The local discontinuous Galerkin finite element method for a multiterm time-fractional initial-boundary value problem

Abstract

In this paper, we study a multiterm time-fractional initial-boundary value problem, whose differential equation contains a sum of Caputo time fractional derivatives with orders in (0, 1). In general, the solution of this kind of problem exhibits a weak regularity at the initial time. Based on the L1 formula on non-uniform meshes for time discretization and the local discontinuous Galerkin (LDG) method for space discretization, fully discrete numerical schemes for one and two space dimensions are constructed. The stability and convergence of the schemes are analyzed. It is shown that the error bounds are $\alpha _1$-robust, that is, they remain valid as $\alpha _1\rightarrow 1^-$, where $\alpha _1$ is the biggest fractional order. Furthermore, a numerical experiment is given to verify the effectiveness of the current method.

1 Introduction

Fractional calculus (fractional differentiation and integration) has received much attention in recent years due to its successful simulation of many phenomena in science and engineering [8, 11,12,13, 20]. Although the analytical solutions of some fractional differential equations can be obtained by means of some special transforms, the complexity involving special functions and infinite series are inconvenient for numerical evaluation. Hence, efficient and accurate numerical approaches are demanded.

In this paper, we will investigate numerical methods and related numerical analysis of the following multiterm time-fractional initial-boundary value problem:

$$\begin{aligned} \sum _{i=1}^l\left[ {q_i}\,_{C}\mathrm{D}_{0,t}^{\alpha _i} u(\mathbf{x},t)\right] - \Delta u(\mathbf{x},t)+ c(\mathbf{x})u(\mathbf{x},t)= f(\mathbf{x},t), \,(\mathbf{x},t)\in \Omega \times (0,T], \end{aligned}$$

(1.1)

with initial and boundary conditions:

$$\begin{aligned}&u|_{t=0} = u_0(\mathbf{x}),\,\mathbf{x}\in {\overline{\Omega }}, \end{aligned}$$

(1.2)

$$\begin{aligned}&u|_{\mathbf{x}\in \partial \Omega }=0,\, t\in (0,T]. \end{aligned}$$

(1.3)

Here, $\Omega \subseteq {\mathbb {R}}^d$($d=1,2$) is a bounded rectangular domain, l is a positive integer, $q_i>0$, $i=1,2,\ldots ,l$, $0<\alpha _l<\ldots<\alpha _2<\alpha _1<1$ are given constants, $c(\mathbf{x})\in C({\overline{\Omega }})$ with $c(\mathbf{x})\ge 0$, source term $f(\mathbf{x},t)\in L^\infty (0,T;L^2(\Omega ))$ and initial value $u_0(\mathbf{x})\in L^2(\Omega )$ are given functions, and $_{C}\mathrm{D}_{0,t}^{\alpha _i}$ is the $\alpha _i$th-order left-sided Caputo derivative operator defined by [13]

$$\begin{aligned} _C\mathrm{D}_{0,t}^{\alpha _i} u(\mathbf{x},t) = \frac{1}{{\Gamma (1 - \alpha _i )}}\int _0^t {{{(t - s )}^{ - \alpha _i }}} \frac{{\partial u}}{{\partial s }}\mathrm{d}s,\,0<\alpha _i<1, \end{aligned}$$

(1.4)

in which $\Gamma (\cdot )$ denotes the usual Gamma function.

The multiterm time-fractional initial-boundary value problem (1.1)–(1.3) has proved to be flexible to describe complex multirate physical processes [25]. So far, several different methods have been developed to solve this problem. Luchko [17] developed the maximum principle for a multiterm time-fractional diffusion equation and constructed a generalized solution by means of the multinomial Mittag-Leffler functions. Wei [22] proposed a fully discrete local discontinuous Galerkin (LDG) method for a class of multiterm time fractional diffusion equations. Zaky [26] used a Legendre spectral quadrature tau method solving the multiterm time-fractional diffusion equations. Very recently, Huang et al. [10] showed that, under proper regularity and compatibility assumptions, the system (1.1)–(1.3) has a unique solution u such that

$$\begin{aligned}&\left| \frac{\partial ^ku(\mathbf{x},t)}{\partial \mathbf{x}^k}\right| \le C\;\;\mathrm{for}\;\;k=0,1,2,3,4, \end{aligned}$$

(1.5)

$$\begin{aligned}&\left| \frac{\partial ^mu(\mathbf{x},t)}{\partial t^m}\right| \le C(1+t^{\alpha _1-m})\;\;\mathrm{for}\;\;m=0,1,2, \end{aligned}$$

(1.6)

where $C>0$ is a bounded constant independent of the variable t but dependent of T. Meng and Stynes [18] presented an L1 finite element method for a multiterm time-fractional initial-boundary value problem.

The result in (1.6) implies that the solution u of the multiterm time-fractional initial-boundary value problem (1.1)–(1.3) exhibits some weak regularity at the starting time and $\frac{\partial u(\cdot ,t)}{\partial t}$ blows up as $t\rightarrow 0^+$. When seeking numerical solutions, the initial layer very likely leads to a loss of accuracy if uniform temporal meshes are used. To tackle such a problem, one can consider the numerical approaches on non-uniform meshes. This is also the topic of the present paper. In this work, we develop a non-uniform L1/LDG method for the multiterm time-fractional initial-boundary value problem (1.1)–(1.3) with weak regular solution. The Caputo time-fractional derivatives are discretized by the L1 finite difference method on non-uniform meshes, and the spatial discretization is performed by using the LDG finite element method. Then the stability and convergence analysis of the proposed numerical scheme are given. However, the obtained error bounds blow up if we consider the limit $\alpha _1\rightarrow 1^-$. Such error bounds are called $\alpha _1$-nonrobust [2]. On the contrary, if the error bound does not blow up as $\alpha _1\rightarrow 1^-$, we call it $\alpha _1$-robust. Therefore, we further investigate the $\alpha _1$-robust stability and convergence of the proposed non-uniform L1/LDG scheme.

The LDG method is a special class of discontinuous Galerkin (DG) finite element methods, introduced first by Cockburn and Shu [4] and has been successful for solving fractional differential equations, e.g., [6, 7, 14,15,16, 19, 23]. The main technique of LDG method is to rewrite higher-order derivative equation into an equivalent system containing only the first derivative, and then discretize it by the standard DG method. More details about the LDG method for high-order time dependent partial differential equations can be found in the review paper [24].

The rest of the paper is organized as follows. In Sects. 2 and 3, we establish the fully discrete non-uniform L1/LDG schemes for the initial-boundary value problem in one and two space dimensions, respectively. The $\alpha _1$-robust and $\alpha _1$-nonrobust stability and convergence analysis are studied too. In Sect. 4, we provide a numerical example to verify the theoretical results. Concluding remarks are given in the last section.

Notations :  Through out this paper we let C be a generic positive constant, which is independent of the mesh sizes and can take different values in different circumstances. We use $\Vert \cdot \Vert$ as the $L^2$-norm on domain $\Omega$ and define the $L^2(\Omega )$ inner product $(u,v)=\iint _\Omega uv\,\mathrm{d}{} \mathbf{x}$.

2 One-dimensional case

In the section, we will develop a non-uniform L1/LDG scheme for the one-dimensional multiterm time-fractional initial-boundary value problem (1.1)–(1.3). The scheme employs the L1 formula with non-uniform meshes for the time-fractional derivative and a LDG method in space. The usual notations are introduced below.

Let ${\mathcal {T}}_h=\left\{ I_{j}=(x_{j-\frac{1}{2}},x_{j+\frac{1}{2}})\right\} _{j=1}^N$ be the partition of ${\Omega }$, where $a=x_{\frac{1}{2}}<x_{\frac{3}{2}}<\cdots <x_{N+\frac{1}{2}}=b$. The cell center and cell length are denoted by $x_{j}=(x_{j-\frac{1}{2}}+x_{j+\frac{1}{2}})/2$ and $h_{j}=x_{j+\frac{1}{2}}-x_{j-\frac{1}{2}}$, respectively. Let $h=\displaystyle \max _{1\le j\le N}h_{j}$ be the length of the largest cell. We use $u_{j+\frac{1}{2}}^{-}$ and $u_{j+\frac{1}{2}}^{+}$ to represent the values of u at the discontinuity point $x_{j+\frac{1}{2}}$, from the left cell, $I_{j}$, and from the right cell, $I_{j+1}$, respectively. The jump value of u at each element boundary is denoted by $[\![u]\!]_{j+\frac{1}{2}}=u^{+}_{j+\frac{1}{2}}-u^{-}_{j+\frac{1}{2}}$. Associated with this mesh, we define the discontinuous finite element space

$$\begin{aligned} V_{h}=\big \{v\in L^{2}(\Omega ):v|_{I_{j}}\in {\mathcal {P}}^{k}(I_{j}),\,v|_{\partial \Omega }=0,\, j=1,\ldots ,N\big \}, \end{aligned}$$

where ${\mathcal {P}}^{k}(I_{j})$ denotes the set of polynomials of degree up to $k\ge 0$ defined on the cell $I_{j}$. For any nonnegative integer m, $H^m(\Omega )$ denotes the usual Sobolev space. Then we define the broken Sobolev space on ${\mathcal {T}}_h$ by

$$\begin{aligned} H^m({\mathcal {T}}_h)=\{v\in L^{2}(\Omega ):\,v|_{I_j}\in H^{m}(I_j),\, \forall j=1,\ldots ,N\}, \end{aligned}$$

which contains the discontinuous finite element space $V_{h}$.

