H^1-stability of an L2-type method on general nonuniform meshes for subdiffusion equation

1. Introduction

The time-fractional diffusion equation was derived from continuous time random walks [12, 3], where a fractional derivative in time is introduced to model the memory effect in diffusing materials.

In the past decade, many numerical methods have been proposed to solve the time-fractional diffusion equation. Some of these methods are on the uniform time meshes. For example, the L1 scheme of $(2 - α)$ -order has been well-developed by Langlands and Henry [8], Sun-Wu [17], and Lin-Xu [10], etc. Alikhanov [1] proposed the L2-1 $_{σ}$ scheme that has second order accuracy in time. An L2 method of $(3 - α)$ -order on uniform meshes is studied in [2] by Gao-Sun-Zhang. In [11], a slightly different L2 fractional-derivative operator is analyzed by Lv-Xu for uniform meshes, where the optimal convergence $(3 - α)$ -order in time is obtained under strong regularity assumptions on the exact solution.

Recently, those methods on nonuniform time meshes for time-fractional diffusion equation have attracted more and more attention, in particular, on the graded meshes. In fact, the exact solution to the time-fractional diffusion equation could have low regularity in general near the initial time $t = 0$ , which would deteriorate the convergence rate of the numerical solutions. This motivates researchers to consider nonuniform time meshes to obtain the desired $(3 - α)$ -order convergence rate under low regularity assumptions on the exact solution. For example, Stynes-Riordan-Gracia [16] prove the sharp error analysis of L1 scheme on graded meshes. Kopteva provides a different framework for the analysis of the L1 scheme on graded meshes in two and three spatial dimensions in [6], and analyzes an L2 scheme of $(3 - α)$ -order on graded meshes in [7]. Note that in these works, the $L^{\infty}$ -stability of numerical solutions is established.

In addition to the L1 and L2 methods on graded meshes, we mention the convolution quadrature methods with corrections which can also overcome the convergence rate problem for time-fractional diffusion equation, see for example [4, 5] and the references therein.

In this work, we consider the $H^{1}$ -stability of an L2 method (the same as [7]) on general nonuniform meshes for subdiffusion equation. Note that in general the time steps could change freely, unlike the graded meshes controlled by the grading parameter. For the L2 fractional-derivative operator denoted by $L_{k}^{α}$ , we prove that the following bilinear form

(1.1)

B_{n} (v, w) = n \sum k = 1 ⟨ L_{k}^{α} v, δ_{k} w ⟩, δ_{k} w = w^{k} - w^{k - 1}, n \geq 1,

is positive semidefinite under mild restrictions on the time step ratio $ρk\coloneqqτk/τk−1$ with $k \geq 2$ ( $τ_{k}$ is the $k$ th time step), see Theorem 3.2 for details. In particular, if

(1.2)

0.4573328 \leq ρ_{k} \leq 3.5615528,

such positive semidefiniteness holds. As a consequence, the $H^{1}$ -stability of the L2 scheme for the subdiffusion equation with homogeneous Dirichlet boundary condition can be derived for all time, see Theorem 4.1. Moreover, in the special case of graded meshes, we show that if the grading parameter

(1.3)

1 < r \leq 3.2016538,

then $B_{n} (v, w)$ is positive semidefinite and consequently the $H^{1}$ -stability of L2 scheme is established.

This work is organized as follows. In Section 2, the derivation, explicit expression and reformulation of L2 fractional-derivative operator are provided. In Section 3, we prove the positive semidefiniteness of the bilinear form $B_{n}$ under some mild restrictions on the time step ratios. In Section 4, we establish the $H^{1}$ -stability of the L2 scheme for the subdiffusion equation, based on the positive semidefiniteness result. In Section 5, the special case of graded meshes is discussed.

2. Discrete fractional-derivative operator

In this part we show the derivation, explicit expression and reformulation of L2 operator on an arbitrary nonuniform mesh.

We consider the L2 approximation of the fractional-derivative operator defined by

(2.1)

\partial_{t}^{α} u = \frac{1}{Γ (1 - α)} \int_{0}^{t} \frac{u^{'} (s)}{(t - s)^{α}} d s .

Take a nonuniform time mesh $0 = t_{0} < t_{1} < \dots < t_{k - 1} < t_{k} < \dots$ with $k \geq 1$ . When $k = 1$

, we use the standard linear Lagrangian polynomial interpolating

{u^{0}, u^{1}}

(2.2)

H_{1}^{1} (t)

\coloneqqt−t1t0−t1u0+t−t0t1−t0u1.

