A Time-Spectral Algorithm for Fractional Wave Problems

Abstract

This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence of this algorithm are derived, and numerical experiments are performed, demonstrating the exponential decay in the temporal discretization error provided the solution is sufficiently smooth.

1 Introduction

Let $1< \gamma < 2$ and let $\Omega \subset \mathbb R^d$ ($d=2,3$) be a polygon/polyhedron. This paper considers the fractional wave problem

$$\begin{aligned} \left\{ \begin{aligned} D_{0+}^\gamma (u-u_0-tu_1) - \Delta u&= f&\text {in } \Omega \times (0,T) , \\ u&= 0&\text {on } \partial \Omega \times (0,T) , \\ u(\cdot ,0)&= u_0&\text {in } \Omega , \\ u_t(\cdot ,0)&= u_1&\text {in } \Omega , \end{aligned} \right. \end{aligned}$$

(1)

where $u_0 \in H_0^1(\Omega )$, $u_1 \in L^2(\Omega )$, and $f \in L^2(\Omega _T)$ with $\Omega _T := \Omega \times (0,T)$. Here $u_t$ is the derivative of u with respect to the time variable t, and $D_{0+}^\gamma$ is a Riemann–Liouville fractional differential operator.

The above problem is a particular case of time fractional diffusion-wave problems, which have attracted a considerable amount of research in the field of numerical analysis in the past twenty years. By now, most of the existing numerical algorithms employ the L1 scheme ([5, 11, 17, 27, 28]), Grünwald-Letnikov discretization ([2, 12, 19, 20, 23, 24]) or fractional linear multi-step method ([8, 21, 26]) to discrete the fractional derivatives. Generally, for those algorithms, the best temporal accuracy are $O(\tau ^2)$ for the fractional diffusion problems and $O(\tau ^{3-\gamma })$ for the fractional wave problems, where $\tau$ is the time step size.

Due to the nonlocal property of fractional differential operator, the memory and computing cost of an accuracy approximation to a fractional diffusion-wave problem is significantly more expensive than that to a corresponding normal diffusion-wave problem. To reduce the cost, high-accuracy algorithms are often preferred, especially those of high accuracy in the time direction. This motivates us to develop high-accuracy numerical algorithms for problem (1). The efforts in this aspect are summarized as follows. Li and Xu [10] proposed a space-time spectral algorithm for the fractional diffusion equation, and then Zheng et al. [29] constructed a high order space-time spectral method for the fractional Fokker–Planck equation. Gao et al. [7] proposed a new scheme to approximate Caputo fractional derivatives of order $\gamma$ ($0<\gamma <1$). Zayernouri and Karniadakis [25] developed an exponentially accurate fractional spectral collocation method. Yang et al. [22] developed a spectral Jacobi collocation method for the time fractional diffusion-wave equation. Recently, Ren et al. [14] investigated the superconvergence of finite element approximation to time fractional wave problems; however, the temporal accuracy order is only $O(\tau ^{3-\gamma })$.

In this paper, using a spectral method in the temporal discretization and a finite element method in the spatial discretization, we design a high-accuracy algorithm for problem (1) and establish its stability and convergence. Our numerical experiments show the exponential decay in the temporal discretization errors, provided the underlying solution is sufficiently smooth.

The rest of this paper is organized as follows. Section 2 introduces some Sobolev spaces and the Riemann–Liouville fractional calculus operators. Section 3 describes a time-spectral algorithm and constructs the basis functions for the temporal discretization. Sections 4 and 5 establish the stability and convergence of the proposed algorithm, and Sect. 6 performs some numerical experiments to demonstrate its high accuracy. Finally, Sect. 7 provides some concluding remarks.

2 Notation

Let us first introduce some Sobolev spaces. For $0< \alpha < \infty$, as usual, $H_0^\alpha (0,T)$, $H^\alpha (0,T)$, $H_0^\alpha (\Omega )$ and $H^\alpha (\Omega )$ are used to denote four standard Sobolev spaces; see [18]. Let X be a separable Hilbert space with an inner product $(\cdot ,\cdot )_X$ and an orthonormal basis $\{ e_k: k \in \mathbb N \}$. For $0< \alpha < \infty$, define

$$\begin{aligned} H^\alpha (0,T; X) := \left\{ v \in L^2(0,T;X):\ \sum _{k=0}^\infty \left\| (v,e_k)_X \right\| _{ H^\alpha (0,T) }^2 < \infty \right\} \end{aligned}$$

and endow this space with the norm

$$\begin{aligned} \left\| \cdot \right\| _{H^\alpha (0,T; X)} := \left( \sum _{k=0}^\infty \left\| (\cdot ,e_k)_X \right\| _{ H^\alpha (0,T) }^2 \right) ^{1/2}, \end{aligned}$$

where $L^2(0,T;X)$ is an X-valued Bochner $L^2$ space. For $v \in H^j(0,T; X)$ with $j \in \mathbb N_{\geqslant 1}$, the symbol $v^{(j)}$ denotes its jth weak derivative:

$$\begin{aligned} v^{(j)}(t) := \sum _{k=0}^\infty c_k^{(j)}(t) e_k, \quad 0< t < T, \end{aligned}$$

where $c_k(\cdot ) := (v(\cdot ), e_k)_X$ and $c_k^{(j)}$ is its jth weak derivative. Conventionally, $v^{(1)}$ and $v^{(2)}$ are also abbreviated to $v'$ and $v''$, respectively.