To obtain the optimal error estimate, we recall two kinds of Gauss-Radau projections ${\mathscr {P}}_h^{\pm }:H^1({\mathcal {T}}_h)\rightarrow V_h$, which were introduced by Castillo et al. [1], i.e., for each j,

$$\begin{aligned} \int _{I_{j}}\left( {\mathscr {P}}_{h}^{+}q(x)-q(x)\right) v_{h}\mathrm{d}x=0,\,\forall v_{h}\in {\mathcal {P}}^{k-1}(I_{j}), \,({\mathscr {P}}_{h}^{+}q)_{j-\frac{1}{2}}^{+}=q(x_{j-\frac{1}{2}}^{+}), \end{aligned}$$

(2.1)

and

$$\begin{aligned} \int _{I_{j}}\left( {\mathscr {P}}_{h}^{-}q(x)-q(x)\right) v_{h}\mathrm{d}x=0,\,\forall v_{h}\in {\mathcal {P}}^{k-1}(I_{j}), \,\,({\mathscr {P}}_{h}^{-}q)_{j+\frac{1}{2}}^{-}=q\left( x_{j+\frac{1}{2}}^{-}\right) . \end{aligned}$$

(2.2)

Denote by $\zeta =q(x)-{\mathbb {P}}_hq(x)$ (${\mathbb {P}}_h={\mathscr {P}}_h^+$ or ${\mathscr {P}}_h^{-}$) the projection error. Then a standard scaling argument as that in [3] yields

$$\begin{aligned} \Vert \zeta \Vert +h\Vert {\zeta }_x\Vert +h^{\frac{1}{2}}\Vert \zeta \Vert _{\Gamma _h}\le C \Vert \zeta \Vert _{H^{k+1}({\mathcal {T}}_h)}h^{k+1}, \end{aligned}$$

(2.3)

where $\Vert \zeta \Vert _{\Gamma _{h}}^{2}=\displaystyle \sum _{j=1}^{N}\left( (\zeta ^{+}|_{j-\frac{1}{2}})^{2} +(\zeta ^{-}|_{j+\frac{1}{2}})^{2}\right)$.

2.1 The fully discrete non-uniform L1/LDG scheme

For a given $T>0$, let $t_n=T(n/M)^r$, $n=0,1,\ldots ,M$ be the mesh points, $r\ge 1$. Denote $\tau _n=t_n-t_{n-1}$, $n=1,\ldots ,M$ be the time mesh sizes. If $r=1$, then the mesh is just uniform. Throughout this paper, we denote $u^n=u(x,t_n)$ if no confusion appears.

The $\mathrm{L}1$ approximation on the non-uniform meshes to the Caputo derivative is given by [10]

$$\begin{aligned} {_{C}\mathrm{D}_{0,t}^{\alpha _i}}u|_{t=t_n}= & {} \frac{1}{\Gamma (2-\alpha _i)}\left[ d_{n,1}^iu^n +\sum _{k=1}^{n-1}(d_{n,k+1}^i-d_{n,k}^i)u^{n-k}-d_{n,n}^iu^0\right] \nonumber \\&+R^n_i\nonumber \\:= & {} \Upsilon _t^{\alpha _i} u^n+R^n_i \end{aligned}$$

(2.4)

for $i=1,\ldots ,l$, where $d_{n,k}^i=\frac{(t_n-t_{n-k})^{1-\alpha _i}-(t_n-t_{n-k+1})^{1-\alpha _i}}{\tau _{n-k+1}}$ and $R^n_i$ is the truncation error. The coefficients $d_{n,k}^i$ have the following properties

$$\begin{aligned}&d_{n,k+1}^i\le d_{n,k}^i, \end{aligned}$$

(2.5)

$$\begin{aligned}&(1-\alpha _i)(t_n-t_{n-k})^{-\alpha _i}\le d_{n,k}^i\le (1-\alpha _i)(t_{n}-t_{n-k+1})^{-\alpha _i}. \end{aligned}$$

(2.6)

Denote

$$\begin{aligned} \Upsilon _t^{\alpha _i} u^n=\frac{1}{\Gamma (2-\alpha _i)}\left[ d_{n,1}^iu^n +\sum _{k=1}^{n-1}(d_{n,k+1}^i-d_{n,k}^i)u^{n-k}-d_{n,n}^iu^0\right] \end{aligned}$$

(2.7)

for $i=1,\ldots ,l,\,n=1,\ldots ,M$.

Let $p=u_x$, then we can get the weak form of system (1.1)–(1.3) at $t_n$ as follows,

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}\left( \displaystyle \sum _{i=1}^lq_i{_{C}\mathrm{D}_{0,t}^{\alpha _i}} u^n,v\right) +(c(x)u^n,v)+( p^n,v_x)\\ &{}\quad -\displaystyle \sum _{j=1}^N\left( p^nv^-|_{j+\frac{1}{2}}-p^nv^+|_{j-\frac{1}{2}}\right) =( f^n,v),\\ &{}(p^n,w)+(u^n,w_{x})-\displaystyle \sum _{j=1}^{N} \left( u^nw^{-}|_{j+\frac{1}{2}}-u^nw^{+}|_{j-\frac{1}{2}}\right) =0, \end{array}\right. } \end{aligned} \end{aligned}$$

(2.8)

where $v,w\in H^1(\Omega )$ are test functions.

The fully discrete non-uniform L1/LDG scheme is defined as follows: find $U_h^n,P_h^n\in V_h$ such that for all test functions $v_h,w_h\in V_h$,

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}\left( \displaystyle \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i} U_h^n,v_h\right) +( c(x)U_h^n,v_h) +(P_h^n,(v_h)_x)\\ &{}\quad -\displaystyle \sum _{j=1}^N\left( {\widehat{P}}_h^nv_h^-|_{j+\frac{1}{2}}-{\widehat{P}}_h^nv_h^+|_{j-\frac{1}{2}}\right) =( f^n,v_h),\\ &{}(P_h^n,w_h)+(U_h^n,(w_h)_{x})-\displaystyle \sum _{j=1}^{N} \left( {\widehat{U}}_h^nw_h^{-}|_{j+\frac{1}{2}} -{\widehat{U}}_h^nw_h^{+}|_{j-\frac{1}{2}}\right) =0. \end{array}\right. } \end{aligned} \end{aligned}$$

(2.9)

The “hat" terms in (2.9) are the boundary terms that emerge from the integration by parts. These are the so-called “numerical fluxes" that are yet to be determined. The freedom in choosing numerical fluxes can be utilized for designing a scheme that enjoys certain stability properties. It turns out that we can take the following choices simply

$$\begin{aligned} {\widehat{U}}_{h}^{n}=(U_{h}^{n})^{-},\,{\widehat{P}}_{h}^{n}=(P_{h}^{n})^{+}. \end{aligned}$$

(2.10)

2.2 $\alpha _1$-Nonrobust error analysis of the non-uniform L1/LDG method

This subsection presents the $\alpha _1$-nonrobust stability and convergence of the scheme (2.9). We give the following $\alpha _1$-nonrobust stability result.

