Pointwise Error Analysis of the Corrected L1 Scheme for the Multi-Term Subdiffusion Equation

Abstract

This paper considers the multi-term subdiffusion equation with weakly singular solutions. In order to use sparser meshes near the initial time, a novel scheme (which we call the corrected L1 scheme) on graded meshes is constructed to estimate the multi-term Caputo fractional derivative by investigating a corrected term for the nonuniform L1 scheme. Combining this nonuniform corrected L1 scheme in the temporal direction and the finite element method (FEM) in the spatial direction, a fully discrete scheme for solving the multi-term subdiffusion equation is developed. The stability result of the developed scheme is given. Furthermore, the optimal pointwise-in-time error estimate of the developed scheme is derived. Finally, several numerical experiments are conducted to verify our theoretical findings.

1. Introduction

Numerous numerical approaches, such as convolution quadrature (CQ) backward differentiation formulas (BDFs) with some corrector terms [1], the nonuniform L1 scheme [2,3,4,5], the nonuniform Alikhanov scheme [6,7], the nonuniform L2 scheme [8], and spectral methods [9], have emerged in recent years to solve time-fractional partial differential equations with a single-term fractional derivative, whose solutions exhibit weak singularity near the initial time. As a generalization of the classical diffusion equation, the subdiffusion equation with a multi-term fractional derivative in time has been extensively utilized in various fields such as physics, engineering, environmental studies, biological systems, and financial modeling [10,11,12,13,14,15].

In this paper, we will consider the numerical simulation of the following multi-term subdiffusion equation:

$$\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}u-{\kappa }^2\Delta u+cu=f(x,t)\forall (x,t)\in Q:=\mathrm{\Omega }\times (0,T],$$

(1)

$$u(x,0)=u_0\left(x\right)\mathrm{for}x\in \mathrm{\Omega },$$

(2)

$${u|}_{\partial \mathrm{\Omega }}=0\mathrm{for}0<t\le T,$$

(3)

where $0<{\alpha }_q<{\alpha }_{q-1}<\dots <{\alpha }_1<1$, $s_i$ are positive constants, c is a non-negative constant, $\mathrm{\Omega }\subset {\mathbb{R}}^d$ for some $d\in \{1,2,3\}$ is a bounded Lipschitz domain [16] (p. 13), $u_0\in C\left(\overline{\mathrm{\Omega }}\right)$, and $f\in C\left(\overline{\mathrm{\Omega }}\right)$. Without loss of generality, we assume that $s_1=1$. The standard Caputo fractional derivative [17,18] $D_{0,t}^{\alpha }$ of (1) is proposed by Michele Caputo in [17], which is defined by

$$D_{0,t}^{\alpha }w\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_0^t{(t-s)}^{-\alpha }{w}^{\prime }\left(s\right)ds\mathrm{for}w\in C^1(0,T].$$

(4)

During recent years, many researchers have begun to focus on the numerical simulation for the multi-term subdiffusion equation with weakly singular solutions. Huang et al. [19] derived a priori bounds on the derivatives of the unknown solution for problem (1)–(3), and proved that there exists a unique solution to problem (1)–(3). Moreover, the nonuniform L1 scheme is constructed to approximate the multi-term temporal fractional derivative. Chen et al. [20] established the pointwise-in-time error estimate of a Grünwald–Letnikov scheme for problem (1)–(3) with initial singularity. Meng and Stynes [21] constructed the barrier function to estimate the errors of the developed L1-FEM scheme for problem (1)–(3), which indicated that the global and local errors in time are sharp. Cao and Chen [22] presented a sharp pointwise-in-time error estimate for the L1-ADI scheme for problem (1)–(3). Hu and Li [23] proposed a fast element-free Galerkin method to solve problem (1)–(3), where the multi-term Alikhanov scheme is used in the temporal direction. Roul et al. [24] constructed L1 schemes on moving meshes to solve problem (1)–(3) and obtained the corresponding convergent result. In addition, several efficient numerical methods have been proposed for the multi-term fourth-order equation, see [25,26].

In fact, the convergent result of [21] showed that global and local errors in time attain $2-{\alpha }_1$ order by choosing $r=(2-{\alpha }_1)/{\alpha }_1$ and $r=(2-{\alpha }_1)$, respectively. To use a smaller grading parameter r to achieve the optimal convergence order, a novel scheme is developed to approximate the multi-term Captuo fractional derivative of problem (1)–(3) by investigating an additional term. By virtue of this novel scheme in time, a fully discrete finite element method is developed for problem (1)–(3). Moreover, the optimal pointwise-in-time error analysis of the developed scheme is established.

The remainder of this paper is organized as follows. In Section 2, a corrected L1 scheme is developed for the multi-term fractional derivative, and a fully discrete finite element method is constructed to solve problem (1)–(3). In Section 3, the stability result and the optimal pointwise-in-time convergence result for the developed scheme are given. Numerical experiments are presented to verify the sharpness of our theoretical findings in Section 4. Finally, a concluding remark is given in Section 5.

Notation. $C_{\mathrm{\Omega }}$ and $C_T$ represent generic constants that are independent of the mesh; they can take different values in different places. The notation $A\lesssim B$ represents $A\le CB$ with a positive constant C independent of the meshes.

2. The Construction of the Corrected L1 Scheme

In this section, a corrected L1 scheme on graded meshes is developed to approximate the Caputo derivative of (1)–(3), and the truncation error of the corrected L1 scheme is given. Using the corrected L1 scheme in time and the finite element method in space, a fully discrete scheme for problem (1)–(3) is developed.

Let r be a grading parameter satisfying $r\ge 1$. Define $t_s:=T{(s/N)}^r$ for $1\le s\le N$ and denote ${\tau }_s:=t_s-t_{s-1}$ for $s=1,2,\dots ,N$, where N is a positive integer. Now we can approximate the Caputo derivative of (4) by the following nonuniform L1 scheme [5] (3.1):

$$D_{0,t}^{{\alpha }_i}{w}^n\approx {\delta }_N^{{\alpha }_i}{v}^n={\mathrm{\Theta }}_{n,1}^{{\alpha }_i}{w}^n-{\mathrm{\Theta }}_{n,n}^{{\alpha }_i}{w}^0+\sum _{j=1}^{n-1}({\mathrm{\Theta }}_{n,j+1}^{{\alpha }_i}-{\mathrm{\Theta }}_{n,j}^{{\alpha }_i})w^{n-j}\mathrm{for}1\le n\le N,$$