Moreover, for $j \in \mathbb N$ we define

$$\begin{aligned} B^j(0,T; X) := \left\{ v \in L^2(0,T;X):\ \sum _{k=0}^\infty \left\| (v,e_k)_X \right\| ^2_{B^j(0,T)} < \infty \right\} \end{aligned}$$

and equip this space with the norm

$$\begin{aligned} \left\| \cdot \right\| _{B^j(0,T; X)} := \left( \sum _{k=0}^\infty \left\| (\cdot ,e_k)_X \right\| _{B^j(0,T)}^2 \right) ^{1/2}, \end{aligned}$$

where the space $B^j(0,T)$ and its norm are respectively given by

$$\begin{aligned} B^j(0,T) := \left\{ v \in L^2(0,T):\ \int _0^T t^i(T-t)^i \left| v^{(i)}(t) \right| ^2 \, \mathrm {d}t < \infty ,\ 0 \leqslant i \leqslant j \right\} \end{aligned}$$

and

$$\begin{aligned} \left\| \cdot \right\| _{B^j(0,T)} := \left( \sum _{i=0}^j \int _0^T t^i(T-t)^i \left| (\cdot )^{(i)}(t) \right| ^2 \, \mathrm {d}t \right) ^{1/2}. \end{aligned}$$

Then we introduce the Riemann–Liouville fractional operators. Let X be a Banach space and let $L^1(0,T;X)$ be an X-valued Bochner $L^1$ space.

Definition 2.1

For $0< \alpha < \infty$, define $I_{0+}^{\alpha ,X}, I_{T-}^{\alpha ,X}:~L^1(0,T;X) \rightarrow L^1(0,T;X)$, respectively, by

$$\begin{aligned} \left( I_{0+}^{\alpha ,X} v\right) (t)&:= \frac{1}{ \Gamma (\alpha ) } \int _0^t (t-s)^{\alpha -1} v(s) \, \mathrm {d}s, \quad 0< t< T, \\ \left( I_{T-}^{\alpha ,X} v\right) (t)&:= \frac{1}{ \Gamma (\alpha ) } \int _t^T (s-t)^{\alpha -1} v(s) \, \mathrm {d}s, \quad 0< t < T, \end{aligned}$$

for all $v \in L^1(0,T;X)$.

Definition 2.2

For $j-1< \alpha < j$ with $j \in \mathbb N_{>0}$, define

$$\begin{aligned} D_{0+}^{\alpha ,X}&:= D^j I_{0+}^{j-\alpha ,X}, \\ D_{T-}^{\alpha ,X}&:= (-1)^j D^j I_{T-}^{j-\alpha ,X}, \end{aligned}$$

where D is the first-order differential operator in the distribution sense.

Above $\Gamma (\cdot )$ is the Gamma function, and, for convenience, we shall simply use $I_{0+}^\alpha$, $I_{T-}^\alpha$, $D_{0+}^\alpha$ and $D_{T-}^\alpha$, without indicating the underlying Banach space X. Each $v \in L^1(\Omega _T)$ also regarded as an element of $L^1(0,T;X)$ with $X=L^1(\Omega )$, and thus $D_{0+}^\alpha v$ and $D_{T-}^\alpha v$ mean $D_{0+}^{\alpha ,X}v$ and $D_{T-}^{\alpha ,X}v$, respectively, for all $0< \alpha < \infty$.

3 Algorithm Definition

Let $\mathcal K_h$ be a triangulation of $\Omega$ consisting of d-simplexes, and let h be the maximum diameter of these simplexes in $\mathcal K_h$. Define

$$\begin{aligned} V_h&:= \left\{ v_h \in H^1(\Omega ):\ v_h|_K \in P_m(K) \text {for all }K \in \mathcal K_h \right\} , \\ \mathring{V}_h&:= \left\{ v_h \in H_0^1(\Omega ):\ v_h|_K \in P_m(K) \text {for all } K \in \mathcal K_h \right\} , \end{aligned}$$

where m is a positive integer and $P_m(K)$ is the set of all polynomials defined on K of degree $\leqslant m$. For $j \in \mathbb N$, define

$$\begin{aligned} P_j[0,T] \otimes \mathring{V}_h := \text {span} \big \{ qv_h:\ v_h \in \mathring{V}_h,\ q \in P_j[0,T] \big \}, \end{aligned}$$

where $P_j[0,T]$ is the set of all polynomials defined on [0, T] of degree $\leqslant j$. Moreover, we introduce a projection operator $R_h: H_0^1(\Omega ) \rightarrow \mathring{V}_h$ by

$$\begin{aligned} \big ( \nabla (I - R_h) v, \nabla v_h \big )_{L^2(\Omega )} = 0, \quad \forall v \in H_0^1(\Omega ), \ \forall v_h \in \mathring{V}_h. \end{aligned}$$

Here and in the rest of this paper, I denotes the identity operator.

Now, let us describe a time-spectral algorithm for problem (1) as follows: seek $U \in P_M[0,T] \otimes \mathring{V}_h$ with $U(0) = R_h u_0$ such that

$$\begin{aligned} \left( D_{0+}^{\gamma _0} (U' - u_{h,1}), D_{T-}^{\gamma _0} V \right) _{ L^2(\Omega _T) } + ( \nabla U,\nabla V )_{ L^2(\Omega _T) } = (f,V)_{ L^2(\Omega _T) } \end{aligned}$$

(2)

for all $V \in P_{M-1}[0,T] \otimes \mathring{V}_h$, where $M \geqslant 2$ is an integer, $\gamma _0 := (\gamma -1)/2$, and $u_{h,1}$ is the $L^2(\Omega )$-projection of $u_1$ onto $V_h$.