Lemma 2.1

The solution $U_h^n$ of the fully discrete scheme (2.9) satisfies

$$\begin{aligned} \Vert U_h^n\Vert \le \Vert U_h^0\Vert +\frac{1}{\sum _{i=1}^l\eta _{n,1}^i}\sum _{j=1}^n\theta _{n,j}\Vert f^j\Vert ,\,n=1,\ldots ,M, \end{aligned}$$

(2.11)

where

$$\begin{aligned} \eta _{n,j}^i=\frac{q_id_{n,j}^i}{\Gamma (2-\alpha _i)},\,i=1,2,\ldots ,l,\,j=1,2,\ldots ,M,\\ \theta _{n,n}=1, \,\theta _{n,j}=\displaystyle \sum _{i=1}^{l} \sum _{k=1}^{n-j}\frac{1}{\sum _{i=1}^l\eta _{n-k,1}^i}(\eta _{n,k}^i-\eta _{n,k+1}^i)\theta _{n-k,j},\,j=1,2,\ldots ,n-1. \end{aligned}$$

Proof

Taking the test functions $(v_h,w_h)=(U_h^n, P_h^n)$, and adding the two equations in (2.9), we obtain

$$\begin{aligned} \begin{aligned}&\left( \sum _{i=1}^l\frac{q_id_{n,1}^i}{\Gamma (2-\alpha _i)}U_h^n,U_h^n\right) +\left( c(x)U_h^n,U_h^n\right) +\Vert P_h^n\Vert ^2\\&\quad =\left( \sum _{i=1}^l\frac{q_id_{n,n}^i}{\Gamma (2-\alpha _i)}U_h^0,U_h^n\right) +\left( \sum _{i=1}^l\frac{q_i}{\Gamma (2-\alpha _i)}\sum _{k=1}^{n-1}(d_{n,k}^i-d_{n,k+1}^i)U_h^{n-k},U_h^n\right) \\&\quad \quad +(f^n,U_h^n). \end{aligned} \end{aligned}$$

(2.12)

By using (2.5) and the Cauchy-Schwarz inequality, we have

$$\begin{aligned} \sum _{i=1}^l\eta _{n,1}^i\Vert U_h^n\Vert \le \sum _{i=1}^l\eta _{n,n}^i\Vert U_h^0\Vert +\sum _{i=1}^l\sum _{k=1}^{n-1}(\eta _{n,k}^i-\eta _{n,k+1}^i)\Vert U_h^{n-k}\Vert +\Vert f^n\Vert . \end{aligned}$$

(2.13)

Now, we prove this lemma by mathematical induction. When $n=1$, (2.13) becomes

$$\begin{aligned} \Vert U_h^1\Vert \le \Vert U_h^0\Vert +\frac{1}{\sum _{i=1}^l\eta _{1,1}^i}\Vert f^1\Vert , \end{aligned}$$

(2.14)

which is identical to (2.11).

Supposing the following estimates hold

$$\begin{aligned} \Vert U_h^m\Vert \le \Vert U_h^0\Vert +\frac{1}{\sum _{i=1}^l\eta _{m,1}^i}\sum _{j=1}^m\theta _{m,j}\Vert f^j\Vert ,\, m=2,\ldots ,s, \end{aligned}$$

(2.15)

we need prove

$$\begin{aligned} \Vert U_h^{s+1}\Vert \le \Vert U_h^0\Vert +\frac{1}{\sum _{i=1}^l\eta _{s+1,1}^i}\sum _{j=1}^{s+1}\theta _{s+1,j}\Vert f^j\Vert . \end{aligned}$$

(2.16)

Letting $n=s+1$ in (2.13) and using (2.15), we have

$$\begin{aligned} \Vert U_h^{s+1}\Vert\le & {} \frac{\sum _{i=1}^l\eta _{s+1,s+1}^i}{\sum _{i=1}^l\eta _{s+1,1}^i}\Vert U_h^0\Vert +\frac{1}{\sum _{i=1}^l\eta _{s+1,1}^i}\Bigg [\sum _{i=1}^l\sum _{k=1}^s(\eta _{s+1,k}^i-\eta _{s+1,k+1}^i)\\&\times \Big (\Vert U_h^0\Vert +\frac{1}{\sum _{i=1}^l\eta _{s+1-k,1}^i}\sum _{j=1}^{s+1-k}\theta _{s+1-k,j}\Vert f^j\Vert \Big )\Bigg ] +\frac{\Vert f^{s+1}\Vert }{\sum _{i=1}^l\eta _{s+1,1}^i}\\= & {} \frac{1}{\sum _{i=1}^l\eta _{s+1,1}^i}\left\{ \Vert f^{s+1}\Vert +\left[ \sum _{i=1}^l\sum _{k=1}^s \frac{\eta _{s+1,k}^i-\eta _{s+1,k+1}^i}{\sum _{i=1}^l\eta _{s+1-k,1}^i}\sum _{j=1}^{s+1-k}\theta _{s+1-k,j}\Vert f^j\Vert \right] \right\} \\&+\left\{ \frac{1}{\sum _{i=1}^l\eta _{s+1,1}^i}\Bigg [\sum _{i=1}^l\sum _{k=1}^s(\eta _{s+1,k}^i-\eta _{s+1,k+1}^i)\Bigg ] +\frac{\sum _{i=1}^l\eta _{s+1,s+1}^i}{\sum _{i=1}^l\eta _{s+1,1}^i}\right\} \Vert U_h^0\Vert \\= & {} \Vert f^{s+1}\Vert +\sum _{j=1}^s\left[ \sum _{i=1}^l\sum _{k=1}^{s+1-j} \frac{\eta _{s+1,k}^i-\eta _{s+1,k+1}^i}{\sum _{i=1}^l\eta _{s+1-k,1}^i}\theta _{s+1-k,j}\right] \Vert f^j\Vert \\&+\frac{1}{\sum _{j=1}^l\eta _{s+1,1}^j}\left\{ \sum _{i=1}^l\sum _{k=1}^s(\eta _{s+1,k}^i-\eta _{s+1,k+1}^i) +\sum _{i=1}^l\eta _{s+1,s+1}^i\right\} \Vert U_h^0\Vert \\= & {} \sum _{j=1}^{s+1}\theta _{s+1,j}\Vert f^j\Vert +\Vert U_h^0\Vert . \end{aligned}$$

This completes the proof of this lemma. $\square$

Lemma 2.2

[10] Let $\beta \le r\alpha _1$, then for $n=1,2,\ldots ,M$, one has

$$\begin{aligned} \frac{1}{\sum _{i=1}^l\eta _{n,1}^i}\sum _{j=1}^nj^{-\beta }\theta _{n,j}\le \Gamma (1-\alpha _1)T^{\alpha _1}M^{-\beta }. \end{aligned}$$

Theorem 2.1

($L^2$-norm stability) The solution $U_h^n$ of the fully discrete scheme (2.9) satisfies

$$\begin{aligned} \Vert U_h^n\Vert \le \Vert U_h^0\Vert + \Gamma (1-\alpha _1)T^{\alpha _1}\max _{1\le j\le n}\left\| f^j\right\| ,\, n=1,\ldots ,M. \end{aligned}$$

(2.17)

Proof

By Lemma 2.2, one has

$$\begin{aligned} \frac{1}{\sum _{i=1}^l\eta _{n,1}^i}\sum _{j=1}^n\theta _{n,j}\le \Gamma (1-\alpha _1)T^{\alpha _1}. \end{aligned}$$

Combining the above estimate with Lemma 2.1 yields the assertion. $\square$

Next, we present the $\alpha _1$-nonrobust convergence analysis. Firstly, we introduce a lemma that will be used later on.

Lemma 2.3

[10] Suppose that the solution u(x, t) of problem (1.1)–(1.3) satisfies (1.6). Then there exists a constant C such that for all $t_n$ one has

$$\begin{aligned} |R^n_i|\le Cn^{-\min \{2-\alpha _1,r\alpha _1\}} \end{aligned}$$

for $i=1,2,\ldots ,l$, $n=1,2,\ldots ,M$.

Theorem 2.2

($L^2$-norm error estimate) Let u be the exact solution of (1.1)–(1.3) and $U_h^n$ be the numerical solution of the fully discrete non-uniform L1/LDG scheme (2.9). Suppose that u satisfies condition (1.6) and $u(\cdot ,t)\in H^{k+1}({\mathcal {T}}_h)$. Then, it holds that

$$\begin{aligned} \Vert u^n-U_h^n\Vert \le C\sqrt{\Gamma (1-\alpha _1)}\left( M^{-\min \{2-\alpha _1,r\alpha _1\}}+h^{k+1}\right) , \end{aligned}$$

(2.18)

where C is a positive constant independent of M and h.

Proof

As usual in the finite element analysis, we denote the error by $e_u^n=u^n-U_h^n$ and $e_p^n=p^n-P_h^n$, respectively, and decompose them into two parts, namely,

$$\begin{aligned} \begin{aligned}&e_{u}^{n}=u^n-U_{h}^{n}={\mathscr {P}}_{h}^{-}e_u^n+\big (u^n-{\mathscr {P}}_h^-u^n\big ):=\xi _u^n+\eta _u^n,\\&e_{p}^{n}=p^n-P_{h}^{n}={\mathscr {P}}_{h}^+e_p^n+\big (p^n-{\mathscr {P}}_h^+ p^n\big ):=\xi _p^n+\eta _p^n. \end{aligned} \end{aligned}$$

(2.19)

Subtracting (2.9) from (2.8), and with the fluxes (2.10), we can obtain the error equation

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}\left( \displaystyle \sum _{i=1}^lq_i\big ({_{C}\mathrm{D}_{0,t}^{\alpha _i}} u^n-\Upsilon _t^{\alpha _i} U_h^n\big ),v_h\right) +(c(x) e_u^n,v_h)+\big (e_p^n,(v_h)_x\big ) \\ &{}\quad -\displaystyle \sum _{j=1}^N\left( (e_p^n)^+v_h^-|_{j+\frac{1}{2}}-(e_p^n)^+v_h^+|_{j-\frac{1}{2}}\right) =0,\\ &{}(e_p^n,w_h)+\big (e_u^n,(w_h)_{x}\big )-\displaystyle \sum _{j=1}^{N} \left( (e_u^n)^-w_h^{-}|_{j+\frac{1}{2}} -(e_u^n)^-w_h^{+}|_{j-\frac{1}{2}}\right) =0. \end{array}\right. } \end{aligned} \end{aligned}$$

