Unconditionally optimal H¹-error estimate of a fast nonuniform L2-1_σ scheme for nonlinear subdiffusion equations

Abstract

This paper is concerned with the unconditionally optimal H¹-error estimate of a fast second-order scheme for solving nonlinear subdiffusion equations on the nonuniform mesh. We use the Galerkin finite element method (FEM) to discretize the spacial direction, the Newton linearization method to approximate the nonlinear term and the sum-of-exponentials (SOE) approximation to speed up the evaluation of Caputo derivative. Our analysis of the unconditionally optimal H¹-error estimate involves the temporal-spatial error splitting approach, an improved discrete fractional Grönwall inequality and error convolution structure. In order to find a suitable test function to estimate H¹-error, we here consider two cases: linear and high-order FEM space, using the time-discrete operator and Laplace operator in the test function respectively. Numerical tests are provided demonstrate the effectiveness and the unconditionally optimal H¹-error convergence of our scheme.

1 Introduction

The subdiffusion equations are widely used to describe various phenomena of anomalous diffusion in control systems, physics, biology [1,2,3], and attract lots of researchers in theoretical and numerical analysis. This paper focuses on the unconditionally optimal H¹-error estimate of a fully discrete scheme for the following nonlinear subdiffusion problem on a bounded convex domain ${\Omega }\subset \mathbb {R}^{d}(d=1,2,3)$:

$$ \begin{array}{@{}rcl@{}} {~}^{\text{C}}\mathcal{D}^{\alpha}_t u(\textbf{x},t)&=&{\Delta} u(\textbf{x},t) + f(u),\qquad\textbf{x}\in{\Omega}, t\in(0,T], \end{array} $$

(1.1)

$$ \begin{array}{@{}rcl@{}} u(\textbf{x},0)&=&u_{0}(\textbf{x}), \qquad \qquad {\kern25pt}\textbf{x}\in\bar{\Omega}, \end{array} $$

(1.2)

$$ \begin{array}{@{}rcl@{}} u(\textbf{x},t)&=&0,\qquad \qquad \qquad{\kern22pt}\textbf{x}\in\partial{\Omega}, t\in[0,T], \end{array} $$

(1.3)

where the Caputo derivative ${~}^{\text {C}}\mathcal {D}^{\alpha }_t$ (0 < α < 1) acting on u is defined as

$$ {~}^{\text{C}}\mathcal{D}^{\alpha}_tu(\textbf{x},t)={{\int}_{0}^{t}}\partial_{s}u(\textbf{x},s)\omega_{1-\alpha}(t-s) \mathrm{d} s \quad \text{ with } \quad \omega_{\beta}(t)={t^{\beta-1}}/{\Gamma(\beta)}. $$

(1.4)

There are three features for problem (1.1)–(1.3):

• the solution u has the initial time singularity;

• the Caputo derivative is nonlocal;

• the problem is nonlinear.

The first feature of problem (1.1)–(1.3) is ubiquitous in nature that the solution u is weakly singular near the initial time t = 0. Generally, the regularity of the solution has the following property [4,5,6,7,8,9,10]:

$$ \|{\partial^{m}_{t}} u(t)\|_{H^{2}({\Omega})}\leq {C}t^{\alpha-m},\quad \text{ for } m = 1,2. $$

(1.5)

We point out that here C generally means a constant, which is independent of mesh sizes h and τ, but it may depend on the given data (such as f, u₀, α, Ω, T). Thus, the initial regularity will become an important consideration for any numerical method to solve the subdiffusion problems. To achieve the optimal convergence rate, the nonuniform/adaptive time step is required, which also brings more complicated and difficult theoretical analysis of numerical schemes comparing with the uniform mesh. A typical nonuniform mesh is the following general graded mesh

$$ \tau_{k}\!\leq\! C_{\gamma}\tau t_{k}^{1 - 1/\gamma} (1\!\leq\! k\!\leq\! N),\quad t_{k}\!\leq \!C_{\gamma} t_{k - 1} \text{and} \tau_{k}/t_{k}\!\leq\! C_{\gamma} \tau_{k - 1} /t_{k - 1} (2\leq k\leq N), $$

(1.6)

where N represents the total number of time steps, γ ≥ 1 is a parameter and $\tau =\max \limits _{1\leq k\leq N} \tau _{k}$ is the maximum step size. The general graded mesh has been successfully applied to various time fractional PDEs, see [8,9,10,11,12,13,14,15,16,17,18]. For instance, Liao et al. [8] obtain the optimal $\mathcal {O}\big (\tau ^{\min \limits \{\gamma \alpha , 2-\alpha \}}\big )$ convergence rate of L1 scheme on the nonuniform mesh. In which, a theoretical framework is proposed by presenting a discrete complementary convolution (DCC) kernel, a discrete fractional Grönwall inequality and an error convolution structure (ECS). Based on this framework, the optimal convergence order of several widely used numerical schemes on the nonuniform mesh is obtained successively, such as the optimal $\mathcal {O}\big (\tau ^{\min \limits \{\gamma \alpha , 2\}}\big )$ convergence rate of L2-1_σ scheme in L²-norm [9] and the optimal $\mathcal {O}\big (\tau ^{\min \limits \{\gamma \alpha , 2-\alpha \}}\big )$ convergence rate of the two-level fast L1 scheme in L²-norm [13].

The second nonlocal feature will lead to the huge computational storage and cost for the long-time or small-mesh simulations, which is especially prohibitive to compute the high-dimensional problem. To circumvent this difficulty of computational complexity, there are generally two alternatives: one is to introduce the fast algorithms to significantly reduce the computational storage and cost; another is to use the high-order schemes to obtain the same accuracy with less time steps. For the fast algorithms, one can refer to [19,20,21,22,23,24,25,26]. For instance, Jiang et al. [20] present the sum-of-exponentials (SOE) approximation to speed up the efficient evaluation of Caputo derivative, which significantly reduces the computational storage and cost. Late, Yan et al. [21] apply the idea of SOE approximate to the second-order L2-1_σ scheme. Baffet et al. [22] combine the kernel compression scheme with a time stepping method. Zhu et al. [23] present a fast L2 scheme with (3-α)-order with the application of SOE approximation. Guo et al. [24] apply the fractional linear multistep method to deal with the tempered fractional integral and derivative operators. For the high-order schemes, one can refer to [27,28,29,30] for the widely used L2 scheme [28] and L2-1_σ scheme [29]. In this paper, we will use the L2-1_σ scheme with the corresponding fast algorithm presented in [20, 21] to speed up the computation of the Caputo derivative on general graded mesh (1.6).

The third feature involves the nonlinearity of the problem itself. The typical methods to numerically deal with the nonlinearity involves pure explicit scheme, fully implicit scheme, and implicit-explicit (or semi-implicit) scheme and so on. The pure explicit scheme is the most easy implementation, but suffers from a CFL condition for the stability. The fully implicit scheme is generally unconditionally stable, but needs extra computational cost for iteratively solving a nonlinear algebraic system. A popular alternative is semi-implicit scheme which discretizes the linear term by implicit scheme and the nonlinear term by a linearized or an explicit scheme. The resulting semi-implicit scheme circumvents the iteration for the fully implicit scheme, but also brings the difficulty of theoretical analysis of the unconditional convergence. The so-called unconditional convergence here means the optimal convergence does not suffer from any restrictions of ratios between temporal and spacial mesh sizes. In this paper, we use the implicit scheme to discretize the linear dispersive term and use the Newton linearization method to approximate the nonlinear term for the time direction, and employ the Galerkin FEM for the spacial direction.

The focus of this paper is on the unconditionally optimal H¹-error estimate for the proposed second order fast scheme to numerically solve problem (1.1) on the nonuniform mesh. To do so, we first carefully use the SOE-based fast L2-1_σ approximation of Caputo derivative [20, 21], which significantly reduce the computational complexity when the total number of the time step is large enough. After that, we use the spatial-temporal splitting approach introduced in [31, 32] to prove our linearization scheme is unconditionally convergent. The idea of the spatial-temporal splitting approach is beginning with proving the boundedness of the solution to the temporal semi-discrete scheme in the infinity norm, and then prove the optimal error estimate of the fully discrete scheme, which successfully evade the ratio between time and space mesh sizes. In the proof process, we consider u is smooth enough in spatial directions. Combined with the original (1.1), it further implies that Δu is zero on ∂Ω. Finally, we use the theoretical framework developed in [8,9,10] to present the error estimate for subdiffusion equations, which involves the discrete time fractional Grönwall inequality, DCC kernel and ECS. This framework can effectively deal with the nonuniform temporal scheme when the initial regularity is considered. Specially for H¹-error estimate, we divide the FEM space into two cases of linear and high-order. Namely, when r = 1 (here r represents the degree of continuous piecewise polynomials), the time-discrete operator is taken in the test function like [33, 34]; when r ≥ 2, the Laplace operator is taken in the test function.

The paper is organized as follows. In Section 2, we introduce the fast L2-1_σ scheme and fully discrete scheme. In Section 3, we first give some necessary lemmas, then split the error into spatial and temporal components for detailed analysis respectively, and present the unconditionally optimal H¹-error estimate. In Section 4, some numerical results are provided to verify our theoretical analysis.