Remark 3.1

In “Appendix A” we define the weak solution of problem (1). The numerical solution obtained by (2) is actually an approximation of the weak solution to problem (1).

Remark 3.2

It is well known that the solution to problem (1) generally has singularity in time, caused by the fractional derivative. However, in view of the basic properties of the operator $D_{0+}^\gamma$, it is anticipated that we can improve the performance of the above algorithm by enlarging $P_M[0,T]$ and $P_{M-1}[0,T]$ by some singular functions, such as $t^\gamma$ for $P_M[0,T]$ and correspondingly $t^{\gamma -1}$ for $P_{M-1}[0,T]$.

The remainder of this section is devoted to the construction of the bases of $P_M[0,T]$ and $P_{M-1}[0,T]$, which is crucial in the implementation of the proposed algorithm. To this purpose, let us first introduce the well-known Jacobi polynomials; see [1, 16] for more details. Given $-1< \alpha , \beta < \infty$, the Jacobi polynomials $\{ J_n^{(\alpha ,\beta )}:\ n \in \mathbb N \}$ are defined by

$$\begin{aligned} J_n^{(\alpha ,\beta )}(t) = w^{-\alpha ,-\beta }(t) \frac{(-1)^n}{2^n n!} \frac{\mathrm {d}^n}{\mathrm {d}t^n} w^{n+\alpha ,n+\beta }(t),\quad -1< t < 1, \ n \in \mathbb N, \end{aligned}$$

where

$$\begin{aligned} w^{r,s}(t) := (1-t)^{r} (1+t)^{s}, \end{aligned}$$

for all $-1< r,s < +\infty$. They form a complete orthogonal basis of $L_{w^{\alpha ,\beta }}^2(-1,1)$, the weighted $L^2$ space with weight function $w^{\alpha ,\beta }$.

Then we construct a basis $\{p_i\}_{i=0}^M$ of $P_M[0,T]$ and a basis $\{q_j\}_{j=0}^{M-1}$ of $P_{M-1}[0,T]$, respectively, by

$$\begin{aligned} \left\{ \begin{aligned}&p_0(t) := 1, \\&p_i(t) := \frac{2t}{T} J_{i-1}^{(-\gamma _0,0)} \left( 2t/T-1 \right) , \quad 1 \leqslant i \leqslant M, \end{aligned} \right. \end{aligned}$$

and

$$\begin{aligned} q_j(t) = J^{(0,-\gamma _0)}_{j}\left( 2t/T-1\right) , \quad 0 \leqslant j \leqslant M-1. \end{aligned}$$

By [3, Lemma 2.5] a straightforward computing yields

$$\begin{aligned} D_{0+}^{\gamma _0} p'_i(t) D_{T-}^{\gamma _0} q_j(t) = t^{-\gamma _0}(T-t)^{-\gamma _0}\zeta _{ij}(t) + t^{1-\gamma _0}(T-t)^{-\gamma _0}\varsigma _{ij}(t), \end{aligned}$$

for all $0 \leqslant i \leqslant M$ and $0 \leqslant j < M$. Above $\zeta _{ij}(t)$ and $\varsigma _{ij}(t)$ are given respectively by

$$\begin{aligned} \zeta _{ij}(t)&= C_{ij} \left( J_{i-1}^{(0,-\gamma _0)} J_{j}^{(-\gamma _0,0)} \right) (2t/T-1), \\ \varsigma _{ij}(t)&= D_{ij} \left( J_{i-2}^{(1,1-\gamma _0)} J_{j}^{(-\gamma _0,0)} \right) (2t/T-1), \end{aligned}$$

where

$$\begin{aligned} C_{ij} := {\left\{ \begin{array}{ll} 0, &{} i = 0, \\ \frac{2}{T} \frac{ \Gamma (i)\Gamma (j+1) }{ \Gamma ( j+1-\gamma _0 )\Gamma (i-\gamma _0) }, &{} i \geqslant 1, \end{array}\right. } \quad D_{ij} := {\left\{ \begin{array}{ll} 0, &{} 0 \leqslant i \leqslant 1, \\ \frac{\Gamma (i+1-\gamma _0)}{\Gamma (i-\gamma _0)T} C_{ij}, &{} i \geqslant 2. \end{array}\right. } \end{aligned}$$

Then $\int _0^T D_{0+}^{\gamma _0} p_i' D_{T-}^{\gamma _0} q_j \, \mathrm {d}t$ is evaluated numerically by a suitable Jacobi-Gauss quadrature rule.

4 Main Results

Let us first introduce the following conventions: u and U are the solutions to problem (1) and (2), respectively; unless otherwise specified, C is a generic positive constant that is independent of any function and is bounded as $M \rightarrow \infty$; $a \lesssim b$ means that there exists a positive constant c, depending only on $\gamma$, T, $\Omega$, m or the shape regular parameter of $\mathcal K_h$, such that $a \leqslant c b$; the symbol $a \sim b$ means $a \lesssim b \lesssim a$. The above shape regular parameter of $\mathcal K_h$ means

$$\begin{aligned} \max \left\{ h_K/\rho _K:\ K \in \mathcal K_h \right\} , \end{aligned}$$

where $h_K$ is the diameter of K, and $\rho _K$ is the diameter of the circle ($d=2$) or ball ($d =3$) inscribed in K.