(2.20)

Substituting (2.19) into (2.20), we have

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}\left( \displaystyle \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i} \xi _u^n,v_h\right) +\big (c(x)\xi _u^n,v_h\big )+\big (\xi _p^n,(v_h)_x\big )\\ &{}\quad -\displaystyle \sum _{j=1}^N\left( (\xi _p^n)^+v_h^-|_{j+\frac{1}{2}} -((\xi _p^n)^+v_h^+|_{j-\frac{1}{2}}\right) \\ &{}\quad =-\left( \displaystyle \sum _{i=1}^lq_iR^n_i,v_h\right) -\left( \displaystyle \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}\eta _u^n,v_h\right) -\big (c(x)\eta _u^n,v_h\big )\\ &{}\quad \quad - \big (\eta _p^n,(v_h)_x\big ) +\displaystyle \sum _{j=1}^N\Big ((\eta _p^n)^+v_h^-|_{j+\frac{1}{2}} -(\eta _p^n)^+v_h^+|_{j-\frac{1}{2}}\Big ),\\ &{}( \xi _p^n,w_h)+\big ( \xi _u^n,(w_h)_x\big ) -\displaystyle \sum _{j=1}^N\left( (\xi _u^n)^-w_h^-|_{j+\frac{1}{2}} -(\xi _u^n)^-w_h^+|_{j-\frac{1}{2}}\right) \\ &{}\quad =-(\eta _p^n,w_h)-\big (\eta _u^n,(w_h)_x\big ) +\displaystyle \sum _{j=1}^N\Big ((\eta _u^n)^-w_h^-|_{j+\frac{1}{2}} -(\eta _u^n)^-w_h^+|_{j-\frac{1}{2}}\Big ). \end{array}\right. } \end{aligned} \end{aligned}$$

(2.21)

Taking the test functions $(v_h,w_h)=(\xi _u^n,\xi _p^n)$ in (2.21) and using the properties (2.1) and (2.2), we get

$$\begin{aligned}&\left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i} \xi _u^n,\xi _u^n\right) +\big (c(x)\xi _u^n,\xi _u^n\big ) +\Vert \xi _p^n\Vert ^2 \nonumber \\&\quad =-\left( \sum _{i=1}^lq_iR^n_i,\xi _u^n\right) -\left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}\eta _u^n,\xi _u^n\right) -\big (c(x)\eta _u^n,\xi _u^n\big )-(\eta _p^n,\xi _p^n),\qquad \quad \end{aligned}$$

(2.22)

which is equivalent to

$$\begin{aligned}&\left( \sum _{i=1}^l\frac{q_id_{n,1}^i}{\Gamma (2-\alpha _i)}\xi _u^n,\xi _u^n\right) +\big (c(x)\xi _u^n,\xi _u^n\big ) +\Vert \xi _p^n\Vert ^2 \nonumber \\&\quad =\left( \sum _{i=1}^l\frac{q_id_{n,n}^i}{\Gamma (2-\alpha _i)}\xi _u^0,\xi _u^n\right) +\left( \sum _{i=1}^l\frac{q_i}{\Gamma (2-\alpha _i)}\sum _{k=1}^{n-1}(d_{n,k}^i-d_{n,k+1}^i)\xi _u^{n-k},\xi _u^n\right) \nonumber \\&\quad \quad -\left( \sum _{i=1}^lq_iR^n_i,\xi _u^n\right) -\left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}\eta _u^n,\xi _u^n\right) -\big (c(x)\eta _u^n,\xi _u^n\big )-(\eta _p^n,\xi _p^n). \end{aligned}$$

(2.23)

Thus, by using Lemma 2.3, we obtain

$$\begin{aligned}&\sum _{i=1}^l\eta _{n,1}^i\Vert \xi _u^n\Vert ^2+\big (c(x)\xi _u^n,\xi _u^n\big )+\Vert \xi _p^n\Vert ^2 \nonumber \\&\quad \le \sum _{i=1}^l\eta _{n,n}^i\Vert \xi _u^0\Vert \Vert \xi _u^n\Vert +\sum _{i=1}^l\sum _{k=1}^{n-1}(\eta _{n,k}^i-\eta _{n,k+1}^i)\Vert \xi _u^{n-k}\Vert \Vert \xi _u^n\Vert +\sum _{i=1}^lq_i\Vert R^n_i\Vert \Vert \xi _u^n\Vert \nonumber \\&\quad \quad +\sum _{i=1}^lq_i\Vert \Upsilon _t^{\alpha _i}\eta _u^n\Vert \Vert \xi _u^n\Vert +\left\| \sqrt{c(x)}\eta _u^n\right\| \left\| \sqrt{c(x)}\xi _u^n\right\| +\Vert \eta _p^n\Vert \Vert \xi _p^n\Vert \nonumber \\&\quad \le \sum _{i=1}^l\sum _{k=1}^{n-1}\frac{1}{2}(\eta _{n,k}^i-\eta _{n,k+1}^i)\Vert \xi _u^{n-k}\Vert ^2 +\frac{1}{2}\sum _{i=1}^l(\eta _{n,1}^i-\eta _{n,n}^i)\Vert \xi _u^n\Vert ^2\nonumber \\&\quad \quad +\frac{C}{\sum _{i=1}^l\eta _{n,n}^i}\left( n^{-2\min \{2-\alpha _1,r\alpha _1\}}+h^{2k+2}\right) +\frac{\sum _{i=1}^l\eta _{n,n}^i}{2}\Vert \xi _u^n\Vert ^2+Ch^{2k+2}\nonumber \\&\quad \quad +\big (c(x)\xi _u^n,\xi _u^n\big )+\Vert \xi _p^n\Vert ^2. \end{aligned}$$

(2.24)

From (2.6), it is easy to get that

$$\begin{aligned} \frac{1}{d_{n,n}^i}\le \frac{T^{\alpha _i}}{1-\alpha _i}n^{r\alpha _i}M^{-r\alpha _i}, \end{aligned}$$

(2.25)

which leads to

$$\begin{aligned} \frac{1}{\sum _{i=1}^l\eta _{n,n}^i}\le \frac{1}{\eta _{n,n}^1}=\frac{\Gamma (2-\alpha _1)}{q_1d_{n,n}^1} \le \frac{\Gamma (1-\alpha _1)}{q_1}T^{\alpha _1}n^{r\alpha _1}M^{-r\alpha _1}. \end{aligned}$$

(2.26)

Combining (2.24) and (2.26), we can derive

$$\begin{aligned} \sum _{i=1}^l\eta _{n,1}^i\Vert \xi _u^n\Vert ^2\le & {} \sum _{i=1}^l\sum _{k=1}^{n-1}(\eta _{n,k}^i-\eta _{n,k+1}^i)\Vert \xi _u^{n-k}\Vert ^2\nonumber \\&+ ~ C\left( h^{2k+2}+M^{-r\alpha _1}n^{-(2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\right) . \end{aligned}$$

(2.27)

Now we prove that the truncation error $\xi _u^n$ satisfies

$$\begin{aligned} \Vert \xi _u^n\Vert ^2\le \frac{C}{\sum _{i=1}^l\eta _{n,1}^i} \sum _{j=1}^n\theta _{n,j}\left( h^{2k+2} +M^{-r\alpha _1}j^{-(2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\right) \end{aligned}$$

(2.28)

for $n=1,\ldots ,M$, where C is the constant in (2.27). We prove (2.28) by mathematical induction. When $n=1$, (2.28) becomes

$$\begin{aligned} \Vert \xi _u^1\Vert ^2\le \frac{C}{\sum _{i=1}^l\eta _{1,1}^i}(h^{2k+2+M^{-r\alpha _1}}). \end{aligned}$$

(2.29)

Supposing the following estimates hold

$$\begin{aligned} \Vert \xi _u^m\Vert ^2\le \frac{C}{\sum _{i=1}^l\eta _{m,1}^i} \sum _{j=1}^n\theta _{m,j}\left( h^{2k+2} +M^{-r\alpha _1}j^{-(2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\right) \end{aligned}$$

(2.30)

for $m=2,3,\ldots ,s$, we only need to prove

$$\begin{aligned} \Vert \xi _u^{s+1}\Vert ^2\le \frac{C}{\sum _{i=1}^l\eta _{s+1,1}^i} \sum _{j=1}^{s+1}\theta _{s+1,j}\left( h^{2k+2} +M^{-r\alpha _1}j^{-(2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\right) .\nonumber \\ \end{aligned}$$

(2.31)