When $k \geq 2$ , for $1 \leq j \leq k - 1$ , we use the standard quadratic Lagrangian polynomial interpolating ${u^{j - 1}, u^{j}, u^{j + 1}}$ :

(2.3)		$H_{2}^{j} (t)$	$\coloneqq(t−tj)(t−tj+1)(tj−1−tj)(tj−1−tj+1)uj−1+(t−tj−1)(t−tj+1)(tj−tj−1)(tj−tj+1)uj$
(2.3)			$+ \frac{(t - t_{j - 1}) (t - t_{j})}{(t_{j + 1} - t_{j - 1}) (t_{j + 1} - t_{j})} u^{j + 1},$

while for $j = k$ , we use the quadratic Lagrangian polynomial $H_{2}^{k - 1} (t)$ defined in (2.3).

Let $τ_{j} = t_{j} - t_{j - 1}$ . At $t = t_{k}$ , the fractional derivative $\partial_{t}^{α} u (t)$ is approximated by the discrete fractional-derivative operator

(2.4)	$L_{1}^{α} u$	$= \frac{u^{1} - u^{0}}{Γ (2 - α) τ_{1}^{α}},$
	$L_{k}^{α} u$	$= \frac{1}{Γ (1 - α)} (k - 1 \sum j = 1 \int_{t_{j - 1}}^{t_{j}} \frac{\partial_{s} H_{2}^{j} (s)}{(t_{k} - s)^{α}} d s + \int_{t_{k - 1}}^{t_{k}} \frac{\partial_{s} H_{2}^{k - 1} (s)}{(t_{k} - s)^{α}} d s)$
		$= \frac{1}{Γ (1 - α)} (k - 1 \sum j = 1 (a_{j}^{k} u^{j - 1} + b_{j}^{k} u^{j} + c_{j}^{k} u^{j + 1}) + a_{k}^{k} u^{k - 2} + b_{k}^{k} u^{k - 1} + c_{k}^{k} u^{k}), k \geq 2,$

where for $1 \leq j \leq k - 1$ ,

(2.5)	$a_{j}^{k}$	$= \int_{t_{j - 1}}^{t_{j}} \frac{2 s - t_{j} - t_{j + 1}}{τ_{j} (τ_{j} + τ_{j + 1})} \frac{1}{(t_{k} - s)^{α}} d s = \int_{0}^{1} \frac{- 2 τ_{j} (1 - θ) - τ_{j + 1}}{(τ_{j} + τ_{j + 1}) (t_{k} - (t_{j - 1} + θ τ_{j}))^{α}} d θ,$
	$b_{j}^{k}$	$= - \int_{t_{j - 1}}^{t_{j}} \frac{2 s - t_{j - 1} - t_{j + 1}}{τ_{j} τ_{j + 1}} \frac{1}{(t_{k} - s)^{α}} d s = - \int_{0}^{1} \frac{2 τ_{j} θ - τ_{j} - τ_{j + 1}}{τ_{j + 1} (t_{k} - (t_{j - 1} + θ τ_{j}))^{α}} d θ,$
	$c_{j}^{k}$	$= \int_{t_{j - 1}}^{t_{j}} \frac{2 s - t_{j - 1} - t_{j}}{τ_{j + 1} (τ_{j} + τ_{j + 1})} \frac{1}{(t_{k} - s)^{α}} d s = \int_{0}^{1} \frac{τ_{j}^{2} (2 θ - 1)}{τ_{j + 1} (τ_{j} + τ_{j + 1}) (t_{k} - (t_{j - 1} + θ τ_{j}))^{α}} d θ,$