Then we introduce an interpolation operator. Let X be a separable Hilbert space and let $P_M[0,T;X]$ be the set of all X-valued polynomials defined on [0, T] of degree $\leqslant M$. Define the interpolation operator

$$\begin{aligned} Q_M^X: H^{1+\gamma _0}(0,T;X) \rightarrow P_M[0,T;X] \end{aligned}$$

as follows: for each $v \in H^{1+\gamma _0}(0,T;X)$, the interpolant $Q_M^X v$ fulfills

$$\begin{aligned} \left\{ \begin{aligned}&\left( Q_M^X v \right) (0) = v(0), \\&\int _0^T D_{0+}^{\gamma _0} \left( v-Q_M^Xv \right) ' D_{T-}^{\gamma _0} q \, \mathrm {d}t = 0, \quad \forall q \in P_{M-1}[0,T]. \end{aligned} \right. \end{aligned}$$

For convenience, we shall use $Q_M$ instead of $Q_M^X$ when no confusion will arise.

Remark 4.1

Let $\{e_k: k \in \mathbb N \}$ be an orthonormal basis of X. For any $v \in H^{\gamma _0}(0,T;X)$, the definition of $H^{\gamma _0}(0,T;X)$ implies that

$$\begin{aligned} (v,e_k)_X \in H^{\gamma _0}(0,T) \quad \text {for each }k \in \mathbb N , \end{aligned}$$

and hence, as Lemma 5.4 (in the next section) indicates

$$\begin{aligned} \left\| D_{0+}^{\gamma _0,\mathbb R}(v,e_k)_X \right\| _{L^2(0,T)} \sim \left\| (v,e_k)_X \right\| _{H^{\gamma _0}(0,T)}, \end{aligned}$$

it is evident that

$$\begin{aligned} \left\| D_{0+}^{\gamma _0,X} v \right\| _{L^2(0,T;X)} = \left( \sum _{k=0}^\infty \left\| D_{0+}^{\gamma _0,\mathbb R} (v,e_k)_X \right\| _{L^2(0,T)}^2 \right) ^\frac{1}{2} \sim \left\| v \right\| _{H^{\gamma _0}(0,T;X)}. \end{aligned}$$

Remark 4.2

Since $Q_M^\mathbb R$ is well-defined by Lemma 5.4, $Q_M^X$ is evidently also well-defined and

$$\begin{aligned} Q_M^X v = \sum _{k=0}^\infty Q_M^\mathbb R (v,e_k)_X e_k, \quad \forall v \in H^{1+\gamma _0}(0,T;X). \end{aligned}$$

Furthermore, we can redefine $Q_M^X$ equivalently as follows: for each $v \in H^{1+\gamma _0}(0,T;X)$, the interpolant $Q_M^X v$ fulfills

$$\begin{aligned} \left\{ \begin{aligned}&\left( Q_M^X v \right) (0) = v(0), \\&\int _0^T \left( D_{0+}^{\gamma _0} \left( v-Q_M^Xv \right) ', D_{T-}^{\gamma _0} q \right) _X \, \mathrm {d}t = 0, \quad \forall q \in P_{M-1}[0,T;X]. \end{aligned} \right. \end{aligned}$$

Finally, we are ready to state the main results of this paper as follows.

Theorem 4.1

Problem (2) has a unique solution U. Moreover,

$$\begin{aligned} \begin{aligned} {}&\left\| U \right\| _{ H^{1+\gamma _0}( 0,T; L^2(\Omega ) ) } + \left\| U(T) \right\| _{ H_0^1(\Omega ) } \\&\quad \lesssim {} \left\| u_0 \right\| _{H_0^1(\Omega )} + \left\| u_1 \right\| _{L^2(\Omega )} + \left\| f \right\| _{ L^2(\Omega _T) }. \end{aligned} \end{aligned}$$

(3)

Theorem 4.2

If $u \in H^2\left( 0,T; H_0^1(\Omega ) \cap H^2(\Omega ) \right)$, then

$$\begin{aligned}&\left\| u - U \right\| _{ H^{1+\gamma _0}( 0,T; L^2(\Omega ) ) } \lesssim \eta _1 + \eta _2 + \eta _3 + \eta _4, \end{aligned}$$

(4)

$$\begin{aligned}&\left\| (u-U)(T) \right\| _{ H_0^1(\Omega ) } \lesssim \eta _1 + \eta _2 + \eta _3 + \eta _5, \end{aligned}$$

(5)

where

$$\begin{aligned}&\eta _1 := \left\| u_1 - u_{h,1} \right\| _{L^2(\Omega )}, \\&\eta _2 := CM^{-1-2\gamma _0} \left\| (I-Q_M)\Delta u \right\| _{ H^{1+\gamma _0}(0,T;L^2(\Omega )) }, \\&\eta _3 := \left\| (I-R_h)u \right\| _{ H^{1+\gamma _0}( 0, T; L^2(\Omega ) ) }, \\&\eta _4 := \left\| (I-Q_MR_h)u \right\| _{H^{1+\gamma _0}(0,T;L^2(\Omega ))}, \\&\eta _5 := \left\| (u - Q_MR_hu)(T) \right\| _{ H_0^1(\Omega ) }. \end{aligned}$$