Letting $n=s+1$ in (2.28) and using the induction hypothesis (2.30), we have

$$\begin{aligned} \Vert \xi _u^{s+1}\Vert ^2\le & {} \frac{1}{\sum _{i=1}^l\eta _{s+1,1}^i} \Big [\sum _{i=1}^l\sum _{k=1}^s(\eta _{s+1,k}^i-\eta _{s+1,k+1}^i)\Vert \xi _u^{s+1-k}\Vert ^2\\&+~C~\Big (h^{2k+2}+M^{-r\alpha _1}(s+1)^{(-2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\Big )\Big ]\\\le & {} \frac{1}{\sum _{i=1}^l\eta _{s+1,1}^i} \Bigg [\sum _{i=1}^l\sum _{k=1}^s(\eta _{s+1,k}^i-\eta _{s+1,k+1}^i) \Bigg (\frac{C}{\sum _{i=1}^l\eta _{s+1-k,1}^i}\\&\times \sum _{j=1}^{s+1-k}\theta _{s+1-k,j} \big (h^{2k+2}+M^{-r\alpha _1}j^{-(2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\big )\Bigg )\\&+~C~\Big (h^{2k+2}+M^{-r\alpha _1}(s+1)^{(-2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\Big )\Bigg ]\\= & {} \frac{1}{\sum _{i=1}^l\eta _{s+1,1}^i}\Bigg [\sum _{j=1}^s\Bigg (\sum _{i=1}^l \sum _{k=1}^{s+1-j}\frac{C(\eta _{s+1,k}^i-\eta _{s+1,k+1}^i)}{\sum _{i=1}^l\eta _{s+1-k,1}^i}\theta _{s+1-k,j}\\&\times \Big (h^{2k+2}+M^{-r\alpha _1}j^{-(2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\Big )\Bigg )\\&+~C~\Big (h^{2k+2}+M^{-r\alpha _1}(s+1)^{(-2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\Big )\Bigg ]\\= & {} \frac{C}{\sum _{i=1}^l\eta _{s+1,1}^i}\sum _{j=1}^{s+1}\theta _{s+1,j} \Big (h^{2k+2}+M^{-r\alpha _1}j^{(-2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\Big ). \end{aligned}$$

Therefore, the estimate (2.28) holds.

Exploiting (2.28) and Lemma 2.2 directly, we obtain

$$\begin{aligned} \Vert \xi _u^n\Vert ^2\le & {} \frac{C}{\sum _{i=1}^l\eta _{n,1}^i}\sum _{j=1}^n\theta _{n,j}h^{2k+2} +\frac{CM^{-r\alpha _1}}{\sum _{i=1}^l\eta _{n,1}^i}\sum _{j=1}^n\theta _{n,j}j^{-(2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\\\le & {} C\Gamma (1-\alpha _1)T^{\alpha _1}h^{2k+2}+CM^{-r\alpha _1}\Gamma (1-\alpha _1)T^{\alpha _1} M^{-(2\min \{2-\alpha _1,r\alpha _1\}-r\alpha _1)}\\\le & {} C\Gamma (1-\alpha _1)\left( h^{2k+2}+M^{-2\min \{2-\alpha _1,r\alpha _1\}}\right) , \end{aligned}$$

which, together with the interpolation property(2.3) and the triangle inequality, completes the proof of this theorem. $\square$

It is clear that the results derived in Theorems 2.1 and 2.2 are $\alpha _1$-nonrobust, i.e., the bounds blow up as $\alpha _1\rightarrow 1^-$. In the following subsection, we present the improved stability and convergence analysis for the scheme (2.9).

2.3 $\alpha _1$-Robust error analysis of the non-uniform L1/LDG method

This subsection is devoted to the $\alpha _1$-robust stability and convergence analysis of non-uniform L1/LDG scheme (2.9) for system (1.1)–(1.3). Let us start by introducing the following lemmas.

Lemma 2.4

([9], Lemma 2) Suppose that the solution u(x, t) of problem (1.1)–(1.3) satisfies (1.6). Then there exists a constant C such that for all $t_n$ one has

$$\begin{aligned} |R^n_i|\le Ct_n^{-\alpha _i}M^{-\min \{2-\alpha _i,r\alpha _1\}} \end{aligned}$$

for $i=1,2,\ldots ,l$, $n=1,2,\ldots ,M$.

Lemma 2.5

([9], Corollary 1) For $n=1,2,\ldots ,M$, one has

$$\begin{aligned} \frac{1}{d_{n,1}}\sum _{j=1}^n\theta _{n,j} \le \sum _{i=1}^l\frac{{t}_n^{\alpha _i}}{q_i\Gamma (1+\alpha _i)}. \end{aligned}$$

Lemma 2.6

([9], Corollary 2) Set $l_M=\frac{1}{\ln M}$. Assume that $M\ge 3$ so $0<l_M<1$. Then

$$\begin{aligned} \frac{1}{d_{n,1}}\sum _{j=1}^n\left( \sum _{i=1}^lq_it_j^{-\alpha _i}\right) \theta _{n,j} \le \frac{le^r\max _{1\le i\le l}\Gamma (1+l_M-\alpha _i)}{\Gamma (1+l_M)}. \end{aligned}$$

Lemma 2.7

([9], Lemma 6) Assume that the sequences $\{\xi ^n\}_{n=1}^\infty$, $\{\eta ^n\}_{n=1}^\infty$ are nonnegative and the grid function $\{v^n:n=0,1,\ldots ,M\}$ satisfies $v^0\ge 0$ and

$$\begin{aligned} v^n\sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}v^n\le \xi ^nv^n+(\eta ^n)^2,\, n=1,2,\ldots ,M. \end{aligned}$$

Then

$$\begin{aligned} v^n\le v^0+\frac{1}{d_{n,1}}\sum _{j=1}^n\theta _{n,j}(\xi ^j+\eta ^j)+\max _{1\le j\le n}\left\{ \eta ^j\right\} ,\, n=1,2,\ldots ,M. \end{aligned}$$

We shall now improve Theorem 2.1 by replacing (2.32) with a bound that is $\alpha _1$-robust.

Theorem 2.3

(Improved $L^2$-norm stability) The solution $U_h^n$ of the fully discrete scheme (2.9) satisfies

$$\begin{aligned} \Vert U_h^n\Vert \le \Vert U_h^0\Vert + \left( \sum _{i=1}^l\frac{{t}_n^{\alpha _i}}{q_i\Gamma (1+\alpha _i)}\right) \max _{1\le j\le n}\Vert f^j\Vert ,\,n=1,\ldots ,M. \end{aligned}$$

(2.32)

Proof

Taking the test functions $(v_h,w_h)=(U_h^n, P_h^n)$, and adding the two equations in (2.9), we have

$$\begin{aligned} \left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}U_h^n,U_h^n\right) +(c(x)U_h^n,U_h^n)+(P_h^n,P_h^n)=(f^n,U_h^n). \end{aligned}$$

(2.33)

It follows from [9, Lemma 3] that

$$\begin{aligned} \left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}v^n,v^n\right) \ge \left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}\Vert v^n\Vert \right) \Vert v^n\Vert \end{aligned}$$

(2.34)

for $n=1,2,\ldots ,M$.

Applying (2.33) and (2.34), as well as Cauchy–Schwarz inequality, we obtain

$$\begin{aligned} \left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}\Vert U_h^n\Vert \right) \Vert U_h^n\Vert \le \Vert f^n\Vert \Vert U_h^n\Vert . \end{aligned}$$

(2.35)

Then an application of Lemmas 2.5 and 2.7 immediately yields

$$\begin{aligned} \Vert U_h^n\Vert\le & {} \Vert U_h^0\Vert +\frac{1}{d_{n,1}}\sum _{j=1}^n\theta _{n,j}\max _{1\le j\le n}\Vert f^j\Vert \\\le & {} \Vert U_h^0\Vert + \left( \sum _{i=1}^l\frac{{t}_n^{\alpha _i}}{q_i\Gamma (1+\alpha _i)}\right) \max _{1\le j\le n}\Vert f^j\Vert , \end{aligned}$$

which completes the proof. $\square$

We also give an $\alpha _1$-robust convergence result of the fully discrete non-uniform L1/LDG scheme (2.9) for (1.1)–(1.3). The conclusion is stated as follows.

Theorem 2.4

(Improved $L^2$-norm error estimate) Let u be the exact solution of (1.1)–(1.3) and $U_h^n$ be the numerical solution of the fully discrete non-uniform L1/LDG scheme (2.9). Suppose that u satisfies condition (1.6) and $u(\cdot ,t)\in H^{k+1}({\mathcal {T}}_h)$. Then, it holds that

$$\begin{aligned} \Vert u^n-U_h^n\Vert \le \frac{C\max _{1\le i\le l}\Gamma (1+l_M-\alpha _i)}{\Gamma (1+l_M)}M^{-\min \{2-\alpha _1,r\alpha _1\}} +Ch^{k+1}, \end{aligned}$$

(2.36)

where C is a positive constant independent of M and h.

