=1}^{i-l}\dfrac{c_p}{\left[ 1+(i-p)M \right] ^s} + \sum _{p=1}^{i-l}\sum _{q=j+1}^{j+l-1}\dfrac{c_p}{\left[ 1+(i-p)M+q-j \right] ^s}\\&\le (l-1)\sum _{i-l}^{p=1}\dfrac{c_p}{\left[ 1+(i-p)M+1 \right] ^s} + \dfrac{c_p}{\alpha _1(lM)^{\alpha _1}}+(l-1)\sum _{i-l}^{p=1}\dfrac{c_p}{\left[ 1+(i-p)M+1 \right] ^s}\\&\le \dfrac{2(l-1)c_p}{\alpha _1(lM+1)^{\alpha _1}}+\dfrac{c_p}{\alpha _1(lM)^{\alpha _1}} \end{aligned} \end{aligned}$$

(A6)

Similarly, we can obtain the other parts of (A5) with

$$\begin{aligned} \begin{aligned}&\sum _{p=i-l+1}^{i+l-1}\sum _{q=1}^{j-l} \dfrac{c_p}{\left[ 1+\left| i-p\right| M +j-q\right] ^s} = \sum _{p=i-l+1}^{i-1}\sum _{q=1}^{j-l}\dfrac{c_p}{\left[ 1+(i-p)M +j-q \right] ^s} + \sum _{q=1}^{j-l}\dfrac{c_p}{\left[ 1+j-q \right] ^s}\\&+\sum _{p=i-l+1}^{i-1}\sum _{q=1}^{j-l}\dfrac{c_p}{\left[ 1+(i-p)M +j-q \right] ^s} \le \dfrac{2(l-1)c_p}{\alpha _1(M+l)^{\alpha _1}} + \dfrac{c_p}{\alpha _1l^{\alpha _1}}, \end{aligned} \end{aligned}$$

(A7)

$$\begin{aligned} \begin{aligned}&\sum _{p=i-l+1}^{i+l-1}\sum _{q=j+l}^{N} \dfrac{c_p}{\left[ 1+\left| i-p\right| M +q-j\right] ^s}\le \dfrac{2(l-1)c_p}{\alpha _1(M+l)^{\alpha _1}} + \dfrac{c_p}{\alpha _1l^{\alpha _1}} \end{aligned} \end{aligned}$$

(A8)

and

$$\begin{aligned} \begin{aligned}&\sum _{p=i+l}^{M}\sum _{q=j-l+1}^{j+l-1} \dfrac{c_p}{\left[ 1+(p-i)M +\left| j-q \right| \right] ^s}\le \dfrac{2(l-1)c_p}{\alpha _1(lM+1)^{\alpha _1}}+\dfrac{c_p}{\alpha _1(lM)^{\alpha _1}}. \end{aligned} \end{aligned}$$

(A9)

It can be seen from (A3)–(A9) that when M tends to infinity, we have

$$\begin{aligned} \begin{aligned} \left\| \left( e_i\otimes e_j \right) ^T \widetilde{A}^{-1}\left( \widetilde{D} - \phi _j^iI_{NM} \right) \right\| _1&\le \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \dfrac{4c_p}{\alpha _1(M+1)^{\alpha _1}l^{\alpha _1}} \\&+\max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \left( \dfrac{2(l-1)c_p}{\alpha _1(lM+1)^{\alpha _1}}+\dfrac{c_p}{\alpha _1(lM)^{\alpha _1}}\right) \\&+\max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \left( \dfrac{4(l-1)c_p}{\alpha _1(M+l)^{\alpha _1}} + \dfrac{2c_p}{\alpha _1l^{\alpha _1}}\right) \\&+\max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \left( \dfrac{2(l-1)c_p}{\alpha _1(lM+1)^{\alpha _1}}+\dfrac{c_p}{\alpha _1(lM)^{\alpha _1}}\right) \\&\le \max _{\begin{array}{c} 1\le p\le M \\ 1\le q\le N \end{array}}\left| \phi _q^p-\phi _j^i \right| \dfrac{2c_p}{\alpha _1l^{\alpha _1}} + \max _{\begin{array}{c} i-l+1\le p\le i+l-1 \\ j-l+1\le q\le j+l-1 \end{array}}\left| \phi _q^p-\phi _j^i \right| \left\| \widetilde{A}^{-1}\right\| _{\infty } \end{aligned} \end{aligned}$$

(A10)

From the mean value theorem for binary functions [42], we have

$$\begin{aligned} \begin{aligned} \max _{\begin{array}{c} i-l+1\le p\le i+l-1 \\ j-l+1\le q\le j+l-1 \end{array}}\left| \phi _q^p-\phi _j^i \right| \le \max _{\begin{array}{c} x \in \left[ a,b\right] \\ y \in \left[ 0,T\right] \end{array}}\left| \phi _x(x,y) \right| lh + \max _{\begin{array}{c} x \in \left[ a,b\right] \\ y \in \left[ 0,T\right] \end{array}}\left| \phi _y(x,y) \right| l\tau \end{aligned} \end{aligned}$$

(A11)

For any given $\varepsilon \ge 0$, let l be the integer satisfying

$$\begin{aligned} l^{\alpha _1}>\dfrac{8c_p}{\alpha _1 \varepsilon }\max _{\begin{array}{c} x \in \left[ a,b\right] \\ y \in \left[ 0,T\right] \end{array}}\left| \phi (x,y) \right| c_4. \end{aligned}$$

(A12)

Then,

$$\begin{aligned} \begin{aligned} \left\| \left( e_i\otimes e_j \right) ^T \widetilde{A}^{-1}\left( \widetilde{D} - \phi _j^iI_{NM} \right) \right\| _1 \le \dfrac{\varepsilon }{2} + \max \limits _{\begin{array}{c} x \in \left[ a,b\right] \\ y \in \left[ 0,T\right] \end{array}}\left| \phi _x(x,y) \right| lhc_4c_5 + \max \limits _{\begin{array}{c} x \in \left[ a,b\right] \\ y \in \left[ 0,T\right] \end{array}}\left| \phi _y(x,y) \right| l\tau c_4c_5. \end{aligned} \end{aligned}$$

(A13)

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. $\square$