Corollary 4.1

If

$$\begin{aligned}&u \in H^2(0,T;H_0^1(\Omega ) \cap H^2(\Omega )) \cap H^{1+\gamma _0}( 0,T;H^{m+1}(\Omega )), \\&u'' \in B^r(0,T;H_0^1(\Omega ) \cap H^2(\Omega )), \end{aligned}$$

then

$$\begin{aligned}&\left\| u - U \right\| _{ H^{1+\gamma _0} ( 0,T; L^2(\Omega ) ) } \lesssim \xi _1 + \xi _2 + \xi _3 + \xi _4, \end{aligned}$$

(6)

$$\begin{aligned}&\left\| (u- U)(T) \right\| _{ H_0^1(\Omega ) } \lesssim \xi _1 + \xi _2 + \xi _3 + \xi _5, \end{aligned}$$

(7)

where $r \in \mathbb N$ and

$$\begin{aligned} \xi _1&:= h^{m+1} \left\| u_1 \right\| _{ H^{m+1}(\Omega ) }, \\ \xi _2&:= CM^{-\gamma _0-2-r} \left\| u'' \right\| _{ B^r( 0,T; H^2(\Omega ) ) }, \\ \xi _3&:= h^{m+1} \left\| u \right\| _{H^{1+\gamma _0}(0,T; H^{m+1}(\Omega ))}, \\ \xi _4&:= CM^{ \gamma _0-1-r } \left\| u'' \right\| _{ B^r( 0,T; L^2(\Omega ) ) } + h^{m+1} \left\| u \right\| _{ H^{1+\gamma _0}( 0,T; H^{m+1}(\Omega ) ) }, \\ \xi _5&:= CM^{-1.5-r} \left\| u'' \right\| _{ B^r( 0,T; H_0^1(\Omega ) ) } + h^m \left\| u(T) \right\| _{ H^{m+1}(\Omega ) }. \end{aligned}$$

5 Proofs

5.1 Preliminaries

Lemma 5.1

If $v \in H_0^1(\Omega ) \cap H^{m+1}(\Omega )$, then

$$\begin{aligned} \left\| (I-R_h)v \right\| _{L^2(\Omega )} + h\left\| (I-R_h)v \right\| _{ H_0^1(\Omega ) } \lesssim h^{m+1} \left\| v \right\| _{ H^{m+1}(\Omega ) }. \end{aligned}$$

Lemma 5.2

If $v \in H^\alpha (0,T)$ with $\alpha > \gamma _0$, then

$$\begin{aligned} \inf _{q \in P_{M-1}[0,T]} \left\| v-q \right\| _{H^{\gamma _0}(0,T)} \leqslant C M^{\gamma _0-\alpha } \left\| v \right\| _{H^\alpha (0,T)}. \end{aligned}$$

If $v \in H^2(0,T)$ such that $v'' \in B^j(0,T)$ with $j \in \mathbb N$, then

$$\begin{aligned} \inf _{ q \in P_{M-1}[0,T] } \left\| v-q \right\| _{ H^{1+\gamma _0}(0,T) } \leqslant C M^{ \gamma _0 - 1 - j } \left\| v'' \right\| _{B^j(0,T)}. \end{aligned}$$

Lemma 5.3

The following properties hold:

• If $0< \alpha , \beta < \infty$, then

  $$\begin{aligned} I_{0+}^\alpha I_{0+}^\beta = I_{0+}^{\alpha +\beta }, \quad I_{T-}^\alpha I_{T-}^\beta = I_{T-}^{\alpha +\beta }. \end{aligned}$$

• If $0< \alpha< \beta < \infty$, then

  $$\begin{aligned} D_{0+}^\beta I_{0+}^\alpha = D_{0+}^{\beta -\alpha }, \quad D_{T-}^\beta I_{T-}^\alpha = D_{T-}^{\beta -\alpha }. \end{aligned}$$

• If $0< \alpha < \infty$, then

  $$\begin{aligned} \left\| I_{0+}^\alpha v \right\| _{L^2(0,T)} \leqslant C \left\| v \right\| _{L^2(0,T)}, \quad \left\| I_{T-}^\alpha v \right\| _{L^2(0,T)} \leqslant C \left\| v \right\| _{L^2(0,T)}, \end{aligned}$$

  where C is a positive constant that only depends on $\alpha$ and T.

• If $0< \alpha < \infty$ and $u,v \in L^2(0,T)$, then

  $$\begin{aligned} ( I_{0+}^\alpha u, v )_{L^2(0,T)} = ( u, I_{T-}^\alpha v )_{L^2(0,T)}. \end{aligned}$$

Lemma 5.4

We have the following properties.

• If $v \in H^{\alpha }(0,T)$ with $0<\alpha <1/2$, then

  $$\begin{aligned} \left\| v \right\| _{H^{\alpha }(0,T)} \sim \left\| {D_{0+}^{\alpha }}v \right\| _{L^2(0,T)} \sim \left\| {D_{T-}^{\alpha }} v \right\| _{L^2(0,T)} \sim \sqrt{ \left( {D_{0+}^{\alpha }}v, {D_{T-}^{\alpha }} v \right) _{L^2(0,T)} }. \end{aligned}$$

• If $v,w \in H^{\alpha }(0,T)$ with $0<\alpha <1/2$, then

  $$\begin{aligned} \left( {D_{0+}^{\alpha }}v, {D_{T-}^{\alpha }} w \right) _{L^2(0,T)}\lesssim \left\| v \right\| _{H^{\alpha }(0,T)} \left\| w \right\| _{H^{\alpha }(0,T)}. \end{aligned}$$

Above, the implicit constants are only depend on $\alpha$ and T.