2 Fast L2-1_σ and fully discrete schemes

We take the general nonuniform by 0 = t₀ < t₁ < t₂ < … < t_N = T with time steps τ_k = t_k − t_k− 1. Set t_k−σ = (1 − σ)t_k + σt_k− 1, σ ∈ [0,1] and the step ratios ρ_k = τ_k/τ_k+ 1. There is a constant ρ > 0 such that the step ratios ρ_k < ρ for 1 ≤ k ≤ N − 1. Set u^k = u(⋅,t_k), u^k−σ = (1 − σ)u^k + σu^k− 1 and the difference operator ∇_τu^k = u^k − u^k− 1 for 1 ≤ k ≤ N. Taking $\sigma = \frac \alpha 2$ here and after, the L2-1_σ scheme is defined by

$$ \begin{array}{@{}rcl@{}} {~}^{\text{C}}\mathcal{D}^{\alpha}_{\tau} u^{n-\sigma} &=&\sum\limits_{k=1}^{n-1}{\int}_{t_{k-1}}^{t_{k}}\partial_{s}\big({\Pi}_{2,k}u(x,s)\big)\omega_{1-\alpha}(t_{n-\sigma}-s) \mathrm{d} s+{\int}_{t_{n-1}}^{t_{n-\sigma}}\partial_{s}\big({\Pi}_{1,n}u(x,s)\big)\omega_{1-\alpha}(t_{n-\sigma}-s) \mathrm{d} s\\ & :=&\sum\limits_{k=1}^{n}A_{n-k}^{(n)}\nabla_{\tau} u^{k}, \end{array} $$

(2.7)

where π_1,ku(x,s) and π_2,ku(x,s) represent the linear interpolation with the nodes t_k− 1, t_k and the quadratic interpolation at t_k, t_k− 1, t_k+ 1 for the variable s, and the discrete convolution kernel $A_{n-k}^{(n)}$ can be calculated by

$$ A_{n-k}^{(n)}=\begin{cases} a_{0}^{(n)}+\rho_{n-1}b_{1}^{(n)},& \ k=n,\\ a_{n-k}^{(n)}+\rho_{k-1}b_{n-k+1}^{(n)}-b_{n-k}^{(n)},& \ 2\leq k\leq n-1,\\ a_{n-1}^{(n)}-b_{n-1}^{(n)},& \ k=1, \end{cases} $$

(2.8)

with

$$ \begin{array}{@{}rcl@{}} a_{0}^{(n)}&=&\frac{1}{\tau_{n}}{\int}_{t_{n-1}}^{t_{n-\sigma}}\omega_{1-\alpha}(t_{n-\sigma}-s) \mathrm{d} s,\quad a_{n-k}^{(n)}=\frac{1}{\tau_{k}}{\int}_{t_{k-1}}^{t_{k}}\omega_{1-\alpha}(t_{n-\sigma}-s) \mathrm{d} s,\\ b_{n-k}^{(n)}&=&\frac{2}{\tau_{k}(\tau_{k}+\tau_{k+1})}{\int}_{t_{k-1}}^{t_{k}}(s-t_{k-\frac12})\omega_{1-\alpha}(t_{n-\sigma}-s) \mathrm{d} s. \end{array} $$

It is known that the direct algorithm of the L2-1_σ scheme (2.7) has the computational complexity with the storage $\mathcal {O}(N)$ and cost $\mathcal {O}(N^{2})$, respectively. This computational complexity will be huge and inadmissible for small time size or long time simulations. It motivates us to consider a fast algorithm of L2-1_σ scheme based on the SOE technique developed in [20, 21] to approximate the kernel t^−α. The resulting fast L2-1_σ scheme only has the complexity of the storage $\mathcal {O}(\log ^{2} N)$ and cost $\mathcal {O}(N \log ^{2} N)$, which significantly reduces the computational complexity for large N. The main methodology of the fast algorithm in [20, 21] is presented as follows.

We first split ${~}^{\text {C}}\mathcal {D}^{\alpha }_t u^{n-\sigma }$ into a local part I and a history part I I, say

$$ {~}^{\text{C}}\mathcal{D}^{\alpha}_t u^{n-\sigma}={\int}_{t_{n-1}}^{t_{n-\sigma}}\partial_{s}u(x,s)\omega_{1-\alpha}(t_{n-\sigma}-s) \mathrm{d} s+ {\int}_{0}^{t_{n-1}}\partial_{s}u(x,s)\omega_{1-\alpha}(t_{n-\sigma}-s) \mathrm{d} s =I + I\!I. $$

(2.9)

The local part I is directly approximated by

$$ I\approx{\int}_{t_{n-1}}^{t_{n-\sigma}}\partial_{s}\big({\Pi}_{1,n}u(x,s)\big)\omega_{1-\alpha}(t_{n-\sigma}-s) \mathrm{d} s =\frac{\nabla_{\tau} u^{n}}{\tau_{n}}{\int}_{t_{n-1}}^{t_{n-\sigma}}\omega_{1-\alpha}(t_{n-\sigma}-s) \mathrm{d} s := a_{0}^{(n)} \nabla_{\tau} u^{n}. $$

(2.10)

To speed up the evaluation of I I, we use the following SOEs approximation for the kernel t^−α.

Lemma 2.1 (20, 21)

For the given parameters α,𝜖,δ and T, one can find a family of points s_i and weights $\omega _{i}\ ; (i=1,2,\ldots ,\mathcal {P}$) such that

$$ \Big |t^{-\alpha}-\sum\limits_{i=1}^{\mathcal{P}}\omega_{i}e^{-s_{i}t}\Big |\leq\epsilon,\quad \forall t\in[\delta,T], $$

(2.11)

where the total number $\mathcal {P}$ of exponentials needed is of order

$$ \mathcal{P}=\mathcal{O}\Big(\big(\log (T/\delta) + \log(\log\epsilon^{-1}) \big)\log\epsilon^{-1}+\big(\log\delta^{-1}+\log(\log\epsilon^{-1})\big)\log\delta^{-1}\Big). $$

(2.12)

Remark 1

In the practical simulations, we generally fix the tolerance error 𝜖 as the machine precision. Once fixing 𝜖, taking $\delta =\min \limits _{1\leq k\leq N}\tau _{k}$ and noting $N = \mathcal {O}(T/\tau )$ in (2.12), we have $\mathcal {P} = \mathcal {O}(\log N)$ for T ≫ 1 and $\mathcal {P} = \mathcal {O}(\log ^{2} N)$ for $T = \mathcal {O}(1)$.

By using the SOEs approximation of t^−α in (2.11), the history part I I can be written as

$$ I\!I\approx \frac{1}{\Gamma(1-\alpha)}\sum\limits_{i=1}^{\mathcal{P}}{\int}^{t_{n-1}}_{0} \partial_{s}\big({\Pi}_{2,n-1}u(x,s)\big)\omega_{i}e^{-s_{i}(t_{n-\sigma}-s)} \mathrm{d} s :=\sum\limits_{i=1}^{\mathcal{P}}H_{i}(t_{n-1}) $$

(2.13)

with the history integral H_i(t₀) = 0 and the following recurrence formula

$$ H_{i}(t_{n-1}) = e^{-s_{i}\tau_{n-\sigma}}H_{i}(t_{n-2})+\frac{1}{\Gamma(1-\alpha)}{\int}_{t_{n-2}}^{t_{n-1}}\partial_{s}\big({\Pi}_{2,n-1}u(x,s)\big)\omega_{i}e^{-s_{i}(t_{n-\sigma}-s)} \mathrm{d} s . $$

(2.14)

Combining (2.9), (2.10) and (2.13), we have the fast L2-1_σ scheme given as

$$ {~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} u^{n-\sigma}=a_{0}^{(n)} \nabla_{\tau} u^{n}+\sum\limits_{i=1}^{\mathcal{P}}H_{i}(t_{n-1}), $$

(2.15)

where H_i(t_n− 1) can be calculated by the recurrence formula (2.14). For further theoretical analysis, we can equivalently reformulate (2.15) into the following convolution form

$$ {~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} u^{n-\sigma} = \sum\limits_{k=1}^{n} B_{n-k}^{(n)}\nabla_{\tau} u^{k}, $$

(2.16)

where

$$ B_{n-k}^{(n)}=\left\{\begin{array}{ll} a_{0}^{(n)}+\sum\limits_{i=1}^{\mathcal{P}}\rho_{n-1}\tilde{b}_{1}^{(n)},& \ k=n,\\ \sum\limits_{i=1}^{\mathcal{P}}e^{-s_{i}(t_{n-\sigma}-t_{k+1-\sigma})} (\tilde{a}_{n-k}^{(k+1)}+e^{-s_{i}\tau_{k+1-\sigma}}\rho_{k-1}\tilde{b}_{n-k+1}^{(k)}-\tilde{b}_{n-k}^{(k+1)}),& \ 2\leq k\leq n-1,\\ \sum\limits_{i=1}^{\mathcal{P}}e^{-s_{i}(t_{n-\sigma}-t_{2-\sigma})}(\tilde{a}_{n-1}^{(2)}-\tilde{b}_{n-1}^{(2)}),& \ k=1, \end{array}\right. $$

(2.17)

where

$$ \begin{array}{@{}rcl@{}} \tilde{a}_{n-k}^{(k+1)}&=&\frac{\omega_{i}}{\tau_{k}{\Gamma}(1-\alpha)}{\int}_{t_{k-1}}^{t_{k}}e^{-s_{i}(t_{k+1-\sigma}-s)} \mathrm{d} s, \\ \tilde{b}_{n-k}^{(k+1)}&=&\frac{2\omega_{i}}{\tau_{k}(\tau_{k}+\tau_{k+1}){\Gamma}(1-\alpha)}{\int}_{t_{k-1}}^{t_{k}}(s-t_{k-\frac12})e^{-s_{i}(t_{k+1-\sigma}-s)} \mathrm{d} s. \end{array} $$

The discrete convolution kernel $B_{n-k}^{(n)}$ has the following properties [35]:

• $B_{n-k-1}^{(n)}-B_{n-k}^{(n)}>0$, for 1 ≤ k ≤ n − 1,

• $B_{n-k}^{(n)}\geq \frac {1}{\pi _{A}}{\int \limits }_{t_{k-1}}^{t_{k}}\frac {\omega _{1-\alpha }(t_{n}-s)}{\tau _{k}} \mathrm {d} s$, π_A = 2, for 1 ≤ k ≤ n,

• $B_{0}^{(n)}\leq \frac {26}{11}{\int \limits }_{t_{n-1}}^{t_{n}}\frac {\omega _{1-\alpha }(t_{n}-s)}{\tau _{n}} \mathrm {d} s\leq \frac {2\tau _{n}^{-\alpha }}{\Gamma (2-\alpha )}$,

where $\epsilon \leq \epsilon _{1}=\min \limits \{\frac {\alpha }{2(1-\alpha )}\omega _{1-\alpha }(T), \frac {1}{26}\omega _{1-\alpha }(T)\}$. We point out that we only consider α is a given constant throughout the paper, and does not consider the case of $\alpha \rightarrow 0$.

For the discretization in space, the continuous Galerkin FEM is used. For brevity, we denote $\|\cdot \|_{\infty }$ as $\|\cdot \|_{W^{0,\infty }}$ and ∥⋅∥_m as $\|\cdot \|_{W^{m,2}}$, where $\|\cdot \|_{W^{m,p}}$ represent the norms for the Sobolev space W^m,p(Ω) [36]. Let $V_{h}\subset {H_{1}^{0}}({\Omega })$ be the space of piecewise polynomials of degree ≤ r corresponding to a conforming (quasi-uniform) triangulation of Ω with maximum element size h. We can get the following fully discrete scheme based on the Newton linearization method for n = 1,2,…,N, namely

$$ \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} U_{h}^{n-\sigma},v\big)=-(\nabla U_{h}^{n-\sigma}, \nabla v)+\big(f(U_{h}^{n-1})+(1-\sigma)f^{\prime}(U_{h}^{n-1})\nabla_{\tau} {U_{h}^{n}},v\big), \quad v\in V_{h}. $$

(2.18)

We first report the unconditionally optimal error estimate of scheme (2.18) as follows.