(5)

where

$${\mathrm{\Theta }}_{n,j}^{{\alpha }_i}:={\displaystyle \frac{{(t_n-t_{n-j})}^{1-{\alpha }_i}-{(t_n-t_{n-j+1})}^{1-{\alpha }_i}}{\mathrm{\Gamma }(2-{\alpha }_i){\tau }_{n-j+1}}}\mathrm{for}1\le j\le n\le N.$$

For $1\le n\le N$, applying (5) yields

$$\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}{w}^n\approx \sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{w}^n={\mathrm{\Theta }}_{n,1}{w}^n-{\mathrm{\Theta }}_{n,n}{w}^0+\sum _{j=1}^{n-1}({\mathrm{\Theta }}_{n,j+1}-{\mathrm{\Theta }}_{n,j})w^{n-j},$$

(6)

where

$${\mathrm{\Theta }}_{n,j}:=\sum _{i=1}^q{s}_i{\mathrm{\Theta }}_{n,j}^{{\alpha }_i}\mathrm{for}1\le j\le n\le N.$$

Suppose that the function $w(·,t)$ satisfies $\parallel {\partial }_t^s{w(·,t)\parallel }_2\le C(1+t^{\sigma -s})$ for $s=0,1,2$ and all $t\in (0,T]$. According to [21] (Lemma 4), we derive the following temporal truncation error:

$$\parallel \sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}w(·,t_n)-\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}w(·,t_n)\parallel \lesssim \mathrm{\Phi }(n,\sigma ):=t_n^{\sigma -{\alpha }_1}{\left({\displaystyle \frac{t_1}{t_n}}\right)}^{min\{\sigma +1,{\displaystyle \frac{2-{\alpha }_1}{r}}\}}.$$

(7)

In order to apply a smaller r than the standard L1 scheme (6) to achieve optimal accuracy, we will construct a corrected L1 scheme by investigating a corrected term. This novel scheme for the Caputo derivative at $t=t_n$ is defined by

$$\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{w}^n:=\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{w}^n+V_n^{\sigma }(w^1-w^0)\mathrm{for}n\ge 1,$$

(8)

where $V_n^{\sigma }$ satisfies

$$\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{w}^n=\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}w\left(t_n\right)\mathrm{for}w=t^{\sigma }.$$

Calculating the above relation yields

$$V_n^{\sigma }=t_1^{-\sigma }\left(\sum _{i=1}^q{s}_i{\displaystyle \frac{\mathrm{\Gamma }(1+\sigma )}{\mathrm{\Gamma }(1+\sigma -{\alpha }_i)}}{t}_n^{\sigma -{\alpha }_i}-{\mathrm{\Theta }}_{n,1}{t}_n^{\sigma }-\sum _{k=1}^{n-1}{t}_{n-k}^{\sigma }({\mathrm{\Theta }}_{n,k+1}-{\mathrm{\Theta }}_{n,k})\right).$$

(9)

Next, an a priori bound of $V_n^{\sigma }$ and the truncation of the corrected L1 scheme will be given in the following lemma.

Lemma 1.

Suppose that $w\left(t\right)=v\left(t\right)+\gamma \left(t\right)$, $0<{\sigma }_1<1$, ${\sigma }_1\le {\sigma }_2$, and

$$v\left(t\right)=a_0+a_1{t}^{{\sigma }_1},\left|{\gamma }^{\left(l\right)}\left(t\right)\right|\le C{t}^{{\sigma }_2-l}forl=0,1,2,$$

(10)

where $a_0,a_1\in \mathbb{R}$. Then we have

$$\begin{array}{cc}\hfill \left|V_n^{{\sigma }_1}\right|& \le C_T{t}_1^{-{\sigma }_1}\mathrm{\Phi }(n,{\sigma }_1)for1\le n\le N,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill \left|\sum _{i=1}^q{q}_i{D}_{0,t}^{{\alpha }_i}w\left(t_n\right)-\sum _{i=1}^q{q}_i{\mathbb{D}}_N^{{\alpha }_i}{w}^n\right|& \le C_T\mathrm{\Phi }(n,{\sigma }_2)+C_T{t}_1^{{\sigma }_2-{\sigma }_1}\mathrm{\Phi }(n,{\sigma }_1)for1\le n\le N,\hfill \end{array}$$

(12)

Proof. 

By utilizing (7) and (9), we have

$$\left|V_n^{{\sigma }_1}\right|=t_1^{-{\sigma }_1}\left|\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}{t}_n^{{\sigma }_1}-\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{\left(t^{{\sigma }_1}\right)}^n\right|\le C_T{t}_1^{-{\sigma }_1}\mathrm{\Phi }(n,{\sigma }_1),$$

which indicates that (11) holds.

Now applying the corrected L1 scheme (8) yields

$$\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}v\left(t_n\right)-\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{v}^n=0.$$

Combining this result with (8), we arrive at

$$\begin{array}{cc}\hfill \left|\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}w\left(t_n\right)-\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{w}^n\right|& =\left|\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}v\left(t_n\right)-\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{v}^n\right.\hfill \\ \hfill & \left.+\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}\gamma \left(t_n\right)-\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{\gamma }^n\right|\hfill \\ \hfill & =\left|\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}\gamma \left(t_n\right)-\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{\gamma }^n-V_n^{{\sigma }_1}{\gamma }^1\right|\hfill \\ \hfill & \le C_T\mathrm{\Phi }(n,{\sigma }_2)+C_T{t}_1^{{\sigma }_2-{\sigma }_1}\mathrm{\Phi }(n,{\sigma }_1),\hfill \end{array}$$

where (7) and (11) are used. □

Partition the domain $\mathrm{\Omega }$ by a quasiuniform mesh with the diameter h. Define $S_h$ as a finite-dimensional subspace of $H_0^1\left(\mathrm{\Omega }\right)$, consisting of multivariate polynomial functions in the variables d ($d\in \{1,2,3\}$), where the maximum degree in each variable does not exceed a given positive integer k.

Since the finite element space $S_h\subset H_0^1\left(\mathrm{\Omega }\right)$ is the space of piecewise polynomials of degree at most k, one can show [27] (Lemma 1.1) that

$$\parallel w-R_{h}w\parallel +h\parallel w-R_h{w\parallel }_1\le C{h}^{k+1}{\left|w\right|}_{k+1}\forall w\in H^{k+1}\left(\mathrm{\Omega }\right)\cap H_0^1\left(\mathrm{\Omega }\right),$$

(13)

where the Ritz projection $R_h$ is defined by $(\nabla R_{h}w,\nabla w_h)=(\nabla w,\nabla w_h)$ for all $w_h\in S_h$.