Lemma 5.5

If $v \in H^2(0,T)$ and $w \in H^1(0,T)$, then

$$\begin{aligned} \left( {D_{0+}^\gamma } (v - v(0) - tv'(0), w \right) _{L^2(0,T)} = \left( {D_{0+}^{\gamma _0}} (v' - v'(0)), {D_{T-}^{\gamma _0}} w \right) _{L^2(0,T)}. \end{aligned}$$

Lemma 5.6

Let X and Y be two separable Hilbert spaces, and let $A: X \rightarrow Y$ be a bounded linear operator. If $v \in H^{1+\gamma _0}(0,T; X)$, then

$$\begin{aligned} A Q_M^X v = Q_M^Y Av. \end{aligned}$$

Lemma 5.1 is standard (see [4]), and Lemma 5.3 follows from [16, Theorems 3.35–3.37] and the basic properties of the interpolation spaces. The proof of Lemma 5.3 is included in [13, 15], and for convenience this lemma will be used implicitly in the forthcoming analysis. Lemma 5.4 is a direct consequence of [6, Lemma 2.4, Theorem 2.13 and Corollary 2.15], and Lemma 5.5 follows from [10, Lemma 2.6]. Finally, by Lemma 5.4 and the standard properties of the interpolation spaces and the Bochner integrals, a rigorous proof of Lemma 5.6 is tedious but straightforward, and so it is omitted here.

Lemma 5.7

If $v \in L^2(0,T)$, then

$$\begin{aligned} \left\| I_{T-}^{2\gamma _0} v \right\| _{H^{2\gamma _0}(0,T)} \lesssim \left\| v \right\| _{L^2(0,T)}. \end{aligned}$$

(8)

Moreover, if $v \in H^{\gamma _0}(0,T)$, then

$$\begin{aligned} \left\| I_{T-}^{2\gamma _0} v \right\| _{H^{3\gamma _0}(0,T)} \lesssim \left\| v \right\| _{H^{\gamma _0}(0,T)}. \end{aligned}$$

(9)

Lemma 5.8

If $v \in H^2(0,T)$ and $w \in H^{\gamma _0}(0,T)$, then

$$\begin{aligned} \big ( (I-Q_M)v, w \big )_{L^2(0,T)} \lesssim C M^{-1-2\gamma _0} \left\| (I-Q_M)v \right\| _{H^{1+\gamma _0}(0,T)} \left\| w \right\| _{H^{\gamma _0}(0,T)}. \end{aligned}$$

(10)

Lemma 5.9

If $v \in H^2(0,T)$ and $v'' \in B^j(0,T)$ with $j \in \mathbb N$, then

$$\begin{aligned} \left\| (I-Q_M)v \right\| _{H^{1+\gamma _0}(0,T)}&\lesssim C M^{\gamma _0-1-j} \left\| v'' \right\| _{B^j(0,T)}, \end{aligned}$$

(11)

$$\begin{aligned} \left\| (I-Q_M)v \right\| _{L^2(0,T)}&\lesssim C M^{-2-j} \left\| v'' \right\| _{B^j(0,T)}, \end{aligned}$$

(12)

$$\begin{aligned} \left\| (I-Q_M)v \right\| _{C[0,T]}&\lesssim C M^{-1.5-j} \left\| v'' \right\| _{B^j(0,T)}. \end{aligned}$$

(13)

Proof of Lemma 5.7

Define

$$\begin{aligned} w(t) := \frac{1}{\Gamma (\gamma _0)} \int _t^\infty (s-t)^{\gamma _0-1} v(s) \, \mathrm {d}s, \quad -\infty< t < \infty , \end{aligned}$$

where v is extended to $\mathbb R \backslash (0,T)$ by zero. Since $0< \gamma _0 < 0.5$, a routine calculation yields $w \in L^2(\mathbb R)$, and then [15, Theorem 7.1] implies

$$\begin{aligned} \mathcal Fw(\xi ) = (-\mathrm {i}\xi )^{-\gamma _0} \mathcal Fv(\xi ), \quad -\infty< \xi < \infty , \end{aligned}$$

where $\mathcal F: L^2(\mathbb R) \rightarrow L^2(\mathbb R)$ is the Fourier transform operator, and $\mathrm {i}$ is the imaginary unit. Therefore, the well-known Plancherel Theorem yields

$$\begin{aligned} \left\| w \right\| _{H^{\gamma _0}(\mathbb R)} \lesssim \left\| v \right\| _{L^2(0,T)}, \end{aligned}$$

and hence

$$\begin{aligned} \left\| I_{T-}^{\gamma _0} v \right\| _{H^{\gamma _0}(0,T)} \lesssim \left\| v \right\| _{L^2(0,T)}. \end{aligned}$$

Furthermore, if $v \in H_0^1(0,T)$ then

$$\begin{aligned} \left\| I_{T-}^{\gamma _0}v \right\| _{ H^{1+\gamma _0}(0, T) } \lesssim \left\| v \right\| _{ H_0^1(0,T)}, \end{aligned}$$

by the evident equality $( I_{T-}^{\gamma _0} v )' = I_{T-}^{\gamma _0} v'$. Consequently, since $H_0^{\gamma _0}(0,T)$ coincides with $H^{\gamma _0}(0,T)$ with equivalent norms, applying [18, Lemma 22.3] gives

$$\begin{aligned} \left\| I_{T-}^{2\gamma _0} v \right\| _{ H^{2\gamma _0}(0,T) } = \left\| I_{T-}^{\gamma _0} I_{T-}^{\gamma _0} v \right\| _{ H^{2\gamma _0}(0,T) } \lesssim \left\| I_{T-}^{\gamma _0} v \right\| _{ H_0^{\gamma _0}(0,T) } \lesssim \left\| v \right\| _{L^2(0,T)}, \end{aligned}$$

namely estimate (8). Analogously, we can obtain (9) and hence conclude the proof of the lemma. $\square$