Theorem 2.1

Assume the problem (1.1)–(1.3) has a unique solution uⁿ which satisfies (1.5) and is smooth enough in spatial directions. Then the fully discrete scheme defined in (2.18) has a unique solution ${U_{h}^{n}}$. If $f\in C^{4}(\mathbb {R})$, there exist τ^∗, h^∗, 𝜖^∗, such that when τ < τ^∗, h < h^∗, 𝜖 < 𝜖^∗ and γα ≤ 2(γ ≥ 1), satisfying

$$ \|u^{n}-{U_{h}^{n}}\|_{1} \leq C^{*}(\tau^{\min\{2,\gamma\alpha\}}+h^{r}+\epsilon), $$

(2.19)

where C^∗ is a positive constant independent of h, τ and 𝜖.

The proof of Theorem 2.1 is presented in the next section.

3 Unconditionally optimal H ¹-error estimate

We here consider the truncation errors caused by the Taylor expansion at t_n−σ and the Newton linearization method, and introduce a discrete fractional Grönwall inequality. After that, we use the temporal-spatial error splitting approach developed in [31] to obtain the unconditionally optimal H¹-error estimate.

3.1 Preliminaries

It is known that the continued kernel ω_α holds the semigroup property ω_α ∗ ω_β = ω_α+β, namely

$$ {{\int}_{s}^{t}}\omega_{\alpha}(t-\mu)\omega_{\beta}(\mu-s) \mathrm{d} \mu=\omega_{\alpha+\beta}(t-s),\quad\forall 0<s<t<\infty, $$

(3.20)

thus we have ω_α ∗ ω_1−α = ω₁ = 1. For the discrete kernel $B_{n-j}^{(n)}$ in (2.17), it generally does not hold the same semigroup property (3.20) as the continued kernel. To preserve the same property, a complementary discrete kernel $P_{n-k}^{(n)}$ proposed in [9] is introduced such that

$$ \sum\limits_{j=k}^{n}P_{n-j}^{(n)}B_{j-k}^{(j)}=1,\quad 1\leq k\leq n. $$

(3.21)

For the given value $B_{j-k}^{(j)}$, we can calculate $P_{n-j}^{(n)}$ from (3.21) by using the following recursive formula:

$$ P_{n-j}^{(n)}=\frac{1}{B_{0}^{(j)}} \left\{\begin{array}{ll} \quad\quad\quad\quad\quad\quad1,& \ j=n,\\ \\ {\sum}_{k=j+1}^{n}(B_{k-j-1}^{(k)}-B_{k-j}^{(k)})P_{n-k}^{(n)},&\ 1\leq j\leq n-1. \end{array}\right. $$

(3.22)

Next, we present several useful Lemmas.

Lemma 3.1 (10)

Let $B_{n-k}^{(n)}$ has properties M1 and M2. For any sequence $(v^{n})_{n=1}^{N}$, it holds

$$ \frac12\sum\limits_{k=1}^{n}B_{n-k}^{(n)}\nabla_{\tau}||v^{k}||_{0}^{2}\leq\big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} v^{n-\sigma},v^{n-\sigma}\big),\quad for\quad 1\leq n\leq N . $$

(3.23)

Lemma 3.2 (34 An improved discrete fractional Grönwall inequality)

Assume $B_{n-k}^{(n)}$ holds the properties of M1 and M2, and $(\xi ^{n})_{n=1}^{N}$, $(\eta ^{n})_{n=1}^{N}$ and $(\lambda _{l})_{l=0}^{N-1}$ are three nonnegative sequences. If the nonnegative sequence $(v^{k})_{k=0}^{N}$ satisfies

$$ \sum\limits_{k=1}^{n}B_{n-k}^{(n)}\nabla_{\tau} (v^{k})^{2}\leq\sum\limits_{k=1}^{n}\lambda_{n-k}(v^{k-\sigma})^{2}+v^{n-\sigma}\xi^{n}+(\eta^{n})^{2}, \qquad 1\leq n\leq N $$

(3.24)

and the maximum step size $\tau \leq 1/{\sqrt [\alpha ]{2\pi _{A}{\Lambda }{\Gamma }(2-\alpha )}}$, then there exists a constant ${\Lambda } \geq {\sum }_{l=0}^{N-1}\lambda _{l}$ such that

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!v^{n}&\leq&2E_{\alpha}(2\max(1,\rho)\pi_{A}{\Lambda} t_{n}^{\alpha})\Big(v^{0}+\max_{1\leq k\leq n}\sum\limits_{j=1}^{k}P_{k-j}^{(k)}\xi^{j}+\sqrt{\pi_{A}{\Gamma}(1-\alpha)}\max_{1\leq k\leq n}\{t_{k}^{\alpha/2}\eta^{k}\}\Big)\\ \!\!\!\!\!\!\!&\leq&2E_{\alpha}(2\max(1,\rho)\pi_{A}{\Lambda} t_{n}^{\alpha})\Big(v^{0}+\pi_{A}{\Gamma}(1-\alpha)\max_{1\leq j\leq n}\{t_{j}^{\alpha}\xi^{j}\}+\sqrt{\pi_{A}{\Gamma}(1-\alpha)}\max_{1\leq k\leq n}\{t_{k}^{\alpha/2}\eta^{k}\}\Big). \end{array} $$

(3.25)

Lemma 3.3 (35)

Assume that v ∈ C³((0,T]) and satisfies (1.5). Denote

$$ R_{t}^{k-\sigma}={~}^{\text{C}}\mathcal{D}^{\alpha}_t v(t_{k-\sigma})-{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} v^{k-\sigma} $$

(3.26)

as the local consistency error of fast L2-1_σ scheme. Then the global consistency error can be bounded by

$$ \begin{array}{@{}rcl@{}} \sum\limits_{k=1}^{n}P_{n-k}^{(n)}|R_{t}^{k-\sigma}|\leq C_{v}\Big(\frac{\tau_{1}^{\alpha}}{\alpha}+\frac{1}{1-\alpha}\max_{2\leq k\leq n}t_{k}^{\alpha}t_{k-1}^{\alpha-3}{\tau_{k}^{3}}\tau_{k-1}^{-\alpha}+\frac{\epsilon}{\alpha}t_{n}^{\alpha}\hat{t}^{2}_{n-1}\Big), \end{array} $$

(3.27)

where $\hat {t}_{n}=\max \limits \{1,t_{n}\}$.

Lemma 3.4

Assume that v ∈ C²((0,T]) and satisfies (1.5), and the nonlinear function $f(v)=f\in C^{4}(\mathbb {R})$. Denote the local truncation error by

$$ R_{f}^{k-\sigma}=f\big(v(t_{k-\sigma})\big)-f(v^{k-1})-(1-\sigma)f^{\prime}(v^{k-1})\nabla_{\tau} v^{k}. $$

(3.28)

Then the global consistency error can bounded by

$$ \sum\limits_{k=1}^{n}P_{n-k}^{(n)}\left|R_{f}^{k-\sigma}\right|+\sum\limits_{k=1}^{n}P_{n-k}^{(n)}\left|\nabla R_{f}^{k-\sigma}\right|+\sum\limits_{k=1}^{n}P_{n-k}^{(n)}\left|{\Delta} R_{f}^{k-\sigma}\right| \leq C_{v}\big(\tau_{1}^{3\alpha}+t_{n}^{\alpha}\max_{2\leq k\leq n}{\tau_{k}^{2}}t_{k-1}^{2(\alpha-1)}\big). $$

(3.29)

The proof is presented in the Appendix for brevity.

Lemma 3.5 (37)

Assume v ∈ C²((0,T]) and satisfies (1.5). Denote by

$$ R_{\sigma}^{k-\sigma}={\Delta} v(t_{k-\sigma})-{\Delta} v^{k-\sigma}. $$

Then it holds

$$ \sum\limits_{k=1}^{n}P_{n-k}^{(n)}\left|R_{\sigma}^{k-\sigma}\right| \leq C_{v}\Big(\frac{\tau_{1}^{2\alpha}}{\alpha}+t_{n}^{\alpha}\max_{2\leq k\leq n}t_{k-1}^{\alpha-2}{\tau_{k}^{2}}\Big). $$

(3.30)

3.2 Analysis of the semi-discrete scheme

We now consider the following semi-discrete problem at time t_n−σ, namely

$$ {~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} U^{n-\sigma}={\Delta} U^{n-\sigma}+f(U^{n-1})+(1-\sigma)f^{\prime}(U^{n-1})\nabla_{\tau} U^{n},\quad n=1,\ldots, N $$

(3.31)

with the initial and boundary conditions

$$ \begin{array}{@{}rcl@{}} U^{0}(x)&=&u_{0}(x), \qquad \qquad x\in\bar{\Omega}, \end{array} $$

(3.32)

$$ \begin{array}{@{}rcl@{}} U^{n}(x)&=&0,\qquad \qquad {\kern16pt}x\in\partial{\Omega},\quad n=1,\ldots,N. \end{array} $$

(3.33)

Set eⁿ = uⁿ − Uⁿ, n = 0,1,…,N. Subtracting (3.31) from (1.1) produces

$$ {~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} e^{n-\sigma}={\Delta} e^{n-\sigma}+E_{1}^{n-\sigma}+R_{\sigma}^{n-\sigma}+R_{t}^{n-\sigma}+R_{f}^{n-\sigma}, $$

(3.34)

where

$$ E_{1}^{n-\sigma}=f(u^{n-1})+(1-\sigma)f^{\prime}(u^{n-1})\nabla_{\tau} u^{n}-f(U^{n-1})-(1-\sigma)f^{\prime}(U^{n-1})\nabla_{\tau} U^{n}. $$

(3.35)

Next, we consider the boundedness of Uⁿ and the error estimate of eⁿ.

Theorem 3.1

The semi-discrete problem (3.31)–(3.33) has a unique solution Uⁿ. Moreover, if $f\in C^{4}(\mathbb {R})$, there exist 𝜖^∗ > 0 and τ^∗∗ > 0 such that when 𝜖 ≤ 𝜖^∗ and τ ≤ τ^∗∗, it holds

$$ \begin{array}{@{}rcl@{}} &&\|e^{n}\|_{2} \leq C_{1}\tau^{\min\{\gamma\alpha,2\}}+C_{2}\epsilon, \end{array} $$

(3.36)

$$ \begin{array}{@{}rcl@{}} &&\|U^{n}\|_{\infty} \leq Q_{1}, \end{array} $$

(3.37)

$$ \begin{array}{@{}rcl@{}} &&\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} U^{n-\sigma}\|_{2} \leq C_{3}, \end{array} $$

(3.38)

where γα ≤ 2, $Q_{1}=\max \limits _{1\leq n\leq N}\|u^{n}\|_{\infty }+1$, C₁,C₂ and C₃ are constants independent of τ and 𝜖.