Now we define the discrete Laplacian operator ${\Delta }_h:S_h\to S_h$ by

$$({\Delta }_{h}w,q)=-(\nabla w,\nabla q)\forall w,q\in S_h.$$

(14)

According to [27] (Equation (1.34)), the operators $R_h$ and ${\Delta }_h$ satisfy the following property:

$${\Delta }_h{R}_{h}w=P_h\Delta w\forall w\in H^2\left(\mathrm{\Omega }\right),$$

(15)

where $P_h$ is the standard $L^2$ projection. Moreover, from [28] (Theorem 4.1), the projection $P_h$ satisfies

$$\parallel \nabla P_{h}w\parallel \le K\parallel \nabla w\parallel \mathrm{for}\mathrm{all}w\in H^1\left(\mathrm{\Omega }\right),$$

(16)

where the positive constant K is independent of h.

Now applying the corrected L1 scheme (8) to approximate ${\sum }_{i=1}^q{s}_i{D}_t^{{\alpha }_i}u$ at each time level $t_n$ and the finite element method in the spatial direction, we obtain our fully discrete method (which we call CL1-FEM). Find $u_h^n\in S_h$ such that

$$\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{u}_h^n-{\kappa }^2{\Delta }_h{u}_h^n+c{u}_h^n=P_h{f}^n\mathrm{for}1\le n\le N,$$

(17)

with $u_h^0=R_h{u}_0$.

3. Optimal Pointwise-in-Time Error Estimate of the CL1-FEM

In this section, we will present the stability result and the optimal pointwise-in-time convergence result of the CL1-FEM (17).

From [29] (Lemma 5.1), we obtain that the nonuniform L1 scheme (6) satisfies the following important property:

$$\left(\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{w}^n,w^n\right)\ge \left(\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}\parallel w^n\parallel \right)\parallel w^n\parallel \mathrm{for}1\le n\le N.$$

(18)

For $1\le j\le n\le N$, define the positive constants ${\mathrm{\Pi }}_{n,j}$ by

$${\mathrm{\Pi }}_{n,n}=1,{\mathrm{\Pi }}_{n,j}=\sum _{s=1}^{n-j}{\displaystyle \frac{1}{{\mathrm{\Theta }}_{n-s,1}}}({\mathrm{\Theta }}_{n,s}-{\mathrm{\Theta }}_{n,s+1}){\mathrm{\Pi }}_{n-s,j}>0.$$

Next we will present the following two lemmas, which will be used in our later error analysis.

Lemma 2

([30] (Lemma 7)). For the arbitrary functions ${\left\{p^j\right\}}_{j=0}^N$, denote $H^0=0$ and $H^n={\mathrm{\Theta }}_{n,1}^{-1}\sum _{j=1}^n{\mathrm{\Pi }}_{n,j}{p}^j$ for $1\le n\le N$. Then one has ${\sum }_{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{H}^n=p^n$ for $1\le n\le N$.

Lemma 3

([21] (Theorem 2 (iii))). Let $\gamma \in R$ and $w^0=0$. Suppose that the non-negative sequence $w^n$ satisfies

$$\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{w}^n\le {(t_1/t_n)}^{\gamma +1},$$

Then one has

$$w^n\le C_T{t}_1{t}_n^{{\alpha }_1-1}\left\{\begin{array}{cc}1\hfill & if\gamma >0,\hfill \\ 1+lnn\hfill & if\gamma =0\forall n\ge 1,\hfill \\ {(t_1/t_n)}^{\gamma }\hfill & if\gamma <0.\hfill \end{array}\right.$$

(19)

According to Lemma 2 and Lemma 3, we obtain the following corollary.

Corollary 1.

For $1\le n\le N$, one has

$${\mathrm{\Theta }}_{n,1}^{-1}\sum _{j=1}^n{\mathrm{\Pi }}_{n,j}\le C_T{t}_n^{{\alpha }_1}.$$

(20)

Proof. 

Define $H^0=1$ and $H^n={\mathrm{\Theta }}_{n,1}^{-1}\sum _{s=1}^n{\mathrm{\Pi }}_{n,j}$ for $1\le n\le N$. Applying Lemma 2 yields ${\sum }_{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{H}^n=1$. By choosing $w^n=H^n$ and $\gamma =-1$ in Lemma 3, we have

$$H^n\le C_T{t}_1{t}_n^{{\alpha }_1-1}{(t_1/t_n)}^{-1}\le C_T{t}_n^{{\alpha }_1},$$

which implies that (20) holds. □

In order to obtain the optimal pointwise-in-time error analysis of our fully discrete scheme (17), we state the following property.

Lemma 4.

For $1\le n\le N$ and any $\sigma \ge {\alpha }_1$, we have

$${\mathrm{\Theta }}_{n,1}^{-1}\sum _{j=1}^n{\mathrm{\Pi }}_{n,j}\mathrm{\Phi }(j,\sigma )\lesssim \mathcal{E}(n,\sigma ):=\left\{\begin{array}{cc}{N}^{-r(\sigma -{\alpha }_1+1)}{t}_n^{{\alpha }_1-1}\hfill & if1\le r<{\displaystyle \frac{(2-{\alpha }_1)}{(\sigma -{\alpha }_1+1)}},\hfill \\ N^{{\alpha }_1-2}{t}_n^{{\alpha }_1-1}\left(1+lnn\right)\hfill & ifr={\displaystyle \frac{(2-{\alpha }_1)}{(\sigma -{\alpha }_1+1)}},\hfill \\ N^{{\alpha }_1-2}{t}_n^{{\alpha }_1-(2-{\alpha }_1)/r}\hfill & ifr>{\displaystyle \frac{(2-{\alpha }_1)}{(\sigma -{\alpha }_1+1)}}.\hfill \end{array}\right.$$

(21)

Proof. 

Set $H^0=0$ and $H^n={\mathrm{\Theta }}_{j,1}^{-1}\sum _{j=1}^n{\mathrm{\Pi }}_{n,j}\mathrm{\Phi }(j,\sigma )$ for $1\le n\le N$. By using Lemma 2, one has

$$\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{H}^n=\mathrm{\Phi }(n,\sigma )\mathrm{for}1\le n\le N.$$

(22)

Actually, (22) implies that (21) holds for the case $\sigma ={\alpha }_1$ by using Lemma 3. Now we will prove the general case $\sigma >{\alpha }_1$. Define ${\tilde{H}}^n=N^{r(\sigma -{\alpha }_1)}{H}^n$ for $1\le n\le N$. Multiplying $N^{r(\sigma -{\alpha }_1)}$ on both sides of (22) yields

$$\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{\tilde{H}}^n=N^{r(\sigma -{\alpha }_1)}\mathrm{\Phi }(n,\sigma )=T^{\sigma -{\alpha }_1}{(\tau /t_n)}^{\gamma +1},$$

where

$$\gamma +1=min\{1+\sigma ,(2-{\alpha }_1)/r-\sigma +{\alpha }_1\}.$$

Next we will prove that (21) holds for $\sigma >{\alpha }_1$ by the following three cases.