Proof of Lemma 5.8

Let $g:= (I-Q_M)v$. Since a straightforward calculation yields

$$\begin{aligned} \left( I_{0+}^{1-\gamma _0}g' \right) (t) = \frac{g'(0)}{ \Gamma (2-\gamma _0) } t^{1-\gamma _0} + \left( I_{0+}^{2-\gamma _0}g'' \right) (t), \quad 0< t < T, \end{aligned}$$

the fact $\gamma _0 < 0.5$ indicates that $I_{0+}^{1-\gamma _0}g' \in H^1(0,T)$ and $(I_{0+}^{1-\gamma _0}g')(0) = 0$. Then using integration by parts gives

$$\begin{aligned} {}&\left( D_{0+}^{\gamma _0} g', I_{T-}^{1+\gamma _0} w \right) _{L^2(0,T)} = \left( \left( I_{0+}^{1-\gamma _0} g' \right) ', I_{T-}^{1+\gamma _0} w \right) _{L^2(0,T)} \\&\quad = -\left( I_{0+}^{1-\gamma _0} g', \left( I_{T-}^{1+\gamma _0} w \right) ' \right) _{L^2(0,T)} = \left( I_{0+}^{1-\gamma _0} g', I_{T-}^{\gamma _0} w \right) _{L^2(0,T)} \\&\quad = \left( g', I_{T-} w \right) _{L^2(0,T)}. \end{aligned}$$

Hence, as the definition of $Q_M$ implies $g(0) = 0$, we obtain

$$\begin{aligned} \left( D_{0+}^{\gamma _0} g', I_{T-}^{1+\gamma _0} w \right) _{L^2(0,T)} = \left( g', I_{T-}w \right) _{L^2(0,T)} = (g, w)_{L^2(0,T)}, \end{aligned}$$

which, combined with the evident equality

$$\begin{aligned} I_{T-}^{1+\gamma _0} w = D_{T-}^{\gamma _0} I_{T-}^{1+2\gamma _0}w, \end{aligned}$$

yields

$$\begin{aligned} \big ( g,w \big )_{L^2(0,T)} = \left( D_{0+}^{\gamma _0} g', D_{T-}^{\gamma _0} I_{T-}^{1+2\gamma _0} w \right) _{L^2(0,T)}. \end{aligned}$$

Therefore, Lemma 5.4, the definition of $Q_M$ and the Cauchy–Schwarz inequality imply

$$\begin{aligned} \big ( g, w \big )_{L^2(0,T)} \lesssim \left\| g \right\| _{H^{1+\gamma _0}(0,T)} \inf _{q \in P_{M-1}[0,T]} \left\| I_{T-}^{1+2\gamma _0}w - q \right\| _{H^{\gamma _0}(0,T)}. \end{aligned}$$

Clearly, to prove (10), by Lemma 5.2 it suffices to show

$$\begin{aligned} \left\| I_{T-}^{1+2\gamma _0}w \right\| _{ H^{1+3\gamma _0}(0,T) } \lesssim \left\| w \right\| _{ H^{\gamma _0}(0,T) }. \end{aligned}$$

Therefore, since

$$\begin{aligned} \left\| I_{T-}^{1+2\gamma _0} w \right\| _{H^{1+3\gamma _0}(0,T)} \lesssim \left\| I_{T-}^{2\gamma _0} w \right\| _{H^{3\gamma _0}(0,T)}, \end{aligned}$$

using Lemma 5.7 completes the proof of Lemma 5.8. $\square$

Proof of Lemma 5.9

Let us first consider (11). For each $p \in P_{M-1}[0,T]$, by Lemma 5.4, the definition of $Q_M$ and the Cauchy–Schwarz inequality, we obtain

$$\begin{aligned} {}&\left\| (Q_Mv)'-p \right\| _{ H^{\gamma _0}(0,T) }^2 \\&\quad \sim \Big ( D_{0+}^{\gamma _0} \big ( (Q_Mv)'-p \big ), D_{T-}^{\gamma _0} \big ( (Q_Mv)'-p \big ) \Big )_{L^2(0,T)} \\&\quad = \Big ( D_{0+}^{\gamma _0} (v'-p), D_{T-}^{\gamma _0} \big ( (Q_Mv)'-p \big ) \Big )_{L^2(0,T)} \\&\quad \lesssim \left\| v'-p \right\| _{H^{\gamma _0}(0,T)} \left\| (Q_Mv)'-p \right\| _{H^{\gamma _0}(0,T)}. \end{aligned}$$

It follows that

$$\begin{aligned} \left\| (Q_Mv)'-p \right\| _{ H^{\gamma _0}(0,T) } \lesssim \left\| v'-p \right\| _{ H^{\gamma _0}(0,T) }, \end{aligned}$$

and so

$$\begin{aligned} \left\| (v-Q_Mv)' \right\| _{ H^{\gamma _0}(0,T) } \lesssim \left\| v'-p \right\| _{ H^{\gamma _0}(0,T) }. \end{aligned}$$

Therefore, as the fact $(v-Q_Mv)(0) = 0$ implies

$$\begin{aligned} \left\| (I-Q_M)v \right\| _{H^{1+\gamma _0}(0,T)} \sim \left\| (v-Q_Mv)' \right\| _{H^{\gamma _0}(0,T)}, \end{aligned}$$

using Lemma 5.2 proves (11).