Proof 1

At each time level, (3.31) is a linear elliptic problem. So it is easy to obtain the existence and uniqueness of the solution Uⁿ. We now use the induction to prove (3.36) and (3.37). For n = 0, it is obvious that (3.36) and (3.37) hold. Then, we assume that (3.36) holds for 0 ≤ k ≤ n − 1, and have

$$ \begin{array}{@{}rcl@{}} \|U^{k}\|_{\infty}&\leq &\|u^{k}\|_{\infty}+\|e^{k}\|_{\infty} \leq \|u^{k}\|_{\infty}+C_{\Omega}\|e^{k}\|_{2}\\ &\leq& \|u^{k}\|_{\infty}+C_{\Omega}(C_{1}\tau^{\min\{\gamma\alpha,2\}}+C_{2}\epsilon) \leq Q_{1}, \end{array} $$

(3.39)

whenever 𝜖 ≤ 𝜖₂ = 1/(2C_ΩC₂) and $\tau <\tau _{a}=(2C_{\Omega } C_{1})^{-\frac {1}{\min \limits \{\gamma \alpha ,2\}}}$. Noting $\|U^{k}\|_{\infty }$ and $\|u^{k}\|_{\infty }$ are bounded for all 0 ≤ k ≤ n − 1, we further have

$$ \begin{array}{@{}rcl@{}} \|E_{1}^{k-\sigma}\|_{2}&\leq& \|f(u^{k-1})+(1-\sigma)f^{\prime}(u^{k-1})\nabla_{\tau} u^{k}-f(U^{k-1})-(1-\sigma)f^{\prime}(U^{k-1})\nabla_{\tau} U^{k}\|_{2}\\ &\leq& \|f(u^{k-1})-f(U^{k-1})\|_{2}+(1-\sigma)\|\big(f^{\prime}(u^{k-1})-f^{\prime}(U^{k-1})\big)u^{k}\|_{2}\\ &&\quad+(1-\sigma)\|f^{\prime}(U^{k-1})(u^{k}-U^{k})\|_{2}+(1-\sigma)\|\big(f^{\prime}(u^{k-1})-f^{\prime}(U^{k-1})\big)u^{k-1}\|_{2}\\ &&\quad+(1-\sigma)\|f^{\prime}(U^{k-1})(u^{k-1}-U^{k-1})\|_{2}\\ &\leq& C_{4}\|e^{k-1}\|_{2}+C_{4}\|e^{k}\|_{2}, \end{array} $$

(3.40)

where C₄ is a constant dependent on u,σ,f,Q₁.

Taking the inner production with e^k−σ both sides of (3.34) for k = n, we have

$$ \begin{array}{@{}rcl@{}} \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} e^{k-\sigma},e^{k-\sigma}\big)&=&({\Delta} e^{k-\sigma},e^{k-\sigma})+(E_{1}^{k-\sigma},e^{k-\sigma})+(R_{\sigma}^{k-\sigma},e^{k-\sigma})+(R_{t}^{k-\sigma},e^{k-\sigma})+(R_{f}^{k-\sigma},e^{k-\sigma})\\ &=&-\|\nabla e^{k-\sigma}\|_{0}^{2}+(E_{1}^{k-\sigma},e^{k-\sigma})+(R_{\sigma}^{k-\sigma}+R_{t}^{k-\sigma}+R_{f}^{k-\sigma},e^{k-\sigma})\\ &\leq& (E_{1}^{k-\sigma},e^{k-\sigma})+(R_{\sigma}^{k-\sigma}+R_{t}^{k-\sigma}+R_{f}^{k-\sigma},e^{k-\sigma})\\ &\leq& \frac{C_{5}}{2}\|e^{k-1}\|^{2}_{0}+\frac{C_{5}}{2}\|e^{k}\|^{2}_{0}+\|R_{\sigma}^{k-\sigma}+R_{t}^{k-\sigma}+R_{f}^{k-\sigma}\|_{0}\|e^{k-\sigma}\|_{0}, \end{array} $$

(3.41)

where C₅ is a constant dependent on u,σ,f,Q₁. By Lemma 3.1 (3.41) can be rewritten as

$$ \frac12\sum\limits_{i=1}^{k}B_{k-i}^{(k)}\nabla_{\tau}||e^{i}||_{0}^{2}\leq \frac{C_{5}}{2}\|e^{k-1}\|^{2}_{0}+\frac{C_{5}}{2}\|e^{k}\|^{2}_{0}+\|R_{\sigma}^{k-\sigma}+R_{t}^{k-\sigma}+R_{f}^{k-\sigma}\|_{0}\|e^{k-\sigma}\|_{0}. $$

(3.42)

Recalling Lemma 3.2 and taking $\tau <\tau _{b}=1/{\sqrt [\alpha ]{8C_{5}{\Gamma }(2-\alpha )}}$, it holds

$$ \|e^{k}\|_{0}\leq 4E_{\alpha}(8 C_{5} \max\{1,\rho\}t_{k}^{\alpha})\big(\max_{1\leq j\leq k}\sum\limits_{i=1}^{j}P_{j-i}^{(j)}\|R_{\sigma}^{i-\sigma}+R_{t}^{i-\sigma}+R_{f}^{i-\sigma}\|_{0}\big). $$

(3.43)

Similarly, taking the inner production with −Δe^k−σ both sides of (3.34) yields

$$ \begin{array}{@{}rcl@{}} && \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} e^{k-\sigma},-{\Delta} e^{k-\sigma}\big)\\&=&({\Delta} e^{k-\sigma},-{\Delta} e^{k-\sigma})+(E_{1}^{k-\sigma},-{\Delta} e^{k-\sigma})+(R_{\sigma}^{k-\sigma}+R_{t}^{k-\sigma}+R_{f}^{k-\sigma},-{\Delta} e^{k-\sigma})\\ &=&-\|{\Delta} e^{k-\sigma}\|_{0}^{2}+(\nabla E_{1}^{k-\sigma},\nabla e^{k-\sigma})+(\nabla(R_{\sigma}^{k-\sigma}+R_{t}^{k-\sigma}+R_{f}^{k-\sigma}),\nabla e^{k-\sigma})\\ &\leq&(\nabla E_{1}^{k-\sigma},\nabla e^{k-\sigma}) +\big(\nabla(R_{\sigma}^{k-\sigma}+R_{t}^{k-\sigma}+R_{f}^{k-\sigma}),\nabla e^{k-\sigma}\big)\\ &\leq& \frac{C_{6}}{2}\|\nabla e^{k-1}\|^{2}_{0}+\frac{C_{6}}{2}\|\nabla e^{k}\|^{2}_{0}+\|\nabla(R_{\sigma}^{k-\sigma} +R_{t}^{k-\sigma}+R_{f}^{k-\sigma})\|_{0}\|\nabla e^{k-\sigma}\|_{0}, \end{array} $$

(3.44)

where C₆ is a constant dependent on u,σ,f,Q₁.

Recalling Lemma 3.2 again and taking $\tau <\tau _{c}=1/{\sqrt [\alpha ]{8C_{6}{\Gamma }(2-\alpha )}}$, it holds

$$ \|\nabla e^{k}\|_{0}\leq 4E_{\alpha}(8C_{6}\max\{1,\rho\}t_{k}^{\alpha})\big(\max_{1\leq j\leq k}\sum\limits_{i=1}^{j}P_{j-i}^{(j)}\|\nabla(R_{\sigma}^{i-\sigma}+R_{t}^{i-\sigma}+R_{f}^{i-\sigma})\|_{0}\big). $$

(3.45)

Next, multiplying (3.34) by Δ²e^k−σ and integrating the result over Ω, we get

$$ \begin{array}{@{}rcl@{}} \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} e^{k-\sigma},{\Delta}^{2} e^{k-\sigma}\big) &=&({\Delta} e^{k-\sigma},{\Delta}^{2} e^{k-\sigma})+(E_{1}^{k-\sigma},{\Delta}^{2} e^{k-\sigma})+(R_{\sigma}^{k-\sigma}+R_{t}^{k-\sigma}+R_{f}^{k-\sigma},{\Delta}^{2} e^{k-\sigma})\\ &=&-\|{\Delta} \nabla e^{k-\sigma}\|_{0}^{2}+({\Delta} {E_{1}^{k}},{\Delta} e^{k-\sigma})+({\Delta}(R_{\sigma}^{k-\sigma}+R_{t}^{k-\sigma}+R_{f}^{k-\sigma}),{\Delta} e^{k-\sigma})\\ &\leq& ({\Delta} E_{1}^{k-\sigma},{\Delta} e^{k-\sigma})+\big({\Delta}(R_{\sigma}^{k-\sigma}+R_{t}^{k-\sigma}+R_{f}^{k-\sigma}),{\Delta} e^{k-\sigma}\big)\\ &\leq& \frac{C_{7}}{2}\|{\Delta} e^{k-1}\|^{2}_{0}+\frac{C_{7}}{2}\|{\Delta} e^{k}\|^{2}_{0}+\|{\Delta}(R_{\sigma}^{k-\sigma}+R_{t}^{k-\sigma}+R_{f}^{k-\sigma})\|_{0}\|{\Delta} e^{{k-\sigma}}\|_{0}, \end{array} $$

(3.46)

where C₇ is a constant dependent on u,σ,f,Q₁.