and

(2.6)	$a_{k}^{k}$	$= \int_{t_{k - 1}}^{t_{k}} \frac{2 s - t_{k - 1} - t_{k}}{τ_{k - 1} (τ_{k - 1} + τ_{k})} \frac{1}{(t_{k} - s)^{α}} d s = \int_{0}^{1} \frac{τ_{k}^{2} (2 θ - 1)}{τ_{k - 1} (τ_{k - 1} + τ_{k}) (t_{k} - (t_{k - 1} + θ τ_{k}))^{α}} d θ,$
	$b_{k}^{k}$	$= - \int_{t_{k - 1}}^{t_{k}} \frac{2 s - t_{k - 2} - t_{k}}{τ_{k - 1} τ_{k}} \frac{1}{(t_{k} - s)^{α}} d s = - \int_{0}^{1} \frac{τ_{k} (2 θ - 1) + τ_{k - 1}}{τ_{k - 1} (t_{k} - (t_{k - 1} + θ τ_{k}))^{α}} d θ,$
	$c_{k}^{k}$	$= \int_{t_{k - 1}}^{t_{k}} \frac{2 s - t_{k - 2} - t_{k - 1}}{τ_{k} (τ_{k - 1} + τ_{k})} \frac{1}{(t_{k} - s)^{α}} d s = \int_{0}^{1} \frac{2 τ_{k} θ + τ_{k - 1}}{(τ_{k - 1} + τ_{k}) (t_{k} - (t_{k - 1} + θ τ_{k}))^{α}} d θ .$

It can be verified that $a_{j}^{k} < 0$ , $b_{j}^{k} > 0$ , $c_{j}^{k} > 0$ for $1 \leq j \leq k - 1$ , and $a_{k}^{k} > 0$ , $b_{k}^{k} < 0$ , $c_{k}^{k} > 0$ . Furthermore,

(2.7)

a_{j}^{k} + b_{j}^{k} + c_{j}^{k} = 0,

always holds for $1 \leq j \leq k$ .

Specifically speaking, we can figure out the explicit expressions of $a_{j}^{k}$ and $c_{j}^{k}$ as follows (note that $b_{j}^{k} = - a_{j}^{k} - c_{j}^{k}$ ): for $1 \leq j \leq k - 1$ ,

(2.8)		$a_{j}^{k}$	$= \frac{τ_{j + 1}}{(1 - α) τ_{j} (τ_{j} + τ_{j + 1})} (t_{k} - t_{j})^{1 - α} - \frac{2 τ_{j} + τ_{j + 1}}{(1 - α) τ_{j} (τ_{j} + τ_{j + 1})} (t_{k} - t_{j - 1})^{1 - α}$
(2.8)			$+ \frac{2}{(2 - α) (1 - α) τ_{j} (τ_{j} + τ_{j + 1})} [(t_{k} - t_{j - 1})^{2 - α} - (t_{k} - t_{j})^{2 - α}],$

(2.9)		$c_{j}^{k}$	$= \frac{1}{(1 - α) τ_{j + 1} (τ_{j} + τ_{j + 1})} [- τ_{j} ((t_{k} - t_{j - 1})^{1 - α} + (t_{k} - t_{j})^{1 - α})$
(2.9)			$+ 2 (2 - α)^{- 1} ((t_{k} - t_{j - 1})^{2 - α} - (t_{k} - t_{j})^{2 - α})],$

while for $j = k$ ,

(2.10)

a_{k}^{k} = \frac{α τ_{k}^{2}}{(2 - α) (1 - α) τ_{k - 1} (τ_{k - 1} + τ_{k}) τ_{k}^{α}},

(2.11)

c_{k}^{k}

= \frac{1}{(1 - α) τ_{k}^{α}} + \frac{α τ_{k}}{(2 - α) (1 - α) (τ_{k - 1} + τ_{k}) τ_{k}^{α}} .

We reformulate the discrete fractional derivative $L_{k}^{α}$ in (2.4) as

(2.12)	$L_{1}^{α} u$	$= \frac{1}{Γ (2 - α) τ_{1}^{α}} δ_{1} u,$
	$L_{k}^{α} u$	$= \frac{1}{Γ (1 - α)} (k - 1 \sum j = 1 (a_{j}^{k} u^{j - 1} + b_{j}^{k} u^{j} + c_{j}^{k} u^{j + 1}) + a_{k}^{k} u^{k - 2} + b_{k}^{k} u^{k - 1} + c_{k}^{k} u^{k})$
		$= \frac{1}{Γ (1 - α)} (k - 1 \sum j = 1 (c_{j}^{k} δ_{j + 1} u - a_{j}^{k} δ_{j} u) + c_{k}^{k} δ_{k} u - a_{k}^{k} δ_{k - 1} u)$
		$= \frac{1}{Γ (1 - α)} ((c_{k}^{k} + c_{k - 1}^{k}) δ_{k} u - a_{k}^{k} δ_{k - 1} u - a_{1}^{k} δ_{1} u + k - 1 \sum j = 2 d_{j}^{k} δ_{j} u), k \geq 2,$

where

(2.13)		$δ_{j} u$	$= u^{j} - u^{j - 1},$
(2.13)		$d_{j}^{k}$	$\coloneqqckj−1−akj.$