(1) If $1\le r\le (2-{\alpha }_1)/(\sigma -{\alpha }_1+1)$, we have $\gamma =min\{{\alpha }_1,(2-{\alpha }_1)/r-\sigma +{\alpha }_1-1\}>0$. Combining $\gamma >0$ with Lemma 3 yields

$${\tilde{H}}^n\le C{T}^{\sigma -{\alpha }_1}{t}_1{t}_n^{{\alpha }_1-1}=C{T}^{1+\sigma -{\alpha }_1}{N}^{-r}{t}_n^{{\alpha }_1-1},$$

which implies

$$H^n\le C{T}^{1+\sigma -{\alpha }_1}{N}^{-r(\sigma -{\alpha }_1)}{N}^{-r}{t}_n^{{\alpha }_1-1}=C{T}^{1+\sigma -{\alpha }_1}{N}^{-r(\sigma -{\alpha }_1+1)}{t}_n^{{\alpha }_1-1}.$$

(2) If $r=(2-{\alpha }_1)/(\sigma -{\alpha }_1+1)$, one has $\gamma =0$. By using $\gamma =0$ and Lemma 3, we arrive at

$${\tilde{H}}^n\le C{T}^{\sigma -{\alpha }_1}{t}_1{t}_n^{{\alpha }_1-1}(1+lnn)=C{T}^{1+\sigma -{\alpha }_1}{N}^{-r}{t}_n^{{\alpha }_1-1}(1+lnn),$$

which indicates

$$H^n\le C{T}^{\sigma -{\alpha }_1}{N}^{-r(\sigma -{\alpha }_1)}{N}^{-r}{t}_n^{{\alpha }_1-1}(1+lnn)=C{T}^{1+\sigma -{\alpha }_1}{N}^{-(2-{\alpha }_1)}{t}_n^{{\alpha }_1-1}(1+lnn).$$

(3) If $r>(2-{\alpha }_1)/(\sigma -{\alpha }_1+1)$, we obtain $\gamma =min\{{\alpha }_1,(2-{\alpha }_1)/r-\sigma +{\alpha }_1-1\}<0$. Applying $\gamma <0$ and Lemma 3, we have

$${\tilde{H}}^n\le C{T}^{\sigma -{\alpha }_1}{t}_1{t}_n^{{\alpha }_1-1}{(t_1/t_n)}^{\gamma },$$

which displays

$$H^n\le C{T}^{1+\sigma -{\alpha }_1}{N}^{-r(\sigma -{\alpha }_1)}{N}^{-r}{t}_n^{{\alpha }_1-1}{(t_1/t_n)}^{\gamma }=C{T}^{1+\sigma -{\alpha }_1}{N}^{-(2-{\alpha }_1)}{t}_n^{\sigma -{\alpha }_1-(2-{\alpha }_1)/r}.$$

Finally, our proof is finished. □

Now a discrete fractional Gronwall inequality of the L1 scheme on graded meshes is presented in the following lemma.

Lemma 5

([31] (Lemma 4.2)). Assume that the sequences ${\left\{{\mu }^n\right\}}_{n=1}^{\infty },{\left\{{\nu }^n\right\}}_{n=1}^{\infty }$ are non-negative and the non-negative grid function $\{w^n:n=0,1,\dots ,N\}$ satisfies $w^0\ge 0$ and

$$\left(\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{w}^n\right)w^n\le {\mu }^n{w}^n+{\left({\nu }^n\right)}^{2}for1\le n\le N.$$

Then, we have

$$w^n\le w^0+{\mathrm{\Theta }}_{n,1}^{-1}\sum _{j=1}^n{\mathrm{\Pi }}_{n,j}\left({\mu }^j+{\nu }^j\right)+\underset{1\le j\le n}{max}\left\{{\nu }^j\right\}for1\le n\le N.$$

Next the stability of the CL1-FEM (17) will be presented in the following theorem.

Theorem 1.

The solution $u_h^n$ of (17) satisfies

$$\parallel \nabla u_h^n\parallel \lesssim \parallel \nabla u_h^0\parallel +K{t}_n^{\alpha }\underset{1\le j\le n}{max}\parallel \nabla f^j\parallel for1\le n\le N.$$

(23)

Proof. 

By taking the inner product on both sides of (17) with $-{\Delta }_h{u}_h^n$, we obtain

$$\left(\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}\nabla u_h^n,\nabla u_h^n\right)+{\kappa }^2\parallel {\Delta }_h{u}_h^1{\parallel }^2+c{\parallel \nabla u_h^n\parallel }^2+V_n^{{\sigma }_1}(\nabla u_h^1-\nabla u_h^0,\nabla u_h^n)=(\nabla P_h{f}^n,\nabla u_h^n),$$

(24)

where (14) is used.

By choosing $n=1$ in (24), we have

$$\sum _{i=1}^q{s}_i{\displaystyle \frac{\mathrm{\Gamma }(1+{\sigma }_1)}{\mathrm{\Gamma }(1+{\sigma }_1-{\alpha }_i)}}{t}_1^{-{\alpha }_i}(\nabla u_h^1-\nabla u_h^0,\nabla u_h^1)+{\kappa }^2\parallel {\Delta }_h{u}_h^1{\parallel }^2+c{\parallel \nabla u_h^1\parallel }^2=(\nabla P_h{f}^1,\nabla u_h^1),$$

(25)

where (9) is used. Now applying a Cauchy–Schwartz inequality yields

$$\parallel \nabla u_h^1\parallel \le \parallel \nabla u_h^0\parallel +\mathrm{\Gamma }(1+{\sigma }_1)t_1^{{\alpha }_1}\parallel \nabla P_h{f}^1\parallel .$$

(26)

If $n\ge 2$, by using a Cauchy–Schwarz inequality and (26), we have

$$\begin{array}{cc}\hfill \sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}\parallel \nabla u_h^n\parallel & \le \parallel \nabla P_h{f}^n\parallel +C_T{t}_1^{-{\sigma }_1}\mathrm{\Phi }(n,{\sigma }_1)\left(\parallel \nabla u_h^1\parallel +\parallel \nabla u_h^0\parallel \right)\hfill \\ \hfill & \le \parallel \nabla P_h{f}^n\parallel +C_T{t}_1^{-{\sigma }_1}\mathrm{\Phi }(n,{\sigma }_1)\left[2\parallel \nabla u_h^0\parallel +\mathrm{\Gamma }(1+{\sigma }_1)t_1^{{\alpha }_1}\parallel \nabla P_h{f}^1\parallel \right]\hfill \\ \hfill & \le 2C_T{t}_1^{-{\sigma }_1}\mathrm{\Phi }(n,{\sigma }_1)\parallel \nabla u_h^0\parallel +(1+C_T\mathrm{\Gamma }(1+{\sigma }_1))\underset{1\le j\le n}{max}\parallel \nabla P_h{f}^j\parallel ,\hfill \end{array}$$

where (18) is used.