Recalling Lemma 3.2 again and taking $\tau <\tau _{d}=1/{\sqrt [\alpha ]{8C_{7}{\Gamma }(2-\alpha )}}$, it holds

$$ \|{\Delta} e^{k}\|_{0}\leq 4E_{\alpha}(8C_{7}\max\{1,\rho\}t_{k}^{\alpha})\big(\max_{1\leq j\leq k}{\sum}_{i=1}^{j}P_{j-i}^{(j)}\|{\Delta}(R_{\sigma}^{i-\sigma}+R_{t}^{i-\sigma}+R_{f}^{i-\sigma})\|_{0}\big). $$

(3.47)

Based on Lemmas 3.3, 3.4 and 3.5, one has

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{i=1}^{j}P_{j-i}^{(j)}\|R_{\sigma}^{i-\sigma}+R_{t}^{i-\sigma}+R_{f}^{i-\sigma}\|_{2}\\ &\leq& \sum\limits_{i=1}^{j}P_{j-i}^{(j)}\|R_{\sigma}^{i-\sigma}\|_{2}+\sum\limits_{i=1}^{j}P_{j-i}^{(j)}\|R_{t}^{i-\sigma}\|_{2}+\sum\limits_{i=1}^{j}P_{j-i}^{(j)}\|R_{f}^{i-\sigma}\|_{2}\\ &\leq& C_{v}\big(\frac{\tau_{1}^{2\alpha}}{\alpha}+t_{j}^{\alpha}\max_{2\leq i\leq j}t_{i-1}^{\alpha-2}{\tau_{i}^{2}}+\frac{\tau_{1}^{\alpha}}{\alpha}+\frac{1}{1-\alpha}\max_{2\leq i\leq j}t_{i}^{\alpha}t_{i-1}^{\alpha-3}{\tau_{i}^{3}}\tau_{i-1}^{-\alpha}+\frac{\epsilon}{\alpha}t_{j}^{\alpha}\hat{t}^{\alpha}_{j-1}+\tau_{1}^{3\alpha}+t_{j}^{\alpha}\max_{2\leq i\leq j}{\tau_{i}^{2}}t_{i-1}^{2(\alpha-1)}\big)\\ &\leq& C_{v}\big(\frac{\tau_{1}^{\alpha}}{\alpha}+t_{j}^{\alpha}\max_{2\leq i\leq n}t_{i-1}^{\alpha-2}{\tau_{i}^{2}}+\frac{1}{1-\alpha}\max_{2\leq i\leq j}t_{i}^{\alpha}t_{1}^{-3}\tau^{3}+\frac{\epsilon}{\alpha}t_{j}^{\alpha}\hat{t}^{\alpha}_{j-1}+t_{j}^{\alpha}\max_{2\leq i\leq j}{\tau_{i}^{2}}t_{i-1}^{2(\alpha-1)}\big)\\ &\leq& C\big(\tau^{\min\{\gamma\alpha,2\}}+\frac{\epsilon}{\alpha} (T\hat{T})^{\alpha}\big), \end{array} $$

(3.48)

where $\hat {T}=\max \limits \{1,T\}$. Then putting (3.48) into (3.43), (3.45) and (3.47), we can obtain

$$ \|e^{k}\|_{2} \leq C_{1}\big(\tau^{\min\{\gamma\alpha,2\}}+\frac{\epsilon}{\alpha} (T\hat{T})^{\alpha}\big) =C_{1}\tau^{\min\{\gamma\alpha,2\}}+C_{2}\epsilon, $$

(3.49)

where $C_{2}=\frac { (T\hat {T})^{\alpha }}{\alpha }C_{1}$ and

$$ C_{1}=4C_{u}\sqrt{E_{\alpha}^{2}(8C_{5}\max\{1,\rho\}T^{\alpha})+E_{\alpha}^{2}(8C_{6}\max\{1,\rho\}T^{\alpha})+E_{\alpha}^{2}(8C_{7}\max\{1,\rho\}T^{\alpha})}. $$

Then, whenever 𝜖 ≤ 𝜖₂ and τ < τ_a, we have

$$ \|U^{k}\|_{\infty}\leq \|u^{k}\|_{\infty}+\|e^{k}\|_{\infty}\leq \|u^{k}\|_{\infty}+C_{\Omega} \big(C_{1}\tau^{\min\{\gamma\alpha,2\}}+C_{2}\epsilon\big)\leq Q_{1}. $$

(3.50)

Thus, the estimates (3.36) and (3.37) hold for n = k by taking $\tau ^{**}=\min \limits \{\tau _{a},\tau _{b},$ τ_c,τ_d} and 𝜖 ≤ 𝜖₂. The mathematical induction is finished.

Based on the above results, we now consider the proof of (3.38). By the definition, we have

$$ \begin{array}{@{}rcl@{}} \|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} e^{n-\sigma}\|_{2} &=&\|B_{0}^{(n)}e^{n}-\sum\limits_{k=1}^{n-1}(B_{n-k-1}^{(n)}-B_{n-k}^{(n)})e^{k}-B_{n-1}^{(n)}e^{0}\|_{2}\\ &=&B_{0}^{(n)}\|e^{n}\|_{2}+\sum\limits_{k=1}^{n-1}(B_{n-k-1}^{(n)}-B_{n-k}^{(n)})\|e^{k}\|_{2}+B_{n-1}^{(n)}\|e^{0}\|_{2}\\ &\leq&\Big(B_{0}^{(n)}+\sum\limits_{k=1}^{n-1}(B_{n-k-1}^{(n)}-B_{n-k}^{(n)})+B_{n-1}^{(n)}\Big)\big(C_{1}\tau^{\min\{\gamma\alpha,2\}}+C_{2}\epsilon\big)\\ &\leq& 2B_{0}^{(n)}\big(C_{1}\tau^{\min\{\gamma\alpha,2\}}+C_{2}\epsilon\big) \leq\frac{48\tau_{n}^{-\alpha}}{11{\Gamma}(2-\alpha)}\big(C_{1}\tau^{\min\{\gamma\alpha,2\}}+C_{2}\epsilon\big)\\ &\leq&\frac{48}{11{\Gamma}(2-\alpha)}\big(C_{1}+C_{2}\big), \end{array} $$

(3.51)

where we apply the properties M3 in the penultimate inequality and take $\epsilon <\epsilon _{3}=\tau ^{\min \limits \{\gamma \alpha ,2\}}$ and γα ≤ 2 in the last inequality. Therefore,

$$ \|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} U^{n-\sigma}\|_{2} \leq \|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} u^{n-\sigma}\|_{2}+\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} e^{n-\sigma}\|_{2}\leq C_{3}. $$

(3.52)

The proof is completed after taking $\epsilon ^{*}=\min \limits \{\epsilon _{1},\epsilon _{2},\epsilon _{3}\}$. □

3.3 Analysis of the fully discrete scheme

We now consider the boundedness of the fully discrete solution ${U_{h}^{n}}$. To analyze the fully discrete scheme (2.18), we introduce the Ritz projection operator $R_{h}: {H^{1}_{0}}({\Omega })\rightarrow V_{h}\subset {H^{1}_{0}}({\Omega })$ by

$$\big(\nabla(v-R_{h}v),\nabla\omega\big)=0,\quad \forall \omega\in V_{h}. $$

For $\forall v\in H^{s}({\Omega })\cap {H^{1}_{0}}({\Omega })$, it holds

$$ \|v-R_{h}v\|_{0}+h\|\nabla(v-R_{h}v)\|_{0}\leq C_{\Omega} h^{s}\|v\|_{s}, \quad 1\leq s\leq r+1. $$

(3.53)

Let

$$ U^{n}-{U_{h}^{n}}=U^{n}-R_{h}U^{n}+R_{h}U^{n}-{U_{h}^{n}}={\rho_{h}^{n}}+{\theta_{h}^{n}},\quad n=0,1,\ldots,N. $$

(3.54)

The weak form of the semi-discrete (3.31) is

$$ \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} U^{n-\sigma},v\big)=-(\nabla U^{n-\sigma},\nabla v)+\big(f(U^{n-1})+(1-\sigma)f^{\prime}(U^{n-1})\nabla_{\tau} U^{n},v\big), \forall v\in V_{h}. $$

(3.55)

Subtracting (2.18) from (3.55), we get

$$ \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} \theta_{h}^{n-\sigma},v\big)=-(\nabla \theta_{h}^{n-\sigma},\nabla v)+(E_{2}^{n-\sigma},v)-\big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} \rho_{h}^{n-\sigma},v\big), $$

(3.56)

where

$$ E_{2}^{n-\sigma}=f(U^{n-1})+(1-\sigma)f^{\prime}(U^{n-1})\nabla_{\tau} U^{n}-f(U_{h}^{n-1})-(1-\sigma)f^{\prime}(U_{h}^{n-1})\nabla_{\tau} {U_{h}^{n}}. $$

(3.57)

Then, based on the boundedness of $||U^{n}||_{\infty }$ in Theorem 3.1, we present the following result.

Theorem 3.2

Suppose the semi-discrete scheme (3.55) has a unique solution Uⁿ and the fully discrete scheme defined in (2.18) has a unique solution ${U_{h}^{n}}$, n = 1,...,N. If $f\in C^{4}(\mathbb {R})$, there exist h^∗ > 0 and τ^∗∗∗ > 0 such that when h < h^∗ and τ < τ^∗∗∗, it holds

$$ \begin{array}{@{}rcl@{}} &&\|{\theta_{h}^{n}}\|_{0} \leq h^{\frac74}, \end{array} $$

(3.58)

$$ \begin{array}{@{}rcl@{}} &&\|{U_{h}^{n}}\|_{\infty} \leq Q_{2}, \end{array} $$

(3.59)

where $Q_{2}=\max \limits _{1\leq n\leq N}\|R_{h}U^{n}\|_{\infty }+1$.

Proof 2

Noting the coefficient matrix of the linear system arising from (2.18) is diagonally dominant, the solution ${U_{h}^{n}}$ of (2.18) exists and is unique. Here, we also apply the mathematical induction to prove (3.58). It is easy to show (3.58) hold for n = 0. Next, we assume (3.58) holds for 1 ≤ k ≤ n − 1, and have

$$ \begin{array}{@{}rcl@{}} \|{U_{h}^{k}}\|_{\infty}&\leq &\|R_{h}U^{k}\|_{\infty}+\|{\theta_{h}^{k}}\|_{\infty} \leq \|R_{h}U^{k}\|_{\infty}+C_{\Omega} h^{-\frac{d}{2}}\|{\theta_{h}^{k}}\|_{0}\\ &\leq& \|R_{h}U^{k}\|_{\infty}+C_{\Omega} h^{-\frac{d}{2}}h^{\frac74} \leq \|R_{h}U^{k}\|_{\infty}+1 \leq Q_{2}, \end{array} $$

(3.60)

whenever $h<h_{1}=C_{\Omega }^{-\frac {4}{7-2d}}$.