To establish the $H^{1}$ -stability of L2-type method for fractional-order parabolic problem, we want to prove the positive semidefiniteness of the following bilinear form

(2.14)

Bn(v,w)\coloneqqn∑k=1⟨Lαkv,δkw⟩,

where $⟨ \cdot, \cdot ⟩$ is the standard $L^{2}$ inner product over $Ω$ and $v, w$ are two functions defined on $[0, \infty) \times Ω$ and $n \geq 1$ . That is to say, we want to prove that for any $v$ defined on $[0, \infty) \times Ω$ and $n \geq 1$ ,

(2.15)

B_{n} (v, v) \geq 0.

3. Positive semidefiniteness of bilinear form $B_{n}$

3.1. Properties of L2 coefficients

We propose the following properties of the L2 coefficients $a_{j}^{k}$ , $c_{j}^{k}$ and $d_{j}^{k}$ in (2.12), which will be useful to establish the positive semidefiniteness of bilinear form $B_{n}$ .

Lemma 3.1 (Properties of $a_{j}^{k}$ , $c_{j}^{k}$ and $d_{j}^{k}$ ).

For the L2 coefficients given in (2.5) and (2.13), given a nonuniform mesh ${τ_{j}}_{j \geq 1}$ , the following properties hold:

$a_{j}^{k} < 0, 1 \leq j \leq k - 1, k \geq 2$ ;
$a_{j}^{k + 1} - a_{j}^{k} > 0, 1 \leq j \leq k - 1, k \geq 2$ ;
$a_{j + 1}^{k} - a_{j}^{k} < 0, 1 \leq j \leq k - 2, k \geq 3$ ;
$a_{j + 1}^{k} - a_{j}^{k} < a_{j + 1}^{k + 1} - a_{j}^{k + 1}, 1 \leq j \leq k - 2, k \geq 3$ ;
$c_{j}^{k} > 0, 1 \leq j \leq k - 1, k \geq 2$ ;
$c_{j}^{k + 1} - c_{j}^{k} < 0, 1 \leq j \leq k - 1, k \geq 2$ ;
$d_{j}^{k} > 0, 2 \leq j \leq k - 1, k \geq 3$ ;
$d_{j}^{k + 1} - d_{j}^{k} < 0, 2 \leq j \leq k - 1, k \geq 3$ .

Furthermore, if the nonuniform mesh ${τ_{j}}_{j \geq 1}$ , with $ρj\coloneqqτj/τj−1$ satisfies

(3.1)

\frac{1}{ρ_{j + 1}} \geq \frac{1}{ρ_{j}^{2} (1 + ρ_{j})} - 3, \forall j \geq 2,

then the following properties of $d_{j}^{k}$ hold:

$d_{j + 1}^{k} - d_{j}^{k} > 0, 2 \leq j \leq k - 2, k \geq 4$ ;
$d_{j + 1}^{k} - d_{j}^{k} > d_{j + 1}^{k + 1} - d_{j}^{k + 1}, 2 \leq j \leq k - 2, k \geq 4$ .

Proof.

We first provide two equivalent forms of $a_{j}^{k}$ in (2.5) as follows:

(3.2)	$a_{j}^{k}$	$= \int_{0}^{1} \frac{- 2 τ_{j} (1 - s) - τ_{j + 1}}{(τ_{j} + τ_{j + 1}) (t_{k} - (t_{j - 1} + s τ_{j}))^{α}} d s$
		$= \frac{1}{τ_{j} + τ_{j + 1}} \int_{0}^{1} (t_{k} - (t_{j - 1} + s τ_{j}))^{- α} d (- τ_{j} s^{2} - (2 τ_{j} + τ_{j + 1}) s)$
		$= - (t_{k} - t_{j})^{- α} + \frac{α τ_{j}}{τ_{j} + τ_{j + 1}} \int_{0}^{1} s (2 τ_{j} + τ_{j + 1} - s τ_{j}) (t_{k} - t_{j - 1} - s τ_{j})^{- α - 1} d s$
		$= - (t_{k} - t_{j})^{- α} + \frac{α τ_{j}}{τ_{j} + τ_{j + 1}} \int_{0}^{1} (τ_{j} + τ_{j + 1} + s τ_{j}) (1 - s) (t_{k} - t_{j} + s τ_{j})^{- α - 1} d s$