Now appealing to the Gronwall inequality of Lemma 5 yields

$$\begin{array}{cc}\hfill \parallel \nabla u_h^n\parallel & \le \parallel \nabla u_h^0\parallel +{\mathrm{\Theta }}_{n,1}^{-1}\sum _{j=1}^n{\mathrm{\Pi }}_{n,j}\left(2C_T{t}_1^{-{\sigma }_1}\mathrm{\Phi }(j,{\sigma }_1)\parallel \nabla u_h^0\parallel \right.\hfill \\ \hfill & \left.+(1+C_T\mathrm{\Gamma }(1+{\sigma }_1))\underset{1\le j\le n}{max}\parallel \nabla P_h{f}^j\parallel \right)\hfill \\ \hfill & \le 2C_T\parallel \nabla u_h^0\parallel +C_T(1+C_T\mathrm{\Gamma }(1+{\sigma }_1))t_n^{{\alpha }_1}\underset{1\le j\le n}{max}\parallel \nabla P_h{f}^j\parallel ,\hfill \end{array}$$

(27)

where Corollary 1 and Lemma 4 are used.

Combining (26) and (27) with (16), our proof is finished. □

Now we split $u^n-u_h^n=(R_h{u}^n-u_h^n)-(R_h{u}^n-u^n)={\xi }^n-{\eta }^n$, where ${\xi }^n:=R_h{u}^n-u_h^n$ and ${\eta }^n:=R_h{u}^n-u^n$. Inequality (13) indicates that the projection error ${\eta }^n$ can be approximated directly. Hence, we will consider the error analysis of ${\xi }^n$.

From (1)–(3) and (17), applying (15) yields

$$\begin{array}{cc}\hfill \sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{\xi }^n-{\kappa }^2{\Delta }_h{\xi }^n+c{\xi }^n& =\left[R_h\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{u}^n-{\kappa }^2{\Delta }_h{R}_h{u}^n+c{u}^n\right]\hfill \\ \hfill & -\left[\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{u}_h^n-{\kappa }^2{\Delta }_h{u}_h^n+c{u}_h^n\right]\hfill \\ \hfill & =(R_h-P_h)\left(\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{u}^n+c{u}^n\right)\hfill \\ \hfill & +P_h\left[\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{u}^n-{\kappa }^2\Delta u^n+c{u}^n\right]-P_h{f}^n\hfill \\ \hfill & =(R_h-P_h)(\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{u}^n+c{u}^n)+P_h{\phi }^n\hfill \\ \hfill & =P_h\left(\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{\eta }^n+c{\eta }^n+{\phi }^n\right),\hfill \end{array}$$

(28)

where ${\phi }^n:={\sum }_{i=1}^q{s}_i(D_N^{{\alpha }_i}{u}^n-D_t^{{\alpha }_i}{u}^n)$. By directly calculating, one has

$$\begin{array}{cc}\hfill \sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{\eta }^n& =\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}{u}^n-\sum _{i=1}^q{s}_i{\mathbb{D}}_N^{{\alpha }_i}{u}^n-\sum _{i=1}^q{s}_i{R}_h(D_{0,t}^{{\alpha }_i}{u}^n-{\mathbb{D}}_N^{{\alpha }_i}{u}^n)+\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}{\eta }^n\hfill \\ \hfill & =-{\phi }^n+R_h{\phi }^n+\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}{\eta }^n.\hfill \end{array}$$

(29)

Inserting (29) into (28) and applying the definition of (8), we obtain the following error equations:

$$\sum _{i=1}^q{s}_i{\delta }_N^{{\alpha }_i}{\xi }^n-{\kappa }^2{\Delta }_h{\xi }^n+c{\xi }^n=P_h\left(\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}{\eta }^n+c{\eta }^n+R_h{\phi }^n\right)-V_n^{{\sigma }_1}({\xi }^1-{\xi }^0).$$

(30)

Actually, Lemma 1 implies that the convergence of the corrected L1 scheme depends on the decomposition of the solution. Imitating [32], we can verify that the solution of problem (1)–(3) satisfies the following decomposition:

$$\begin{array}{cc}\hfill & u=v+w,v=a_0\left(x\right)+a_1\left(x\right)t^{{\sigma }_1},\parallel a_k\parallel \lesssim 1,k=0,1,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & w(x,0)=0,\parallel {\partial }_t^{l}w(·,t)\parallel \le C_T{t}^{{\sigma }_2-l},l=0,1,2,3,\hfill \end{array}$$

(32)

where ${\sigma }_1={\alpha }_1$ and ${\sigma }_2=2{\alpha }_1$.

Next a pointwise-in-time error estimate for our fully discrete CL1-FEM (17) will be stated in the following theorem.

Theorem 2

(Error estimate for the CL1-FEM). At each time level $t_n$, let $u^n$ and $u_h^n$ be the solutions of (1)–(3) and (17), respectively. For $1\le n\le N$, we obtain

$$\parallel u^n-u_h^n\parallel +\parallel \nabla u^n-\nabla R_h{u}_h^n\parallel \lesssim \left(1+t_n^{{\alpha }_1}\right)h^{k+1}+\mathcal{E}(n,2{\alpha }_1)+t_1^{{\alpha }_1}\mathcal{E}(n,{\alpha }_1),$$

(33)

where $\mathcal{E}(n,2{\alpha }_1)$ and $\mathcal{E}(n,{\alpha }_1)$ are defined in (21).

Proof. 