Proof

As shown in Theorem 2.2, one has

$$\begin{aligned}&\left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i} \xi _u^n,\xi _u^n\right) +\big (c(x)\xi _u^n,\xi _u^n\big ) +\Vert \xi _p^n\Vert ^2\\&\quad =-\left( \sum _{i=1}^lq_iR^n_i,\xi _u^n\right) -\left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}\eta _u^n,\xi _u^n\right) -\big (c(x)\eta _u^n,\xi _u^n\big )-(\eta _p^n,\xi _p^n). \end{aligned}$$

By using (2.34), we obtain that

$$\begin{aligned}&\left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i} \Vert \xi _u^n\Vert \right) \Vert \xi _u^n\Vert +\big (c(x)\xi _u^n,\xi _u^n\big ) +\Vert \xi _p^n\Vert ^2 \nonumber \\&=-\left( \sum _{i=1}^lq_iR^n_i,\xi _u^n\right) -\left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}\eta _u^n,\xi _u^n\right) -\big (c(x)\eta _u^n,\xi _u^n\big )-(\eta _p^n,\xi _p^n)\nonumber \\&\le \sum _{i=1}^lq_i\Vert R_i^n\Vert \Vert \xi _u^n\Vert +\sum _{i=1}^lq_i\Vert \Upsilon _t^{\alpha _i}\eta _u^n\Vert \Vert \xi _u^n\Vert +\left\| \sqrt{c(x)}\eta _u^n\right\| \left\| \sqrt{c(x)}\xi _u^n\right\| \nonumber \\&\quad +\Vert \eta _p^n\Vert \Vert \xi _p^n\Vert \nonumber \\&\le \left( C\sum _{i=1}^lq_it_n^{-\alpha _i}M^{-\min \{2-\alpha _i,r\alpha _1\}}+C\sum _{i=1}^lq_ih^{k+1}\right) \Vert \xi _u^n\Vert +Ch^{2k+2}\nonumber \\&\quad +\left\| \sqrt{c(x)}\xi _u^n\right\| ^2+\Vert \xi _p^n\Vert ^2, \end{aligned}$$

(2.37)

where we invoked (2.3) and Lemma 2.4 for the last inequality.

Consequently, applying Lemmas 2.5–2.7, we can get

$$\begin{aligned} \Vert \xi _u^n\Vert\le & {} \Vert \xi _u^0\Vert +\frac{C}{d_{n,1}}\sum _{j=1}^n \left( \sum _{i=1}^lq_it_j^{-\alpha _i}M^{-\min \{2-\alpha _i,r\alpha _1\}}+h^{k+1}\right) \theta _{n,j} +Ch^{k+1}\\\le & {} \frac{Ce^r\max _{1\le i\le l}\Gamma (1+l_M-\alpha _i)}{\Gamma (1+l_M)}M^{-\min \{2-\alpha _1,r\alpha _1\}} +Ch^{k+1}. \end{aligned}$$

Finally, by using the triangle inequality and the interpolation property (2.3) again, we can complete the proof of Theorem 2.4. $\square$

3 Two-dimensional case

For a bounded rectangular domain $\Omega =(a_1,b_1)\times (a_2,b_2)\subset {\mathbb {R}}^2$, we divide it into a Cartesian grid ${\mathcal {T}}_h=\{K\}$ consisting of $N_x\times N_y$ rectangular elements $K:=I_i\times J_j=(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}) \times (y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})$, $i=1,\ldots ,N_x$, $j=1,\ldots ,N_y$, where $a_1=x_{\frac{1}{2}}<x_{\frac{3}{2}}<\cdots <x_{N_x+\frac{1}{2}}=b_1$ and $a_2=y_{\frac{1}{2}}<y_{\frac{3}{2}}<\cdots <y_{N_y+\frac{1}{2}}=b_2$. Denoting $\Delta x_i=x_{i+\frac{1}{2}}-x_{i-\frac{1}{2}}$ and $\Delta y_j=y_{j+\frac{1}{2}}-y_{j-\frac{1}{2}}$, respectively. Then the maximal length of all edges is defined by $h=\max _{1\le i\le N_x, 1\le j\le N_y}(\Delta x_i,\Delta y_j)$. We assume that the mesh ${\mathcal {T}}_h$ is quasi-uniform in the sense that there exist constants $C_1,C_2>0$ such that $h\le C_1\Delta x_i$ and $h\le C_2\Delta y_j$ for all $K\in {\mathcal {T}}_h$. Then the finite element space is defined by

$$\begin{aligned} \begin{aligned}&V_{h}=\{v_h\in L^{2}(\Omega ):v_h|_{K}\in {\mathcal {Q}}^{k}(K),\,v_h|_{\partial \Omega }=0,\, \forall K\in {\mathcal {T}}_h\},\\&\mathbf{V}_h=\left\{ \mathbf{w}_h\in L^2(\Omega )^2: \mathbf{w}_h|_K\in {\mathcal {Q}}^k(K)^2,\,\mathbf{w}_h|_{\partial \Omega }=\mathbf{0 },\,\forall K\in {\mathcal {T}}_h\right\} , \end{aligned} \end{aligned}$$

(3.1)

where ${\mathcal {Q}}^{k}(K)$ denotes the space of polynomials of degrees at most k defined on K.

We use a fixed vector $\mathbf{I}=(1,1)^\top$ to uniquely define the inflow and outflow boundaries of $\Omega$, namely,

$$\begin{aligned} {\partial \Omega }^-=\{(x,y)\in \partial \Omega :\, \mathbf{I}\cdot \mathbf \mathbf{n}< 0\},\, {\partial \Omega }^+=\{(x,y)\in \partial \Omega :\,\mathbf{I}\cdot \mathbf \mathbf{n}> 0\}, \end{aligned}$$

where $\mathbf \mathbf{n}$ is the outward unit normal vector of $\Omega$. Similarly, we denote $\partial K^-$ and $\partial K^+$ the inflow and outflow boundaries of K, respectively, i.e.,

$$\begin{aligned} {\partial K}^-=\{(x,y)\in \partial K:\, \mathbf{I}\cdot \mathbf \mathbf{n}< 0\},\, {\partial K}^+=\{(x,y)\in \partial K:\,\mathbf{I}\cdot \mathbf \mathbf{n}> 0\}. \end{aligned}$$

If two elements $K_1$ and $K_2$ are neighbours and share one common side e, i.e., $e=\partial K_1\cap \partial K_2$, then there are two traces for any function defined on e. We denote

$$\begin{aligned} u^+=u|_{{\partial K}_2^-\cap e},\;u^-=u|_{{\partial K}_1^+\cap e},\,[\![u]\!]_e=u^+-u^-,\;[\![u]\!]_{\partial \Omega }=u|_{\partial \Omega }. \end{aligned}$$

For each $h>0$, ${\mathcal {E}}_B$ denotes the set of all boundary edges of the mesh ${\mathcal {T}}_h$ on $\partial \Omega$, ${\mathcal {E}}_I$ denotes the set of all interior edges of the mesh ${\mathcal {T}}_h$ in $\Omega$, and ${\mathcal {E}}$ denotes the union of all edges, i.e., ${\mathcal {E}}={\mathcal {E}}_B\cup {\mathcal {E}}_I$. The $L^2$ norm and $L^2$ inner product on the edges ${\partial K}^{\pm }$ are given by

$$\begin{aligned} \Vert u\Vert _{{\partial K}^\pm }^2=(u,u)_{{\partial K}^\pm },\;(u,v)_{{\partial K}^\pm }=\int _{{\partial K}^\pm }u^\mp (s)\, v^\mp (s)\mathrm{d}s. \end{aligned}$$

The norms on the whole outflow and inflow boundaries ${\mathcal {E}}$ are denoted by

$$\begin{aligned} \Vert u\Vert _{{\mathcal {E}}}^2=\sum _{e\in {\mathcal {E}}}\Vert u\Vert _e^2. \end{aligned}$$

We define the broken Sobolev space V on ${\mathcal {T}}_h$ by

$$\begin{aligned} \begin{aligned} V=\left\{ v\in L^2(\Omega ): v|_K\in H^1(K),\,\forall K\in {\mathcal {T}}_h\right\} , \end{aligned} \end{aligned}$$

and denote by ${\mathbf {V}}=V\times V$.