Similar to the estimate (3.40) of $E_{1}^{k-\sigma }$, we use the boundedness of $||U^{k}||_{\infty }$ and $||{U_{h}^{k}}||_{\infty }$ for all 0 ≤ k ≤ n − 1 to obtain

$$ \begin{array}{@{}rcl@{}} \|E_{2}^{k-\sigma}\|_{0}&\leq& \|f(U^{k-1})+(1-\sigma)f^{\prime}(U^{k-1})\nabla_{\tau} U^{k}-f(U_{h}^{k-1})-(1-\sigma)f^{\prime}(U_{h}^{k-1})\nabla_{\tau} {U_{h}^{k}}\|_{0}\\ &\leq& \|f(U^{k-1})-f(U_{h}^{k-1})\|_{0}+(1-\sigma)\|\big(f^{\prime}(U^{k-1})-f^{\prime}(U_{h}^{k-1})\big)U^{k}\|_{0}\\ &&\quad+(1-\sigma)\|f^{\prime}(U_{h}^{k-1})(U^{k}-{U_{h}^{k}})\|_{0}+(1-\sigma)\|\big(f^{\prime}(U^{k-1})-f^{\prime}(U_{h}^{k-1})\big)U^{k-1}\|_{0}\\ &&\quad+(1-\sigma)\|f^{\prime}(U_{h}^{k-1})(U^{k-1}-U_{h}^{k-1})\|_{0}\\ &\leq& C_{8}\|U^{k-1}-U_{h}^{k-1}\|_{0}+C_{8}\|U^{k}-{U_{h}^{k}}\|_{0}\\ &\leq& C_{8}\|\theta_{h}^{k-1}\|_{0}+C_{8}\|{\theta_{h}^{k}}\|_{0}+2C_{8}C_{\Omega} h^{2}, \end{array} $$

(3.61)

where C₈ is a constant dependent on σ,f,Q₂ and (3.53) is used in the last inequality. Setting k = n and $v=\theta _{h}^{k-\sigma }$ in (3.56), we obtain

$$ \begin{array}{@{}rcl@{}} \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\theta_{h}^{k-\sigma},\theta_{h}^{k-\sigma}\big) &=&-(\nabla\theta_{h}^{k-\sigma},\nabla\theta_{h}^{k-\sigma})+(E_{2}^{k-\sigma},\theta_{h}^{k-\sigma})-\big({~}^{\text{F}} \mathcal{D}^{\alpha}_{\tau}\rho_{h}^{k-\sigma},\theta_{h}^{k-\sigma}\big)\\ &\leq&-\|\nabla\theta_{h}^{k-\sigma}\|^{2}_{0}+\frac{C_{9}}{2}\|{\theta_{h}^{k}}\|^{2}_{0}+\frac{C_{9}}{2}\|\theta_{h}^{k-1}\|^{2}_{0} +\big(2C_{8}C_{\Omega} h^{2}+\|{~}^{\text{F}}\mathcal{D}^{{\alpha}}_{\tau}\rho_{h}^{k-\sigma}\|_{0}\big)\|\theta_{h}^{k-\sigma}\|_{0}\\ &\leq& \frac{C_{9}}{2}\|{\theta_{h}^{k}}\|^{2}_{0}+\frac{C_{9}}{2}\|\theta_{h}^{k-1}\|^{2}_{0} +\big(2C_{8}C_{\Omega} h^{2}+C_{\Omega}\|{~}^{\text{F}}\mathcal{D}^{{\alpha}}_{\tau}U^{k-\sigma}\|_{2}h^{2}\big)\|\theta_{h}^{k-\sigma}\|_{0}, \end{array} $$

(3.62)

where C₉ is a constant dependent on σ,f,Q₂. Applying Lemma 3.2, we have

$$ \begin{array}{@{}rcl@{}} \|{\theta_{h}^{k}}\|_{0} &\leq& 2E_{\alpha}(8C_{9}\max\{1,\rho\}t_{k}^{\alpha})\Big(\|{\theta_{h}^{0}}\|_{0}+4t_{k}^{\alpha}{\Gamma}(1-\alpha)\big(2C_{8}C_{\Omega} h^{2}+\max_{1\leq i\leq k}C_{\Omega}\|{~}^{\text{F}}\mathcal{D}^{{\alpha}}_{\tau}U^{i-\sigma}\|_{2}h^{2}\big)\Big)\\ &\leq&2E_{\alpha}(8C_{9}\max\{1,\rho\}t_{k}^{\alpha})\Big(C_{\Omega}+4T^{\alpha}{\Gamma}(1-\alpha)\big(2C_{8}C_{\Omega}+\max_{1\leq i\leq k}C_{\Omega}\|{~}^{\text{F}}\mathcal{D}^{{\alpha}}_{\tau}U^{i-\sigma}\|_{2}\big)\Big)h^{2}\\ &\leq& h^{\frac74}, \end{array} $$

(3.63)

when $\tau <\tau _{e}=1/{\sqrt [\alpha ]{8C_{9}{\Gamma }(2-\alpha )}}$ and

$$ h<h_{2}=\Big(2E_{\alpha}(8C_{9}\max\{1,\rho\}t_{k}^{\alpha})\big(C_{\Omega}+4T^{\alpha}{\Gamma}(1-\alpha)(2C_{8}C_{\Omega}+\max_{1\leq i\leq k}C_{\Omega}\|{~}^{\text{F}}\mathcal{D}^{{\alpha}}_{\tau}U^{i-\sigma}\|_{2})\big)\Big)^{-4}. $$

Then, when h < h₁, we can verify that

$$ \|{U_{h}^{k}}\|_{\infty} \leq \|R_{h}U^{k}\|_{\infty}+\|{\theta_{h}^{k}}\|_{\infty}\leq \|R_{h}U^{k}\|_{\infty}+C_{\Omega} h^{-\frac{d}{2}}\|{\theta_{h}^{k}}\|_{0}\leq Q_{2}. $$

(3.64)

Thus, the estimates (3.58) and (3.59) hold for n = k by taking $h^{*}=\min \limits \{h_{1},h_{2}\}$ and τ^∗∗∗ = τ_e. The proof is completed. □

3.4 The proof of Theorem 2.1

Noting the boundedness of $||U^{n}||_{\infty }$ in Theorem 3.1, we can use the method in [33, 34] to get the estimate of $\|u^{n}-{U_{h}^{n}}\|_{1}$ for the linear element (i.e., r = 1). The method in [33, 34] will become too complicated to be used for high-order elements (i.e., r ≥ 2). Hence, the following proof will be split into two cases: one is for r = 1 and another one is for r ≥ 2.

Proof 3

We first prove the case of r = 1 by taking $v={~}^{\text {F}}\mathcal {D}^{\alpha }_{\tau }\theta _{h}^{n-\sigma }$ in (3.56 ) to get

$$ \begin{array}{@{}rcl@{}} \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\theta_{h}^{n-\sigma},{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\theta_{h}^{n-\sigma}\big)&=&-\big(\nabla\theta_{h}^{n-\sigma},{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\nabla\theta_{h}^{n-\sigma}\big) +\big(E_{2}^{n-\sigma},{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\theta_{h}^{n-\sigma}\big) -\big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\rho_{h}^{n-\sigma},{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\theta_{h}^{n-\sigma}\big) \\ &\leq&-\big(\nabla\theta_{h}^{n-\sigma},{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\nabla\theta_{h}^{n-\sigma}\big)+\frac{C_{10}}{2}\|{\theta_{h}^{n}}\|^{2}_{0}+\frac{C_{10}}{2}\|\theta_{h}^{n-1}\|^{2}_{0} +\frac{C_{10}}{2}h^{4}\\ &\quad&+\frac12\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\theta_{h}^{n-\sigma}\|_{0}^{2}+\frac12\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\rho_{h}^{n-\sigma}\|_{0}^{2} +\frac12\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\theta_{h}^{n-\sigma}\|_{0}^{2}, \end{array} $$

(3.65)

where C₁₀ is a constant dependent on σ,f,Q₂. Rearranging (3.65) produces

$$ \begin{array}{@{}rcl@{}} \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\nabla\theta_{h}^{n-\sigma},\nabla\theta_{h}^{n-\sigma}\big)&\leq&\frac{C_{10}}{2}\|{\theta_{h}^{n}}\|^{2}_{0}+\frac{C_{10}}{2}\|\theta_{h}^{n-1}\|^{2}_{0} +\frac{C_{10}}{2}h^{4} +\frac12\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\rho_{h}^{n-\sigma}\|_{0}^{2}\\ &\leq&\frac{C_{10}}{2}\|{\theta_{h}^{n}}\|^{2}_{0}+\frac{C_{10}}{2}\|\theta_{h}^{n-1}\|^{2}_{0} +\frac{C_{10}}{2}h^{4}+\frac{1}{2}(C_{\Omega}\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} U^{n-\sigma}\|_{2}h^{2})^{2}\\ &\leq&\frac{C_{10}C_{\Omega}}{2}\|{\nabla\theta_{h}^{n}}\|^{2}_{0}+\frac{C_{10}C_{\Omega}}{2}\|\nabla\theta_{h}^{n-1}\|^{2}_{0}+\frac{C_{10}}{2}h^{4}+\frac{1}{2} (C_{\Omega}\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} U^{n-\sigma}\|_{2}h^{2})^{2}. \end{array} $$

From Lemma 3.2, it holds

$$ \begin{array}{@{}rcl@{}} \|{\nabla\theta_{h}^{n}}\|_{0}&\leq& 2E_{\alpha}(8C_{10}C_{\Omega}\max\{1,\rho\}t_{n}^{\alpha})\\ &&\quad \Big(\|{\nabla\theta_{h}^{0}}\|_{0}+\sqrt{2{\Gamma}(1-\alpha)}t_{n}^{\frac\alpha2}\big(\max_{1\leq k\leq n}C_{\Omega}\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} U^{k-\sigma}\|_{2}+\sqrt{C_{10}}\big)h^{2}\Big), \end{array} $$

(3.66)

when $\tau \leq \tau _{f}=1/{\sqrt [\alpha ]{8C_{10}C_{\Omega }{\Gamma }(2-\alpha )}}$. Therefore, we have

$$ \begin{array}{@{}rcl@{}} \|u^{n}-{U_{h}^{n}}\|_{1}&\leq& \| u^{n}-U^{n}\|_{1} + \| U^{n}- R_{h}U^{n}\|_{1} + \| R_{h}U^{n}- {U_{h}^{n}}\|_{1}\\ &=& \| e^{n}\|_{1} + \| {\rho_{h}^{n}}\|_{1} + \| {\theta_{h}^{n}}\|_{1} \leq C^{*}\big(\tau^{\min\{\gamma\alpha,2\}}+h+\epsilon\big). \end{array} $$

(3.67)

Then, we prove the case of r ≥ 2 by considering the exact solution uⁿ satisfies

$$ \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} u^{n-\sigma},v\big)=-(\nabla u^{n-\sigma},\nabla v)+\big(f(u^{n-1})+(1-\sigma)f^{\prime}(u^{n-1}) \nabla_{\tau} u^{n},v\big), \quad \forall v\in V_{h}. $$

(3.68)

Set

$$ u^{n}-{U_{h}^{n}}=u^{n}-R_{h}u^{n}+R_{h}u^{n}-{U_{h}^{n}}={\xi_{h}^{n}}+{\eta_{h}^{n}},\quad n=0,1,\ldots,N. $$

(3.69)

Subtracting (2.18) from (3.68), we get

$$ \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} \eta_{h}^{n-\sigma},v\big)=-(\nabla \eta_{h}^{n-\sigma},\nabla v)+(E_{3}^{n-\sigma},v)+(R_{t}^{n-\sigma}+R_{\sigma}^{n-\sigma}+R_{f}^{n-\sigma},v)-\big({~}^{\text{F}}\mathcal{D}^{{\alpha}}_{\tau} \xi_{h}^{n-\sigma},v\big), $$