Multiplying (30) by $-{\Delta }_h{\xi }^n$ and integrating it over $\mathrm{\Omega }$, one has

$$\begin{array}{cc}\hfill (\sum _{i=1}^q{s}_i{\delta }_t^{{\alpha }_i}\nabla {\xi }^n,\nabla {\xi }^n)+{\kappa }^2\parallel {\Delta }_h{\xi }^n{\parallel }^2+c{\parallel \nabla {\xi }^n\parallel }^2& =(\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}{\eta }^n+c{\eta }^n+R_h{\phi }^n,-{\Delta }_h{\xi }^n)\hfill \\ \hfill & -V_n^{{\sigma }_1}(\nabla {\xi }^1-\nabla {\xi }^0,\nabla {\xi }^n).\hfill \end{array}$$

(34)

By taking $n=1$ in (34) and applying ${\xi }^0=0$, we have

$$\left({\mathrm{\Theta }}_{1,1}+W_1^{{\sigma }_1}\right)\parallel \nabla {\xi }^1{\parallel }^2\le {\displaystyle \frac{1}{4{\kappa }^2}}{∥\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}{\eta }^1+c{\eta }^1∥}^2+\parallel \nabla R_h{\phi }^1\parallel \parallel \nabla {\xi }^1\parallel .$$

(35)

where a Cauchy–Schwartz inequality and Young’s inequality are used.

Furthermore, applying the definition of $V_1^{{\sigma }_1}$, (12), and (13), we arrive at

$$\begin{array}{c}\hfill \sum _{i=1}^q{s}_i{\displaystyle \frac{\mathrm{\Gamma }(1+{\sigma }_1)}{\mathrm{\Gamma }(1+{\sigma }_1-{\alpha }_i)}}{t}_1^{-{\alpha }_i}\parallel \nabla {\xi }^1{\parallel }^2\le {\displaystyle \frac{C_{\mathrm{\Omega }}^2}{4{\kappa }^2}}{h}^4+C_T{t}_1^{{\sigma }_2-{\alpha }_1}\parallel \nabla {\xi }^1\parallel ,\end{array}$$

which means that

$${\left[\parallel \nabla {\xi }^1\parallel -{\displaystyle \frac{C_T{t}_1^{{\sigma }_2}}{\mathrm{\Gamma }(1+{\sigma }_1)}}\right]}^2\le {\displaystyle \frac{C_{\mathrm{\Omega }}^2{t}_1^{{\alpha }_1}}{4{\kappa }^2\mathrm{\Gamma }(1+{\sigma }_1)}}{h}^{2(k+1)}+{\left({\displaystyle \frac{C_T{t}_1^{{\sigma }_2}}{\mathrm{\Gamma }(1+{\sigma }_1)}}\right)}^2.$$

Hence, we have

$$\parallel \nabla {\xi }^1\parallel \le \sqrt{{\displaystyle \frac{C_{\mathrm{\Omega }}^2{t}_1^{{\alpha }_1}}{4{\kappa }^2\mathrm{\Gamma }(1+{\sigma }_1)}}}{h}^{k+1}+{\displaystyle \frac{2C_T}{\mathrm{\Gamma }(1+{\sigma }_1)}}{N}^{-r{\sigma }_2}.$$

By choosing $n\ge 2$ in (34) and applying a Cauchy–Schwartz inequality, one has

$$\begin{array}{cc}\hfill \sum _{i=1}^q{s}_i\parallel {\delta }_t^{{\alpha }_i}\nabla {\xi }^n\parallel \parallel \nabla {\xi }^n\parallel & \le {\displaystyle \frac{1}{4{\kappa }^2}}{∥\sum _{i=1}^q{s}_i{D}_{0,t}^{{\alpha }_i}{\eta }^n+c{\eta }^n∥}^2+\parallel \nabla R_h{\phi }^n\parallel \parallel \nabla {\xi }^n\parallel +|V_n^{{\sigma }_1}|\parallel \nabla {\xi }^1\parallel \parallel \nabla {\xi }^n\parallel \hfill \\ \hfill & \le {\displaystyle \frac{C_{\mathrm{\Omega }}^2}{4{\kappa }^2}}{h}^{2(k+1)}+C_T\left(\mathrm{\Phi }(n,{\sigma }_2)+t_1^{{\sigma }_2-{\sigma }_1}\mathrm{\Phi }(n,{\sigma }_1)\right)\parallel \nabla {\xi }^n\parallel \hfill \\ \hfill & +C_T{t}_1^{-{\sigma }_1}\mathrm{\Phi }(n,{\sigma }_1)\left[\sqrt{{\displaystyle \frac{C_{\mathrm{\Omega }}{t}_1^{{\alpha }_1}}{\mathrm{\Gamma }(1+{\sigma }_1)}}}{h}^{k+1}+{\displaystyle \frac{2C_T}{\mathrm{\Gamma }(1+{\sigma }_1)}}{N}^{-r{\sigma }_2}\right]\parallel \nabla {\xi }^n\parallel \hfill \\ \hfill & \le {\displaystyle \frac{C_{\mathrm{\Omega }}^2}{4{\kappa }^2}}{h}^{2(k+1)}+C_T\left(\mathrm{\Phi }(n,{\sigma }_2)+t_1^{{\sigma }_2-{\sigma }_1}\mathrm{\Phi }(n,{\sigma }_1)+h^{k+1}\right)\parallel \nabla {\xi }^n\parallel ,\hfill \end{array}$$

where Young’s inequality, (12), and (13) are used.

Furthermore, applying Lemma 5 yields

$$\begin{array}{cc}\hfill \parallel \nabla {\xi }^n\parallel & \le \parallel \nabla {\xi }^0\parallel +{\mathrm{\Theta }}_{n,1}^{-1}\sum _{j=1}^n{\mathrm{\Pi }}_{n,j}{C}_T\left(\left(\mathrm{\Phi }(j,{\sigma }_2)+t_1^{{\sigma }_2-{\sigma }_1}\mathrm{\Phi }(j,{\sigma }_1)+h^{k+1}\right)+{\displaystyle \frac{C_{\mathrm{\Omega }}}{2\kappa }}{h}^{k+1}\right)\hfill \\ \hfill & +\underset{1\le j\le n}{max}{\displaystyle \frac{C_{\mathrm{\Omega }}}{2\kappa }}{h}^{k+1}\hfill \\ \hfill & \le C_T{\mathrm{\Theta }}_{n,1}^{-1}\sum _{j=1}^n{\mathrm{\Pi }}_{n,j}\mathrm{\Phi }(j,{\sigma }_2)+C_T{t}_1^{{\sigma }_2-{\sigma }_1}{\mathrm{\Theta }}_{n,1}^{-1}\sum _{j=1}^n{\mathrm{\Pi }}_{n,j}\mathrm{\Phi }(j,{\sigma }_1)\hfill \\ \hfill & +\left(1+{\displaystyle \frac{C_{\mathrm{\Omega }}}{2\kappa }}\right)h^{k+1}{\mathrm{\Theta }}_{n,1}^{-1}\sum _{j=1}^n{\mathrm{\Pi }}_{n,j}+\underset{1\le j\le n}{max}{\displaystyle \frac{C_{\mathrm{\Omega }}}{2\kappa }}{h}^{k+1}\hfill \\ \hfill & \le 2C_T{t}_n^{{\alpha }_1}\left(1+{\displaystyle \frac{C_{\mathrm{\Omega }}}{2\kappa }}\right)h^{k+1}+C_T\mathcal{E}(n,{\sigma }_2)+C_T{t}_1^{{\sigma }_2-{\sigma }_1}\mathcal{E}(n,{\sigma }_1),\hfill \end{array}$$