3.1 The fully discrete non-uniform L1/LDG scheme

We still use the L1 method on non-uniform meshes (see (2.4)) as time discretization and LDG method as space discretization. As the usual treatment, we firstly rewrite (1.1)–(1.3) into a system of the first order derivatives

$$\begin{aligned} {\left\{ \begin{array}{ll} &{} \sum _{i=1}^l\left[ q_i{_{C}\mathrm{D}_{0,t}^{\alpha _i}} u \right] -\nabla \cdot {\mathbf {p}}+cu = f(x,y,t),\,(x,y,t)\in \Omega \times (0,T], \\ &{} {\mathbf {p}}=\nabla u,\,(x,y,t)\in \Omega \times (0,T],\\ &{}u|_{t=0} = {u_0}(x,y),\,(x,y)\in {\overline{\Omega }},\\ &{}u|_{(x,y)\in \partial \Omega }=0,\,t\in (0,T], \end{array}\right. } \end{aligned}$$

(3.2)

Then we can define the semi-discrete LDG scheme for (1.1)–(1.3) as follows: find $(U_h,{\mathbf {P}}_h)\in V_h\times {\mathbf {V}}_h$ such that for all test functions $(v_h,{\mathbf {w}}_h)\in V_h\times {\mathbf {V}}_h$, we have

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}\left( \displaystyle \sum _{i=1}^lq_i{_{C}\mathrm{D}_{0,t}^{\alpha _i}} U_h,v_h\right) +(cU_h,v_h) ={\mathcal {L}}(\mathbf{P}_h,v_h)+( f,v_h),\\ &{}(\mathbf{P}_h,\mathbf{w}_h)={\mathcal {K}}(U_h,\mathbf{w}_h), \end{array}\right. } \end{aligned} \end{aligned}$$

(3.3)

where

$$\begin{aligned} {\mathcal {L}}(\mathbf{P}_h, v_h)=-(\mathbf{P}_h,\nabla v_h)+\sum _{K\in {\mathcal {T}}_h}(\widehat{\mathbf{P}_h}\cdot \mathbf {\mathbf{n}},v_h)_{\partial K}, \end{aligned}$$

(3.4)

$$\begin{aligned} {\mathcal {K}}(U_h, \mathbf{w}_h)=-(U_h,\nabla \cdot \mathbf{w}_h)+\sum _{K\in {\mathcal {T}}_h}(\widehat{U_h},\mathbf{w}_h\cdot \mathbf {\mathbf{n}})_{\partial K}. \end{aligned}$$

(3.5)

Similar to the one-dimensional case, the numerical fluxes $\widehat{U_h}$, $\widehat{\mathbf{P}_h}$ can be chosen as

$$\begin{aligned} \widehat{U_h}=U_h^-,\;\widehat{\mathbf{P}_h}=\mathbf{P}_h^+. \end{aligned}$$

(3.6)

Let $(U_{h}^n,\mathbf{P}_h^n)\in V_h\times {\mathbf {V}}_h$ be the approximation of $( u(x,y,t_n),\mathbf{p}(x,y,t_n))$. Then we define the fully discrete non-uniform L1/LDG scheme as follows: find $(U_{h}^n,{ \mathbf{P}}_h^n)\in V_h\times {\mathbf {V}}_h$ such that for all test functions $( v_h,\mathbf{w}_h)\in V_h\times {\mathbf {V}}_h$,

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}\left( \displaystyle \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i} U^n_h,v_h\right) +(cU^n_h,v_h) ={\mathcal {L}}(\mathbf{P}_h^n,v_h)+( f^n,v_h), \\ &{}(\mathbf{P}_h^n,\mathbf{w}_h)={\mathcal {K}}(U_h^n,\mathbf{w}_h). \end{array}\right. } \end{aligned} \end{aligned}$$

(3.7)

Here the notation $\Upsilon _t^{\alpha _i} U^n_h$ is defined in (2.4).

3.2 $\alpha _1$-Nonrobust error analysis of the non-uniform L1/LDG method

The fully discrete non-uniform L1/LDG scheme (3.7) for the two-dimensional multiterm time-fractional initial-boundary value problem satisfies the following $\alpha _1$-nonrobust stability. First of all, we introduce a lemma that will be used later on.

Lemma 3.1

[21] For any $v_h\in V_h$ and ${\mathbf {w}}_h\in {\mathbf {V}}_{h}$, there holds the equality

$$\begin{aligned} {\mathcal {L}}({\mathbf {w}}_h, v_h)=-{\mathcal {K}}( v_h,{\mathbf {w}}_h). \end{aligned}$$

Theorem 3.1

($L^2$-norm stability) The solution $U_h^n$ of the fully discrete scheme (3.7) satisfies

$$\begin{aligned} \Vert U_h^n\Vert \le \Vert U_h^0\Vert + \Gamma (1-\alpha _1)T^{\alpha _1}\max _{1\le j\le n}\left\| f^j\right\| ,\, n=1,\ldots ,M. \end{aligned}$$

(3.8)

Proof

Taking the test function $v_h=U_h^n$ and $w_h=\mathbf{P}_h^n$ in (3.7) and using Lemma 3.1, we obtain

$$\begin{aligned} \begin{aligned}&\left( \sum _{i=1}^l\frac{q_id_{n,1}^i}{\Gamma (2-\alpha _i)}U_h^n,U_h^n\right) +\left( cU_h^n,U_h^n\right) +\Vert \mathbf{P}_h^n\Vert ^2\\&\quad =\left( \sum _{i=1}^l\frac{q_id_{n,n}^i}{\Gamma (2-\alpha _i)}U_h^0,U_h^n\right) +\left( \sum _{i=1}^l\frac{q_i}{\Gamma (2-\alpha _i)}\sum _{k=1}^{n-1}(d_{n,k}^i-d_{n,k+1}^i)U_h^{n-k},U_h^n\right) \\&\quad \quad +(f^n,U_h^n). \end{aligned} \end{aligned}$$

(3.9)

By using an analysis similar to that in (2.11) and in Theorem 2.1, we can complete the proof of this theorem. $\square$

Now we present the $\alpha _1$-nonrobust convergence analysis and give the detailed proof. To obtain the optimal error estimate for the non-uniform L1/LDG scheme (3.7), we would like to use the elliptic projection introduced in [5] to eliminate the element boundary errors. Let $u\in V$ and ${\mathbf {q}}=\nabla u$, define the elliptic projection $({\mathcal {P}}_hu, {\mathcal {P}}_h{\mathbf {q}})\in V_h\times {\mathbf {V}}_h$ as: for any $(v_h,{\mathbf {w}}_h)\in V_h\times {\mathbf {V}}_h$, it holds that

$$\begin{aligned} {\mathcal {L}}({\mathbf {q}},v_h) ={\mathcal {L}}({\mathcal {P}}_h{\mathbf {q}},v_h), \end{aligned}$$

(3.10)

$$\begin{aligned} ({\mathcal {P}}_h{\mathbf {q}},{\mathbf {w}}_h) ={\mathcal {K}}({\mathcal {P}}_hu,{\mathbf {w}}_h), \end{aligned}$$

(3.11)

$$\begin{aligned} (u-{\mathcal {P}}_hu,1)=0. \end{aligned}$$

(3.12)

The elliptic projection defined above uniquely exists and satisfies the following approximation properties.

Lemma 3.2

[21] Assume $u\in H^{k+2}(\Omega )$, then there exists a constant C depending on the regularity of u such that

$$\begin{aligned} \Vert u-{\mathcal {P}}_hu\Vert +h^{\frac{1}{2}}\Vert u-{\mathcal {P}}_hu\Vert _{\mathcal {E}}\le Ch^{k+1}. \end{aligned}$$

(3.13)

Theorem 3.2

($L^2$-norm error estimate) Let u be the exact solution of (1.1)–(1.3) and $U_h^n$ be the numerical solution of the fully discrete non-uniform L1/LDG scheme (3.7). Suppose that u satisfies condition (1.6) and $u(\cdot ,t)\in H^{k+2}(\Omega )$. Then, it holds that

$$\begin{aligned} \Vert u^n-U_h^n\Vert \le C\sqrt{\Gamma (1-\alpha _1)}\left( M^{-\min \{2-\alpha _1,r\alpha _1\}}+h^{k+1}\right) , \end{aligned}$$

(3.14)

where C is a positive constant independent of M and h.