(3.70)

where

$$ E_{3}^{n-\sigma}=f(u^{n-1})+f^{\prime}(u^{n-1})\nabla_{\tau} u^{n} -f(U_{h}^{n-1})-(1-\sigma)f^{\prime}(U_{h}^{n-1}) \nabla_{\tau} {U_{h}^{n}}. $$

(3.71)

It is obvious that $||u^{n}||_{\infty }$ and $||{U_{h}^{n}}||_{\infty }$ are bounded for 1 ≤ n ≤ N. So we obtain

$$ \begin{array}{@{}rcl@{}} \|\nabla E_{3}^{n-\sigma}\|_{0}&\leq& \|\nabla\big(f(u^{n-1})+(1-\sigma)f^{\prime}(u^{n-1})\nabla_{\tau} u^{n}\big)-\nabla\big(f(U_{h}^{n-1})+(1-\sigma)f^{\prime}(U_{h}^{n-1})\nabla_{\tau} {U_{h}^{n}}\big)\|_{0}\\ &\leq& \|\nabla\big(f(u^{n-1})-f(U_{h}^{n-1})\big)\|_{0}+(1-\sigma)\|\nabla\big((f^{\prime}(u^{n-1})-f^{\prime}(U_{h}^{n-1}))u^{n}\big)\|_{0}\\ &&\quad+(1-\sigma)\|\nabla\big(f^{\prime}(U_{h}^{n-1})(u^{n}-{U_{h}^{n}})\big)\|_{0}+(1-\sigma)\|\nabla\big((f^{\prime}(u^{n-1})-f^{\prime}(U_{h}^{n-1}))u^{n-1}\big)\|_{0}\\ &&\quad+(1-\sigma)\|\nabla\big(f^{\prime}(U_{h}^{n-1})(u^{n-1}-U_{h}^{n-1})\big)\|_{0}\\ &\leq& C_{11}\|\nabla(u^{n-1}-U_{h}^{n-1})\|_{0}+C_{11}\|\nabla(u^{n}-{U_{h}^{n}})\|_{0}\\ &\leq& C_{11}\|\nabla\eta_{h}^{n-1}\|_{0}+C_{11}\|{\nabla\eta_{h}^{n}}\|_{0}+C_{11}\|\nabla\xi_{h}^{n-1}\|_{0}+C_{11}\|{\nabla\xi_{h}^{n}}\|_{0}\\ &\leq& C_{11}\|\nabla\eta_{h}^{n-1}\|_{0}+C_{11}\|{\nabla\eta_{h}^{n}}\|_{0}+2C_{11}C_{\Omega} h^{r}, \end{array} $$

(3.72)

where C₁₁ is a constant dependent on u, σ, f and Q₂. Next, taking $v=-{\Delta }\eta _{h}^{n-\sigma }$ in (3.68) , we get

$$ \begin{array}{@{}rcl@{}} \big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} \eta_{h}^{n-\sigma},-{\Delta}\eta_{h}^{n-\sigma}\big)&=&-(\nabla \eta_{h}^{n-\sigma},-\nabla {\Delta}\eta_{h}^{n-\sigma})+(E_{3}^{n-\sigma},-{\Delta}\eta_{h}^{n-\sigma})\\ &&\quad+(R_{t}^{n-\sigma}+R_{\sigma}^{n-\sigma}+R_{f}^{n-\sigma},-{\Delta}\eta_{h}^{n-\sigma})-\big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} \xi_{h}^{n-\sigma},-{\Delta}\eta_{h}^{n-\sigma}\big)\\ &=&-\|{\Delta}\eta_{h}^{n-\sigma}\|_{0}^{2}+(\nabla E_{3}^{n-\sigma},\nabla\eta_{h}^{n-\sigma})\\ &&\quad+(\nabla R_{t}^{n-\sigma}+\nabla R_{\sigma}^{n-\sigma}+\nabla R_{f}^{n-\sigma},\nabla\eta_{h}^{n-\sigma})-\big({~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau} \nabla\xi_{h}^{n-\sigma},\nabla\eta_{h}^{n-\sigma}\big)\\ &\leq&\frac{C_{12}}{2}\|{\nabla\eta_{h}^{k}}\|^{2}_{0}+\frac{C_{12}}{2}\|\nabla\eta_{h}^{n-1}\|^{2}_{0}+\big(2C_{11}C_{\Omega} h^{r}\\ &&\quad+\|\nabla (R_{t}^{n-\sigma}+ R_{\sigma}^{n-\sigma}+ R_{f}^{n-\sigma})\|_{0}+\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\nabla\xi_{h}^{n-\sigma}\|_{0}\big)\|\nabla\eta_{h}^{n-\sigma}\|_{0} \\ &\leq&\frac{C_{12}}{2}\|{\nabla\eta_{h}^{k}}\|^{2}_{0}+\frac{C_{12}}{2}\|\nabla\eta_{h}^{n-1}\|^{2}_{0}+\big(2C_{11}C_{\Omega} h^{r}+\|\nabla (R_{t}^{n-\sigma}+ R_{\sigma}^{n-\sigma}\\ &&\quad+ R_{f}^{n-\sigma})\|_{0}+C_{\Omega}\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\nabla u^{n-\sigma}\|_{r+1}h^{r}\big)\|\nabla\eta_{h}^{n-\sigma}\|_{0}, \end{array} $$

(3.73)

where C₁₂ is a constant dependent on u, σ, f and Q₂. By Lemma 3.2 and (3.48), it holds

$$ \begin{array}{@{}rcl@{}} \|{\nabla\eta_{h}^{n}}\|_{0} &\leq& 2E_{\alpha}(8C_{12}t_{n}^{\alpha})\Big(\|{\nabla\eta_{h}^{0}}\|^{2}_{0}+4t_{n}^{\alpha}{\Gamma}(1-\alpha)\big(2C_{11}C_{\Omega} h^{r}+\max_{1\leq i\leq n}C_{\Omega}\|{~}^{\text{F}}\mathcal{D}^{{\alpha}}_{\tau}\nabla u^{i}\|_{r+1}h^{r}\big)\\ &&\quad+\max_{1\leq j\leq n}{\sum}_{i=1}^{j}P_{j-i}^{(j)}\|\nabla(R_{t}^{i-\sigma}+ R_{\sigma}^{i-\sigma}+ R_{f}^{i-\sigma})\|_{0}\Big) \\ &\leq& 2E_{\alpha}(8C_{12}t_{n}^{\alpha})\Big(C_{\Omega}+4T^{\alpha}{\Gamma}(1-\alpha)\big(2C_{11}C_{\Omega} h^{r}+\max_{1\leq i\leq n}C_{\Omega}\|{~}^{\text{F}}\mathcal{D}^{\alpha}_{\tau}\nabla u^{i}\|_{r+1}h^{r}\big)\\ &&\quad+C\big(\tau^{\min\{\gamma\alpha,2\}}+\frac\epsilon\alpha (T\hat{T})^{\alpha} \big)\Big), \end{array} $$

(3.74)

when $\tau \leq \tau _{g}=1/{\sqrt [\alpha ]{8C_{12}{\Gamma }(2-\alpha )}}$. Therefore, we have

$$ \|u^{n}-{U_{h}^{n}}\|_{1} \leq \| u^{n}- R_{h}u^{n}\|_{1} + \| R_{h}u^{n}- {U_{h}^{n}}\|_{1} = \| {\xi_{h}^{n}}\|_{1} + \| {\eta_{h}^{n}}\|_{1} \leq C^{*}\big(\tau^{\min\{\gamma\alpha,2\}}+h^{r}+\epsilon\big), $$

(3.75)

whenever γα ≤ 2, $\tau ^{*}=\min \limits \{\tau ^{**},\tau ^{***},\tau _{f},\tau _{g}\}$, $h^{*}=\min \limits \{h_{1},h_{2}\}$ and $\epsilon ^{*}=\min \limits \{\epsilon _{1},\epsilon _{2},\epsilon _{3}\}$. Thus the proof of Theorem 2.1 is completed. □

4 Numerical examples

We now provide two examples to demonstrate the unconditionally optimal convergence orders with $\mathcal {O}(\tau ^{\min \limits \{\gamma \alpha ,2\}})$ in time and $\mathcal {O}(h^{r})$ in space as presented in Theorem 2.1. Here we consider the graded meshes t_k = (k/N)^γ(k = 1,…,N), and divide the space Ω into M parts with quasi-uniform quadrilateral partition $\mathcal {T}_{i} (i=1,\cdots ,M)$ with maximum mesh size $h=\max \limits _{1\leq i\leq M}\{\text {diam} \mathcal {T}_{i}\}$. The error is calculated by H¹-norm in space and the maximum norm in time.

Example 4.1

We first consider the following nonlinear subdiffusion equation

$${~}^{\text{C}}\mathcal{D}^{\alpha}_t u={\Delta} u + \sin(u) + g(x,y,t) ,\quad (x,y)\in (0,1)^{2}, \quad t\in(0,1].$$

As a benchmark solution to investigate the convergence orders in time and space by using the linear and quadratic elements, the exact solution is constructed by u(x,y,t) = (t² + t^α)x(1 − x)y(1 − y).

Tables 1 and 2 show the temporal errors and convergence orders by taking N = 8,16,32,64 and M = ⌈N^γα⌉ with linear element for α = 0.5 and α = 0.8, respectively. Table 3 presents the numerical results of the linear and quadratic elements for α = 0.5, γ = 2, N = 10⁴, M = 12,24,48,96, which illustrates the r-degree finite element method has r-order accuracy.

Table 1 Errors and convergence orders of temporal directions for Example 4.1

Table 2 Errors and convergence orders of temporal directions for Example 4.1

Table 3 Errors and convergence orders of spatial directions when α = 0.5, γ = 2 and N = 10⁴ for Example 4.1

The CPU time of the direct algorithm (2.7) and the fast algorithm (2.16) is given in Table 4, which is calculated by using the linear finite element with M = 15, α = 0.5, γ = 2 and changing N from 1000 to 16000. One can see that the fast L2-1_σ scheme can speed up the simulations significantly as N is larger.

Table 4 Computational time with M = 15, α = 0.5, γ = 2 for Example 4.1

If a numerical method is unconditionally convergent, given a N, the error should be tended to a constant as M is taken larger and larger. The phenomenon is displayed in Fig. 1 by respectively taking the linear and quadratic elements with α = 0.5, γ = 2 for different M and N. Figure 1 shows that our error estimates are unconditionally convergent.