(36)

where Corollary 1, Lemma 4, and ${\xi }^0=u_h^0-R_h{u}_0=0$ are used.

Now applying the Poincare inequality, we arrive at

$$\begin{array}{cc}\hfill \parallel u^n-u_h^n\parallel & \le \parallel u^n-R_h{u}^n\parallel +\parallel R_h{u}^n-u_h^n\parallel \hfill \\ \hfill & \le \parallel u^n-R_h{u}^n\parallel +C_{\mathrm{\Omega }}\parallel \nabla R_h{u}^n-\nabla u_h^n\parallel \hfill \\ \hfill & \le 2C_T{t}_n^{{\alpha }_1}\left(1+{\displaystyle \frac{C_{\mathrm{\Omega }}}{2\kappa }}\right)h^{k+1}+C_T\mathcal{E}(n,{\sigma }_2)+C_T{t}_1^{{\sigma }_2-{\sigma }_1}\mathcal{E}(n,{\sigma }_1),\hfill \end{array}$$

where (36) and (13) are used.

Finally, applying ${\sigma }_1={\alpha }_1$ and ${\sigma }_2=2{\alpha }_1$, our proof is complete. □

Next, we state the local and global errors in the following two corollaries.

Corollary 2

(Local errors). Suppose that $t^{*}$ is a fixed positive constant. Then, we have

$$\underset{t^{*}\le t_n\le T}{max}\parallel u^n-u_h^n\parallel \lesssim h^{k+1}+N^{-min\{r(1+{\alpha }_1),2-{\alpha }_1\}}.$$

Proof. 

Applying Theorem 2 and $1\le r<max\{(2-{\alpha }_1)/(1+{\alpha }_1),1\}$ yields

$$\underset{t^{*}\le t_n\le T}{max}\parallel u^n-u_h^n\parallel \lesssim \left(1+t_n^{{\alpha }_1}\right)h^{k+1}+N^{-r(1+{\alpha }_1)}+t_1^{{\alpha }_1}{N}^{-r}\lesssim h^{k+1}+N^{-r(1+{\alpha }_1)}.$$

If $r\ge max\{(2-{\alpha }_1)/(1+{\alpha }_1),1\}$, one has

$$\begin{array}{cc}\hfill \underset{t^{*}\le t_n\le T}{max}\parallel u^n-u_h^n\parallel & \lesssim h^{k+1}+N^{-(2-{\alpha }_1)}+t_1^{{\alpha }_1}{N}^{-min\{r,2-{\alpha }_1\}}\hfill \\ \hfill & \lesssim h^{k+1}+N^{-(2-{\alpha }_1)}.\hfill \end{array}$$

Combining the results of the above two cases, our proof is complete. □

Corollary 3

(Global errors). Let $r\ge 1$. Then we obtain

$$\underset{1\le n\le N}{max}\parallel u^n-u_h^n\parallel \lesssim h^{k+1}+N^{-min\{2r{\alpha }_1,2-{\alpha }_1\}}.$$

Proof. 

Imitating the proof of Corollary 2, one has

$$\begin{array}{cc}\hfill \underset{1\le n\le N}{max}\parallel u^n-u_h^n\parallel & \lesssim \left(1+t_n^{{\alpha }_1}\right)h^{k+1}+N^{-min\{2r{\alpha }_1,2-{\alpha }_1\}}+t_1^{{\alpha }_1}{N}^{-min\{r{\alpha }_1,2-{\alpha }_1\}}\hfill \\ \hfill & \lesssim h^{k+1}+N^{-min\{2r{\alpha }_1,2-{\alpha }_1\}}.\hfill \end{array}$$

□

4. Numerical Experiments

In this section, the CL1-FEM (17) is used to solve the following two-dimensional initial-boundary value problem, whose solutions satisfy (31). All the algorithms are implemented using MATLAB 2024a, running on a 2.80 GHz PC with 8 GB RAM and a Windows 11 operating system.

In our computation, the linear element and the bilinear element on a uniform rectangular partition of $\mathrm{\Omega }$ with $M+1$ nodes in each direction are utilized in space for $\mathrm{\Omega }\subset \mathbb{R}$ and $\mathrm{\Omega }\subset {\mathbb{R}}^2$, respectively. By taking $r=max\{(2-{\alpha }_1)/(1+{\alpha }_1),1\}$ and $r=max\{(2-{\alpha }_1)/\left(2{\alpha }_1\right),1\}$, Corollaries 2 and 3 imply that the local error ${max}_{0.2\le t_n\le 1}\parallel u^n-u_h^n\parallel$ and the global error ${max}_{1\le n\le N}\parallel u^n-u_h^n\parallel$ attain $O\left(h^2+N^{-(2-{\alpha }_1)}\right)$ convergence, respectively.

In the following two examples, we only focus on the convergence order in time.

Example 1.

Consider the initial-boundary value problem (1)–(3) with $s_1=1$, $s_2=0.1$, ${\alpha }_2=0.2$, ${\kappa }^2=0.1$, $c=0$, $\mathrm{\Omega }=(0,0.5)$, $f(x,t)=0$, and $u_0\left(x\right)=sin\left(2\pi x\right)$.

Since the exact solution of this example is unknown, we will adopt the two-mesh principle given in [33] to verify the convergent result of Corollaries 2 and 3.

Table 1. Local errors ${max}_{0.2\le t_n\le 1}\parallel u^n-u_h^n\parallel$ and rates of convergence for Example 1.