Proof

Denote

$$\begin{aligned} \begin{aligned}&e_{u}^{n}=u^n-U_{h}^{n}={\mathcal {P}}_hu^n-U_h^n+(u^n-{\mathcal {P}}_hu^n)=\xi _u^n+\eta _u^n,\\&\mathbf{e}_{\mathbf{p}}^{n}=\mathbf{p}^n-\mathbf{P}_{h}^{n}={\mathcal {P}}_h\mathbf{p}^n-\mathbf{P}_h^n +(\mathbf{p}^n-{\mathcal {P}}_h\mathbf{p}^n)={{\varvec{\xi }}}_\mathbf{p}^n+{{\varvec{\eta }}}_\mathbf{p}^n. \end{aligned} \end{aligned}$$

(3.15)

From (3.2), we can get the weak form of (1.1)–(1.3) at $t_n$ as follows,

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}\left( \displaystyle \sum _{i=1}^lq_i{_{C}\mathrm{D}_{0,t}^{\alpha _i}} u^n,v_h\right) +(cu^n,v_h) ={\mathcal {L}}(\mathbf{p}^n,v_h)+( f^n,v_h), \\ &{}(\mathbf{p}^n,\mathbf{w}_h)={\mathcal {K}}(u^n,\mathbf{w}_h). \end{array}\right. } \end{aligned} \end{aligned}$$

(3.16)

Applying the property of elliptic projection (3.10)–(3.12), it holds that

$$\begin{aligned} {\left\{ \begin{array}{ll} &{}{\mathcal {L}}({{\varvec{\eta }}}_\mathbf{p}^n,v_h)={\mathcal {L}}(\mathbf{p}^n-{\mathcal {P}}_h{\mathbf {p}}^n,v_h),\\ &{}({{\varvec{\eta }}}_\mathbf{p}^n,\mathbf{w}_h)={\mathcal {K}}(\eta _u^n,\mathbf{w}_h). \end{array}\right. } \end{aligned}$$

(3.17)

Subtracting (3.7) from (3.16) and noticing (3.17), we obtain the error equation

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} &{}\left( \displaystyle \sum _{i=1}^lq_i\big ({_{C}\mathrm{D}_{0,t}^{\alpha _i}} u^n-\Upsilon _t^{\alpha _i} U_h^n\big ),v_h\right) +(c e_u^n,v_h)={\mathcal {L}}({{\varvec{\xi }}}_\mathbf{p}^n, v_h),\\ &{}({{\varvec{\xi }}}_\mathbf{p}^n,{\mathbf {w}}_h)={\mathcal {K}}(\xi _u^n,{\mathbf {w}}_h). \end{array}\right. } \end{aligned} \end{aligned}$$

(3.18)

Taking the test function $(v_h,{\mathbf {w}}_h)=(\xi _u^n,{{\varvec{\xi }}}_\mathbf{p}^n)$ in (3.18) and using Lemma 3.1, we get

$$\begin{aligned}&\left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i} \xi _u^n,\xi _u^n\right) +\big (c\xi _u^n,\xi _u^n\big ) +\Vert {{\varvec{\xi }}}_\mathbf{p}^n\Vert ^2 \nonumber \\&\quad =-\left( \sum _{i=1}^lq_iR^n_i,\xi _u^n\right) -\left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}\eta _u^n,\xi _u^n\right) -\big (c\eta _u^n,\xi _u^n\big ). \end{aligned}$$

(3.19)

Repeating similar arguments as Theorem 2.2 (see the proof of (2.29) and (2.31)), we can use mathematical induction to obtain the error estimate

$$\begin{aligned} \Vert u^n-U_h^n\Vert \le C\sqrt{\Gamma (1-\alpha _1)}\left( M^{-\min \{2-\alpha _1,r\alpha _1\}}+h^{k+1}\right) . \end{aligned}$$

The proof is thus completed. $\square$

3.3 $\alpha _1$-Robust error analysis of the non-uniform L1/LDG method

We are now ready to state the $\alpha _1$-robust stability and convergence analysis of non-uniform L1/LDG scheme (3.7) for system (1.1)–(1.3).

Theorem 3.3

(Improved $L^2$-norm stability) The solution $U_h^n$ of the fully discrete scheme (3.7) satisfies

$$\begin{aligned} \Vert U_h^n\Vert \le \Vert U_h^0\Vert + \left( \sum _{i=1}^l\frac{{t}_n^{\alpha _i}}{q_i\Gamma (1+\alpha _i)}\right) \max _{1\le j\le n}\Vert f^j\Vert ,\,n=1,\ldots ,M. \end{aligned}$$

(3.20)

Proof

Taking the test function $(v_h,w_h)=(U_h^n,\mathbf{P}_h^n)$ in (3.7) and applying Lemma 3.1, we can get

$$\begin{aligned} \left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}U_h^n,U_h^n\right) +(cU_h^n,U_h^n)+(\mathbf{P}_h^n,\mathbf{P}_h^n)=(f^n,U_h^n). \end{aligned}$$

(3.21)

Then, similar to the proof of Theorem 2.3, the $L^2$-norm stability (3.20) can be obtained immediately. This finishes the proof. $\square$

Next, we state the $\alpha _1$-robust convergence result of the fully discrete non-uniform L1/LDG scheme (3.7).

Theorem 3.4

(Improved $L^2$-norm error estimate) Let u be the exact solution of (1.1)–(1.3) and $U_h^n$ be the numerical solution of the fully discrete non-uniform L1/LDG scheme (3.7). Suppose that u satisfies condition (1.6) and $u(\cdot ,t)\in H^{k+2}(\Omega )$. Then, it holds that

$$\begin{aligned} \Vert u^n-U_h^n\Vert \le \frac{C\max _{1\le i\le l}\Gamma (1+l_M-\alpha _i)}{\Gamma (1+l_M)}M^{-\min \{2-\alpha _1,r\alpha _1\}} +Ch^{k+1}, \end{aligned}$$

(3.22)

where C is a positive constant independent of M and h.

Proof

Set

$$\begin{aligned} \begin{aligned}&e_{u}^{n}=u^n-U_{h}^{n}={\mathcal {P}}_hu^n-U_h^n+(u^n-{\mathcal {P}}_hu^n)=\xi _u^n+\eta _u^n, \\&\mathbf{e}_{\mathbf{p}}^{n}=\mathbf{p}^n-\mathbf{P}_{h}^{n}={\mathcal {P}}_h\mathbf{p}^n-\mathbf{P}_h^n +(\mathbf{p}^n-{\mathcal {P}}_h\mathbf{p}^n)={{\varvec{\xi }}}_\mathbf{p}^n+{{\varvec{\eta }}}_\mathbf{p}^n. \end{aligned} \end{aligned}$$

By the similar techniques used in the proof of Theorem 3.2, it holds that

$$\begin{aligned}&\left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i} \xi _u^n,\xi _u^n\right) +\big (c\xi _u^n,\xi _u^n\big ) +\Vert {{\varvec{\xi }}}_\mathbf{p}^n\Vert ^2 \\&\quad =-\left( \sum _{i=1}^lq_iR^n_i,\xi _u^n\right) -\left( \sum _{i=1}^lq_i\Upsilon _t^{\alpha _i}\eta _u^n,\xi _u^n\right) -\big (c\eta _u^n,\xi _u^n\big ). \end{aligned}$$

Then, repeating similar arguments as Theorem 2.4, we can obtain (3.22). The proof is thus completed. $\square$

4 Numerical examples

In this section, we present a numerical example to validate our theoretical results.

Example 4.1

Consider the following three-term time-fractional diffusion equation

$$\begin{aligned} {\left\{ \begin{array}{ll} &{} _{C}\mathrm{D}_{0,t}^{\alpha _1} u+0.1 _{C}\mathrm{D}_{0,t}^{0.1} u +0.1 _{C}\mathrm{D}_{0,t}^{0.2} u -u_{xx}+u = f(x,t),\,(x,t)\in (0,1)\times (0,1], \\ &{} u(x,0) = 0,\,x\in (0,1),\\ &{}u(0,t)=u(1,t)=0,\,t\in (0,1], \end{array}\right. } \end{aligned}$$

(4.1)

where $0<\alpha _1<1$. The source term f(x, t) is chosen such that the exact solution of the problem is $u=(t^{\alpha _1}+t^3)\sin (2\pi x)$.

The $L^2$ and $L^\infty$ numerical errors and orders with different $\alpha _1$ at $T=1$ are given in Tables 1–5. From these results, we conclude that the non-uniform L1/LDG scheme (2.9) for the three-term time-fractional diffusion equation in Example 4.1 can achieve $\min \{2-\alpha _1,r\alpha _1\}$-th order convergence in time and $(k+1)$-th order convergence in space, which are in line with the theoretical rate established in Theorem 2.4.

Table 1 The time convergence results for Example 4.1 at $T=1$ with $k=1$, $M=N$, and $r=1$

Table 2 The time convergence results for Example 4.1 at $T=1$ with $k=1$, $M=N$, and $r=\frac{1}{\alpha }$

Table 3 The time convergence results for Example 4.1 at $T=1$ with $k=1$, $M=N$, and $r=\frac{2-\alpha }{\alpha }$

Table 4 The time convergence results for Example 4.1 at $T=1$ with $k=1$, $M=N$, and $r=\frac{2(2-\alpha )}{\alpha }$

Table 5 The spatial convergence results for Example 4.1 at $T=1$ with $M=500$, $r=\frac{2-\alpha }{\alpha }$, and $k=1$

5 Concluding remarks

In this paper, we have studied the multiterm time-fractional initial-boundary value problem. Considering the weak regularity of the solution at the starting time, we use the L1 scheme with non-uniform meshes to discretize the time fractional derivative, and the classical LDG method for the space derivative. Numerical stability and convergence of the established schemes are analyzed. Such stability and convergence results are proved to be $\alpha _1$-robust. Finally, a numerical example is given to confirm the theoretical results.