and

(3.3)	$a_{j}^{k}$	$= \int_{0}^{1} \frac{- 2 τ_{j} (1 - s) - τ_{j + 1}}{(τ_{j} + τ_{j + 1}) (t_{k} - (t_{j - 1} + (1 - s) τ_{j}))^{α}} d s$
		$= \int_{0}^{1} \frac{- 2 τ_{j} s - τ_{j + 1}}{(τ_{j} + τ_{j + 1}) (t_{k} - t_{j} + s τ_{j})^{α}} d s$
		$= \frac{1}{τ_{j} + τ_{j + 1}} \int_{0}^{1} (t_{k} - t_{j} + s τ_{j})^{- α} d (- τ_{j} s^{2} - τ_{j + 1} s)$
		$= - (t_{k} - t_{j - 1})^{- α} - \frac{α τ_{j}}{τ_{j} + τ_{j + 1}} \int_{0}^{1} s (s τ_{j} + τ_{j + 1}) (t_{k} - t_{j} + s τ_{j})^{- α - 1} d s$
		$= - (t_{k} - t_{j - 1})^{- α} - \frac{α τ_{j}}{τ_{j} + τ_{j + 1}} \int_{0}^{1} (τ_{j} + τ_{j + 1} - s τ_{j}) (1 - s) (t_{k} - t_{j - 1} - s τ_{j})^{- α - 1} d s .$

It is easy to see $a_{j}^{k} < 0$ , i.e., (P1) holds.

Combining (3.2) and (3.3), we have

(3.4)		$a_{j + 1}^{k} - a_{j}^{k}$
	$=$	$- \frac{α τ_{j}}{τ_{j} + τ_{j + 1}} \int_{0}^{1} (τ_{j} + τ_{j + 1} + s τ_{j}) (1 - s) (t_{k} - t_{j} + s τ_{j})^{- α - 1} d s$
		$- \frac{α τ_{j + 1}}{τ_{j + 1} + τ_{j + 2}} \int_{0}^{1} (τ_{j + 1} + τ_{j + 2} - s τ_{j + 1}) (1 - s) (t_{k} - t_{j} - s τ_{j + 1})^{- α - 1} d s$
	$<$	$0,$

where we use the form (3.2) for $a_{j}^{k}$ and the form (3.3) for $a_{j + 1}^{k}$ . Therefore (P3) holds. Moreover, for any fixed $s$ ,

(3.5)

(t_{k} - t_{j - 1})^{- α}, (t_{k} - t_{j - 1} - s τ_{j})^{- α - 1}, (t_{k} - t_{j} + s τ_{j})^{- α - 1} and (t_{k} - t_{j} - s τ_{j + 1})^{- α - 1}

decrease w.r.t. $k$ . As a consequence, (3.3) and (3.4) result in

(3.6)			$a_{j}^{k + 1} - a_{j}^{k} > 0,$
(3.6)			$(a_{j + 1}^{k + 1} - a_{j}^{k + 1}) - (a_{j + 1}^{k} - a_{j}^{k}) > 0,$

i.e., the properties (P2) and (P4) hold.

We now turn to prove the properties of $c_{j}^{k}$ and $d_{j}^{k} = c_{j - 1}^{k} - a_{j}^{k}$ . For $c_{j}^{k}$ in (2.5), we have

(3.7)	$c_{j}^{k}$	$= \int_{0}^{1} \frac{τ_{j}^{2} (2 s - 1)}{τ_{j + 1} (τ_{j} + τ_{j + 1}) (t_{k} - (t_{j - 1} + s τ_{j}))^{α}} d s$
		$= \frac{τ_{j}^{2}}{τ_{j + 1} (τ_{j} + τ_{j + 1})} \int_{0}^{1} (t_{k} - (t_{j - 1} + s τ_{j}))^{- α} d (s^{2} - s)$
		$= \frac{α τ_{j}^{3}}{τ_{j + 1} (τ_{j} + τ_{j + 1})} \int_{0}^{1} s (1 - s) (t_{k} - t_{j} + s τ_{j})^{- α - 1} d s$
		$> 0.$

This is the property (P5). Since $a_{j}^{k} < 0$ , we have $d_{j}^{k} = c_{j - 1}^{k} - a_{j}^{k} > 0$ for $j \geq 2$ and the property (P7) holds. For any fixed $s$ ,

(3.8)

(t_{k} - t_{j} + s τ_{j})^{- α - 1}

decreases w.r.t. $k$ , implying that

(3.9)

c_{j}^{k + 1} - c_{j}^{k} < 0,

i.e., the property (P6). Combining this with property (P2), the property (P8) holds.