${\mathit{\alpha }}_1$	$\mathit{N}=160$	$\mathit{N}=320$	$\mathit{N}=640$	$\mathit{N}=1280$
$0.4$	$1.9793\times {10}^{-4}$	$7.2326\times {10}^{-5}$	$2.5963\times {10}^{-5}$	$9.2368\times {10}^{-6}$
		1.4780	1.4910	1.4962
$0.6$	$5.6695\times {10}^{-4}$	$2.3521\times {10}^{-4}$	$9.5452\times {10}^{-5}$	$3.8557\times {10}^{-5}$
		1.2692	1.3011	1.3077
$0.8$	$5.3020\times {10}^{-4}$	$2.2683\times {10}^{-4}$	$9.6740\times {10}^{-5}$	$4.1321\times {10}^{-5}$
		1.2248	1.2294	1.2272

and

Table 2. Global errors ${max}_{1\le n\le N}\parallel u^n-u_h^n\parallel$ and rates of convergence for Example 1.

${\mathit{\alpha }}_1$	$\mathit{N}=160$	$\mathit{N}=320$	$\mathit{N}=640$	$\mathit{N}=1280$
$0.4$	$1.7375\times {10}^{-4}$	$6.3972\times {10}^{-5}$	$2.2996\times {10}^{-5}$	$8.1906\times {10}^{-6}$
		1.4415	1.4759	1.4894
$0.6$	$5.6860\times {10}^{-4}$	$2.4184\times {10}^{-4}$	$9.9170\times {10}^{-5}$	$4.0033\times {10}^{-5}$
		1.2333	1.2861	1.3087
$0.8$	$4.3199\times {10}^{-4}$	$1.6446\times {10}^{-4}$	$7.9663\times {10}^{-5}$	$3.3996\times {10}^{-5}$
		1.2121	1.2269	1.2285

display the local error ${max}_{0.2\le t_n\le 1}\parallel u^n-u_h^n\parallel$ and the global error ${max}_{1\le n\le N}\parallel u^n-u_h^n\parallel$ for different ${\alpha }_1$, where $N=M$ is taken to eliminate the effect from the spatial error. From these two tables, we discover that the local error ${max}_{0.2\le t_n\le 1}\parallel u^n-u_h^n\parallel$ and the global error ${max}_{1\le n\le N}\parallel u^n-u_h^n\parallel$ attain $2-{\alpha }_1$ order in the temporal direction, in agreement with Corollaries 2 and 3. Moreover, Corollaries 2 and 3 indicate that the errors in the temporal direction are controlled by the term $N^{-(2-{\alpha }_1)}$. This means that the errors in the temporal direction become larger as the constant ${\alpha }_1$ increases. The errors in Table 1 and Table 2 verify this phenomenon.

Example 2.

Consider the multi-term subdiffusion equation with $s_1=1$, $s_2=s_3=0.1$, ${\alpha }_2=0.2$, ${\alpha }_3=0.1$, ${\kappa }^2=0.1$, $c=0.1$, and $\mathrm{\Omega }=(0,2)\times (0,2)$. The function $f(x,y,t)$ is chosen such that the exact solution of this problem is $u(x,y,t)=\left(t^{{\alpha }_1}+2t^3\right)sin\left(x\right)sin\left(y\right)(2-x)(2-y)$, which displays a weak singularity at $t=0$.

Table 3. Local errors ${max}_{0.2\le t_n\le 1}\parallel u^n-u_h^n\parallel$ and rates of convergence for Example 2.

${\mathit{\alpha }}_1$	$\mathit{N}=40$	$\mathit{N}=80$	$\mathit{N}=160$	$\mathit{N}=320$
$0.4$	$4.7189\times {10}^{-3}$	$1.4181\times {10}^{-3}$	$4.3307\times {10}^{-4}$	$1.3413\times {10}^{-4}$
		1.7344	1.7112	1.6909
$0.6$	$8.6545\times {10}^{-3}$	$3.0409\times {10}^{-3}$	$1.0927\times {10}^{-3}$	$3.9912\times {10}^{-4}$
		1.5089	1.4765	1.4530
$0.8$	$1.9655\times {10}^{-2}$	$8.2042\times {10}^{-3}$	$3.4842\times {10}^{-3}$	$1.4949\times {10}^{-3}$
		1.2604	1.2354	1.2207

and

Table 4. Global errors ${max}_{1\le n\le N}\parallel u^n-u_h^n\parallel$ and rates of convergence for Example 2.

${\mathit{\alpha }}_1$	$\mathit{N}=40$	$\mathit{N}=80$	$\mathit{N}=160$	$\mathit{N}=320$
$0.4$	$7.6427\times {10}^{-3}$	$2.4548\times {10}^{-3}$	$7.9246\times {10}^{-4}$	$2.5687\times {10}^{-4}$
		1.6384	1.6311	1.6253
$0.6$	$9.7782\times {10}^{-3}$	$3.4835\times {10}^{-3}$	$1.2644\times {10}^{-3}$	$4.6515\times {10}^{-4}$
		1.4890	1.4620	1.4427
$0.8$	$1.9655\times {10}^{-2}$	$8.2041\times {10}^{-3}$	$3.4842\times {10}^{-3}$	$1.4949\times {10}^{-3}$
		1.2604	1.2354	1.2207

present the local error ${max}_{0.2\le t_n\le 1}\parallel u^n-u_h^n\parallel$ and the global error ${max}_{1\le n\le N}\parallel u^n-u_h^n\parallel$ in the temporal direction for the CL1-FEM (17) with different ${\alpha }_1$, respectively. In these two tables, $M=N$ is taken so that the temporal error dominates the spatial error. The orders of convergence displayed show that rates of convergence for the local error ${max}_{0.2\le t_n\le 1}\parallel u^n-u_h^n\parallel$ and the global error ${max}_{1\le n\le N}\parallel u^n-u_h^n\parallel$ are $N^{-(2-{\alpha }_1)}$, consistent with Corollaries 2 and 3. Figure 1 presents local errors and global errors for CL1-FEM (17) and L1-FEM (which is given in [29]). In Figure 1, we observe that the convergent order (local errors and global errors) for CL1-FEM (17) is higher than that for L1-FEM of [29].

5. Conclusions

In order to use a smaller grading parameter r to achieve the optimal convergence order, a corrected L1 scheme is constructed to approximate the Caputo derivative by introducing a corrected term of the nonuniform L1 scheme. By virtue of this novel scheme in time and the finite element method in space, a fully discrete CL1-FEM is developed for problem (1)–(3). The stability analysis and the optimal pointwise-in-time error analysis of the proposed scheme are presented. Furthermore, numerical examples are supplied to verify the theoretical findings. In the future, we will try to develop the corrected Alikhanov scheme for problem (1)–(3).