Fig. 1

H¹-errors of linear and quadratic finite elements with α = 0.5 and γ = 2 for Example 4.1

Example 4.2

Consider the following two-dimensional nonlinear subdiffusion equation

$$ {~}^{\text{C}}\mathcal{D}^{\alpha}_t u={\Delta} u + u(1-u^{2}) + g(x,y,t) ,\quad (x,y)\in (0,1)^{2}, \quad t\in(0,1], $$

with $u(x,y,t)=(1+t^{\alpha })\sin \limits (\pi x)\sin \limits (\pi y)$ to investigate the convergence order in time and space by using the linear and quadratic elements.

Tables 5 and 6 show the errors and convergence orders of temporal directions by taking N = 8,16,32,64 and M = ⌈N^γα⌉ with linear element for α = 0.5 and α = 0.8, respectively. Table 7 presents the numerical results of the linear and quadratic elements for α = 0.5, γ = 2, N = 10⁴, M = 8,16,32,64, which illustrates the r-degree finite element method has r-order accuracy again.

Table 5 Errors and convergence orders of temporal directions for Example 4.2

Table 6 Errors and convergence orders of temporal directions for Example 4.2

Table 7 Errors and convergence orders of spatial directions when α = 0.5, γ = 2 and N = 10⁴ for Example 4.2

In Table 8, the computational time of the direct algorithm (2.7) is compared with the fast algorithm (2.16) with M = 10, α = 0.3 and γ = 2 for N from 1000 to 16000. One can see that the smaller the time step, the more effective the fast algorithm.

Table 8 Computational time with M = 10, α = 0.3 and γ = 2 for Example 4.2

In Fig. 2, the errors of the linear (on the left) and quadratic (on the right) elements are shown with a fixed N and increasing M for α = 0.3, γ = 2. The figure shows that the errors tend to different constants which illustrates our theoretical analysis is unconditionally convergent.

Fig. 2

H¹-errors of linear and quadratic finite elements with α = 0.3 and γ = 2 for Example 4.2

5 Conclusion

The unconditionally optimal H¹-error estimate of the SOE-based fast L2-1_σ scheme is presented to numerically solve the nonlinear subdiffusion problem (1.1)–(1.3) on the nonuniform mesh. The SOE approximation for Caputo derivative can efficiently reduce the computational storage and cost when time steps are large. To deal with the initial singularity of the solution, a nonuniform mesh is used to have the globally optimal convergence, which also bring the complication of theoretical analysis. Thus, we used a modified discrete fractional Grönwall inequality to present the stability analysis, and introduced the ECS to express the truncation error of SOE-based fast L2-1_σ scheme. Combining with DCC kernels and ECS, the global error analysis is significantly simplified. For the nonlinear term, we consider a linearized scheme by approximating it with a Newton linearization method and approximate the dissipative term with implicit scheme. Then, we use the spatial-temporal splitting approach to prove that the proposed scheme is unconditionally convergent. Numerical tests are given to verify the effectiveness and optimal convergence of our scheme.

Appendix: . The proof of Lemma 3.4

Proof 4

By the Taylor expansion, we obtain

$$ \begin{array}{@{}rcl@{}} R_{f}^{k-\sigma}&=&{\int}_{v^{k-1}}^{v^{k-\sigma}}(v^{k-\sigma}-\mu)f^{\prime\prime}(\mu) \mathrm{d} \mu\\ &=&{{\int}_{0}^{1}}\left[v^{k-\sigma}-v^{k-1}-s(v^{k-\sigma}-v^{k-1})\right]f^{\prime\prime}\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right)(v^{k-\sigma}-v^{k-1}) \mathrm{d} s\\ &=&(1-\sigma)^{2}(v^{k}-v^{k-1})^{2}{{\int}_{0}^{1}}(1-s)f^{\prime\prime}\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right) \mathrm{d} s,\\ \nabla R_{f}^{k-\sigma}&=&(1-\sigma)^{2}(v^{k}-v^{k-1})^{2}{{\int}_{0}^{1}}(1-s)f^{\prime\prime\prime}\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right)\nabla\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right) \mathrm{d} s\\ &&\quad+2(1-\sigma)^{2}(v^{k}-v^{k-1})\nabla(v^{k}-v^{k-1}){{\int}_{0}^{1}}(1-s)f^{\prime\prime}\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right) \mathrm{d} s,\\ {\Delta} R_{f}^{k-\sigma}&=&2(1-\sigma)^{2}(v^{k}-v^{k-1})\nabla(v^{k}-v^{k-1}){{\int}_{0}^{1}}(1-s)f^{\prime\prime\prime}\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right)\\ &&\quad\quad\nabla\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right) \mathrm{d}s\\ &&\quad+(1-\sigma)^{2}(v^{k}-v^{k-1})^{2}{{\int}_{0}^{1}}(1-s)f^{(4)}\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right)\left( \nabla(v^{k-1}+s(v^{k-\sigma}-v^{k-1}))\right)^{2} \mathrm{d} s\\ &&\quad+(1-\sigma)^{2}(v^{k}-v^{k-1})^{2}{{\int}_{0}^{1}}(1-s)f^{\prime\prime\prime}\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right){\Delta}\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right) \mathrm{d} s\\ &&\quad+2(1-\sigma)^{2}\left( \nabla(v^{k}-v^{k-1})\right)^{2}{{\int}_{0}^{1}}(1-s)f^{\prime\prime}\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right) \mathrm{d} s \end{array} $$

$$ \begin{array}{@{}rcl@{}} &&\quad+2(1-\sigma)^{2}(v^{k}-v^{k-1}){\Delta}(v^{k}-v^{k-1}){{\int}_{0}^{1}}(1-s)f^{\prime\prime}\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right) \mathrm{d} s\\ &&\quad+2(1-\sigma)^{2}(v^{k}-v^{k-1})\nabla(v^{k}-v^{k-1}){{\int}_{0}^{1}}(1-s)f^{\prime\prime\prime}\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right)\\ &&\quad\quad\nabla\left( v^{k-1}+s(v^{k-\sigma}-v^{k-1})\right) \mathrm{d} s. \end{array} $$

By the condition (1.5) of v, we have

$$ \begin{array}{@{}rcl@{}} \left|R_{f}^{k-\sigma}\right|&\leq& C_{v}\left( {\int}_{t_{k-1}}^{t_{k}}\left|v^{\prime}(t)\right|dt\right)^{2}\leq C_{v}\left( {\int}_{t_{k-1}}^{t_{k}}(1+t^{\alpha-1})dt\right)^{2}\leq\begin{cases} C_{v}({\tau_{1}^{2}}+\frac{\tau_{1}^{2\alpha}}{\alpha^{2}}),& \ k=1,\\ C_{v}({\tau_{k}^{2}}+{\tau_{k}^{2}}t_{k-1}^{2(\alpha-1)}),& \ 2\leq k\leq n, \end{cases} \end{array} $$

$$ \begin{array}{@{}rcl@{}} \left|\nabla R_{f}^{k-\sigma}\right|&\leq& C_{v}\left( {\int}_{t_{k-1}}^{t_{k}}|v^{\prime}(t)|dt\right)^{2}+C_{v}{\int}_{t_{k-1}}^{t_{k}}|v^{\prime}(t)|dt{\int}_{t_{k-1}}^{t_{k}}|\nabla v^{\prime}(t)|dt\leq \begin{cases} C_{v}({\tau_{1}^{2}}+\frac{\tau_{1}^{2\alpha}}{\alpha^{2}}),& \ k=1,\\ C_{v}({\tau_{k}^{2}}+{\tau_{k}^{2}}t_{k-1}^{2(\alpha-1)}),& \ 2\leq k\leq n, \end{cases} \end{array} $$

$$ \begin{array}{@{}rcl@{}} \left|{\Delta} R_{f}^{k-\sigma}\right|&\leq& C_{v}{\int}_{t_{k-1}}^{t_{k}}|v^{\prime}(t)|dt{\int}_{t_{k-1}}^{t_{k}}|\nabla v^{\prime}(t)|dt+C_{v}\left( {\int}_{t_{k-1}}^{t_{k}}|v^{\prime}(t)|dt\right)^{2}+C_{v}\left( {\int}_{t_{k-1}}^{t_{k}}|v^{\prime}(t)|dt\right)^{2}\\ &&\quad+C_{v}\left( {\int}_{t_{k-1}}^{t_{k}}|\nabla v^{\prime}(t)|dt\right)^{2}+C_{v}{\int}_{t_{k-1}}^{t_{k}}|v^{\prime}(t)|dt{\int}_{t_{k-1}}^{t_{k}}|{\Delta} v^{\prime}(t)|dt+C_{v}{\int}_{t_{k-1}}^{t_{k}}|v^{\prime}(t)|dt{\int}_{t_{k-1}}^{t_{k}}|\nabla v^{\prime}(t)|dt\\ &\leq&\begin{cases} C_{v}({\tau_{1}^{2}}+\frac{\tau_{1}^{2\alpha}}{\alpha^{2}}),& \ k=1,\\ C_{v}({\tau_{k}^{2}}+{\tau_{k}^{2}}t_{k-1}^{2(\alpha-1)}),& \ 2\leq k\leq n. \end{cases} \end{array} $$

It can be further obtained that

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{k=1}^{n}P_{n-k}^{(n)}\left|R_{f}^{k-\sigma}\right|+{\sum}_{k=1}^{n}P_{n-k}^{(n)}\left|\nabla R_{f}^{k-\sigma}\right|+\sum\limits_{k=1}^{n}P_{n-k}^{(n)}\left|{\Delta} R_{f}^{k-\sigma}\right|\\ &\leq& P_{n-1}^{(n)}\left|R_{v}^{1-\sigma}\right|+\max_{2\leq k\leq n}\left|R_{f}^{k-\sigma}\right|\sum\limits_{k=2}^{n}P_{n-k}^{(n)}+P_{n-1}^{(n)}\left|\nabla R_{v}^{1-\sigma}\right|+\max_{2\leq k\leq n}\left|\nabla R_{f}^{k-\sigma}\right|{\sum}_{k=2}^{n}P_{n-k}^{(n)}\\ &&\quad+P_{n-1}^{(n)}\left|{\Delta} R_{v}^{1-\sigma}\right|+\max_{2\leq k\leq n}\left|{\Delta} R_{f}^{k-\sigma}\right|{\sum}_{k=2}^{n}P_{n-k}^{(n)}\\ &\leq& C_{v}\left[\tau_{1}^{3\alpha}+\max_{2\leq k\leq n}t_{n}^{\alpha}\left( {\tau_{k}^{2}}+{\tau_{k}^{2}}t_{k-1}^{2\alpha-2}\right)\right], \end{array} $$

where C_v in different places represents different constant. The proof is completed. □