We now prove the property (P9). Combining (3.4) and (3.7) gives

(3.10)		$d_{j + 1}^{k} - d_{j}^{k} = (c_{j}^{k} - c_{j - 1}^{k}) - (a_{j + 1}^{k} - a_{j}^{k})$
	$=$	$\frac{α τ_{j}^{3}}{τ_{j + 1} (τ_{j} + τ_{j + 1})} \int_{0}^{1} s (1 - s) (t_{k} - t_{j} + s τ_{j})^{- α - 1} d s$
		$- \frac{α τ_{j - 1}^{3}}{τ_{j} (τ_{j - 1} + τ_{j})} \int_{0}^{1} s (1 - s) (t_{k} - t_{j - 1} + s τ_{j - 1})^{- α - 1} d s$
		$+ \frac{α τ_{j}}{τ_{j} + τ_{j + 1}} \int_{0}^{1} (τ_{j} + τ_{j + 1} + s τ_{j}) (1 - s) (t_{k} - t_{j} + s τ_{j})^{- α - 1} d s$
		$+ \frac{α τ_{j + 1}}{τ_{j + 1} + τ_{j + 2}} \int_{0}^{1} (τ_{j + 1} + τ_{j + 2} - s τ_{j + 1}) (1 - s) (t_{k} - t_{j} - s τ_{j + 1})^{- α - 1} d s .$

Note that for any fixed $j$ ,

(3.11)

(t_{k} - t_{j} + s τ_{j})^{- α - 1} \geq 0

decreases w.r.t. $s$ , and

(3.12)

\int_{0}^{1} (1 - 3 s) (1 - s) = 0,

which imply

(3.13)

(τ_{j} + τ_{j + 1}) \int_{0}^{1} (1 - 3 s) (1 - s) (t_{k} - t_{j} + s τ_{j})^{- α - 1} d s \geq 0,

i.e.,

(3.14)			$\int_{0}^{1} (τ_{j} + τ_{j + 1} + s τ_{j}) (1 - s) (t_{k} - t_{j} + s τ_{j})^{- α - 1} d s$
(3.14)			$\geq \int_{0}^{1} (4 τ_{j} + 3 τ_{j + 1}) s (1 - s) (t_{k} - t_{j} + s τ_{j})^{- α - 1} d s .$

Using (3.14) and the fact

(3.15)

(t_{k} - t_{j} + s τ_{j})^{- α - 1}

> (t_{k} - t_{j - 1} + s τ_{j - 1})^{- α - 1},

we can derive from (3.10) that

(3.16)		$d_{j + 1}^{k} - d_{j}^{k} >$	$α (\frac{τ_{j}^{3}}{τ_{j + 1} (τ_{j} + τ_{j + 1})} - \frac{τ_{j - 1}^{3}}{τ_{j} (τ_{j - 1} + τ_{j})}$
(3.16)			$+ \frac{(4 τ_{j} + 3 τ_{j + 1}) τ_{j}}{τ_{j} + τ_{j + 1}}) \int_{0}^{1} s (1 - s) (t_{k} - t_{j} + s τ_{j})^{- α - 1} d s .$

To ensure the property (P9), we need

(3.17)			$\frac{τ_{j}^{3}}{τ_{j + 1} (τ_{j} + τ_{j + 1})} - \frac{τ_{j - 1}^{3}}{τ_{j} (τ_{j - 1} + τ_{j})} + \frac{(4 τ_{j} + 3 τ_{j + 1}) τ_{j}}{τ_{j} + τ_{j + 1}} \geq 0,$
			$⟺ \frac{1}{ρ_{j + 1} (1 + ρ_{j + 1})} - \frac{1}{ρ_{j}^{2} (1 + ρ_{j})} + \frac{4 + 3 ρ_{j + 1}}{1 + ρ_{j + 1}} \geq 0,$
			$⟺ \frac{1}{ρ_{j + 1}} \geq \frac{1}{ρ_{j}^{2} (1 + ρ_{j})} - 3,$

that is the condition (3.1).

We now prove the last property (P10). From (3.10), we can derive that

(3.18)		$(d_{j + 1}^{k} - d_{j}^{k}) - (d_{j + 1}^{k + 1} - d_{j}^{k + 1})$
	$=$	$\frac{α τ_{j}^{3}}{τ_{j + 1} (τ_{j} + τ_{j + 1})} \int_{0}^{1} s (1 - s) [(t_{k} - t_{j} + s τ_{j})^{- α - 1} - (t_{k + 1} - t_{j} + s τ_{j})^{- α - 1}] d s$
		$- \frac{α τ_{j - 1}^{3}}{τ_{j} (τ_{j - 1} + τ_{j})} \int_{0}^{1} s (1 - s) [(t_{k} - t_{j - 1} + s τ_{j - 1})^{- α - 1} - (t_{k + 1} - t_{j - 1} + s τ_{j - 1})^{- α - 1}] d s$
		$+ \frac{α τ_{j}}{τ_{j} + τ_{j + 1}} \int_{0}^{1} (τ_{j} + τ_{j + 1} + s τ_{j}) (1 - s) [(t_{k} - t_{j} + s τ_{j})^{- α - 1} - (t_{k + 1} - t_{j} + s τ_{j})^{- α - 1}] d s$
		$+ \frac{α τ_{j + 1}}{τ_{j + 1} + τ_{j + 2}} \int_{0}^{1} (τ_{j + 1} + τ_{j + 2} - s τ_{j + 1}) (1 - s) [(t_{k} - t_{j} - s τ_{j + 1})^{- α - 1}$
		$- (t_{k + 1} - t_{j} - s τ_{j + 1})^{- α - 1}] d s .$

The convexity of the function $t^{- 1 - α}$ gives

(3.19)			$(t_{k} - t_{j} + s τ_{j})^{- α - 1} - (t_{k + 1} - t_{j} + s τ_{j})^{- α - 1}$
(3.19)			$> (t_{k} - t_{j - 1} + s τ_{j - 1})^{- α - 1} - (t_{k + 1} - t_{j - 1} + s τ_{j - 1})^{- α - 1},$

and for fixed $j$ , it is easy to see that

(3.20)

(t_{k} - t_{j} + s τ_{j})^{- α - 1} - (t_{k + 1} - t_{j} + s τ_{j})^{- α - 1} > 0

decreases w.r.t. $s$ . Then we can get the following result similar to (3.16):

(3.21)

(d_{j + 1}^{k} - d^{k}

H^1-stability of an L2-type method on general nonuniform meshes for subdiffusion equation

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Highly efficient and energy dissipative schemes for the time fractional Allen-Cahn equation

A sharp α-robust L1 scheme on graded meshes for two-dimensional time tempered fractional Fokker-Planck equation

gPAV-Based Unconditionally Energy-Stable Schemes for the Cahn-Hilliard Equation: Stability and Error Analysis

Highly efficient and accurate schemes for time fractional Allen-Cahn equation by using extended SAV approach

Backward difference formula: The energy technique for subdiffusion equation

On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

1. Introduction

2. Discrete fractional-derivative operator

3. Positive semidefiniteness of bilinear form $B_{n}$

3.1. Properties of L2 coefficients

Lemma 3.1 (Properties of $a_{j}^{k}$ , $c_{j}^{k}$ and $d_{j}^{k}$ ).

Proof.

H^1-stability of an L2-type method on general nonuniform meshes for subdiffusion equation

Related Research

On stability and convergence of L2-1_σ method on general nonuniform meshes for subdiffusion equation

Highly efficient and energy dissipative schemes for the time fractional Allen-Cahn equation

A sharp α-robust L1 scheme on graded meshes for two-dimensional time tempered fractional Fokker-Planck equation

gPAV-Based Unconditionally Energy-Stable Schemes for the Cahn-Hilliard Equation: Stability and Error Analysis

Highly efficient and accurate schemes for time fractional Allen-Cahn equation by using extended SAV approach

Backward difference formula: The energy technique for subdiffusion equation

On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

1. Introduction

2. Discrete fractional-derivative operator

3. Positive semidefiniteness of bilinear form Bn

3.1. Properties of L2 coefficients

Lemma 3.1 (Properties of akj, ckj and dkj).

Proof.

3. Positive semidefiniteness of bilinear form $B_{n}$

Lemma 3.1 (Properties of $a_{j}^{k}$ , $c_{j}^{k}$ and $d_{j}^{k}$ ).