Next let us consider (12, 13). Proceeding as in the proof of Lemma 5.8 gives

$$\begin{aligned} {}&\left\| (I-Q_M)v \right\| _{L^2(0,T)}^2 \\&\quad \lesssim \left\| (I-Q_M)v \right\| _{H^{1+\gamma _0}(0,T)} \inf _{ q \in P_{M-1}[0,T] } \left\| I_{T-}^{1+2\gamma _0} (I-Q_M)v-q \right\| _{ H^{\gamma _0}(0,T) } \\&\quad \lesssim C M^{-1-\gamma _0} \left\| (I-Q_M)v \right\| _{H^{1+\gamma _0}(0,T)} \left\| (I-Q_M)v \right\| _{L^2(0,T)}, \end{aligned}$$

which proves (12) by (11). Then, combining (11,12) and applying [18, Lemma 22.3] yield

$$\begin{aligned} \left\| (I-Q_M)v \right\| _{H^1(0,T)} \lesssim C M^{-1-j} \left\| v'' \right\| _{B^j(0,T)}, \end{aligned}$$

so that (13) follows from (12) and the Gagliardo–Nirenberg interpolation inequality, namely,

$$\begin{aligned} \left\| w \right\| _{C[0,T]} \lesssim \left\| w \right\| _{L^2(0,T)}^\frac{1}{2} \left\| w \right\| _{H^1(0,T)}^\frac{1}{2}, \quad \forall w \in H^1(0,T). \end{aligned}$$

This concludes the proof of Lemma 5.9. $\square$

Remark 5.1

Assume that $P_M[0,T]$ and $P_{M-1}[0,T]$ are respectively replaced by

$$\begin{aligned} P_M[0,T] + \left\{ cw^{1+2\gamma _0}:\ c \in \mathbb R \right\} \quad \text {and}\quad P_{M-1}[0,T] + \left\{ c w^{2\gamma _0}:\ c \in \mathbb R \right\} , \end{aligned}$$

where $w(t) := T-t,\ 0< t < T$. For each $v \in H^{1+\gamma _0}(0,T)$, the definition of $Q_M$ implies

$$\begin{aligned} \int _0^T D_{0+}^{\gamma _0}(v-Q_Mv)' D_{T-}^{\gamma _0} w^{2\gamma _0} \, \mathrm {d}t = 0, \end{aligned}$$

and then, as in the previous remark, a straightforward computing yields

$$\begin{aligned} (v-Q_Mv)(T) = 0. \end{aligned}$$

Correspondingly, we can improve Corollary 4.1 by

$$\begin{aligned} \xi _5 := h^m \left\| u(T) \right\| _{H^{m+1}(\Omega )}. \end{aligned}$$

5.2 Proofs of Theorems 4.1 and 4.2 and Corollary 4.1

Proof of Theorem 4.1

Since (3) contains the unique existence of U, it suffices to prove the former. Observe first that integration by parts yields

$$\begin{aligned} 2 ( \nabla U, \nabla U' )_{L^2(\Omega _T)} = \left\| U(T) \right\| _{H_0^1(\Omega )}^2 - \left\| U(0) \right\| _{H_0^1(\Omega )}^2 \end{aligned}$$

and that Lemma 5.4 implies

$$\begin{aligned} \left\| D_{0+}^{\gamma _0} u_{h,1} \right\| _{ L^2(\Omega _T) } \sim \left\| u_{h,1} \right\| _{ H^{\gamma _0}(0,T;L^2(\Omega ) } \sim \left\| u_{h,1} \right\| _{L^2(\Omega )}, \\ \left( D_{0+}^{\gamma _0} U', D_{T-}^{\gamma _0} U' \right) _{ L^2(\Omega _T) } \sim \left\| U' \right\| _{ H^{\gamma _0}( 0,T; L^2(\Omega ) ) }^2 \sim \left\| D_{T-}^{\gamma _0} U' \right\| _{ L^2(\Omega _T) }^2. \end{aligned}$$

Moreover, the fact that $u_{h,1}$ is the $L^2(\Omega )$-projection of $u_1$ onto $V_h$ gives

$$\begin{aligned} \left\| u_{h,1} \right\| _{L^2(\Omega )} \leqslant \left\| u_1 \right\| _{L^2(\Omega )}. \end{aligned}$$

Consequently, by the Cauchy–Schwarz inequality and the Young’s inequality with $\epsilon$, inserting $V := U'$ into (2) yields

$$\begin{aligned} {}&\left\| U' \right\| _{ H^{\gamma _0}( 0,T; L^2(\Omega ) ) } + \left\| U(T) \right\| _{ H_0^1(\Omega ) } \\&\quad \lesssim \left\| U(0) \right\| _{ H_0^1(\Omega ) } + \left\| u_1 \right\| _{L^2(\Omega )} + \left\| f \right\| _{ L^2(\Omega _T) }, \end{aligned}$$

which, combined with the estimate

$$\begin{aligned} \left\| U \right\| _{ H^{1+\gamma _0}( 0,T; L^2(\Omega ) ) } \sim \left\| U(0) \right\| _{L^2(\Omega )} + \left\| U' \right\| _{ H^{\gamma _0}( 0,T; L^2(\Omega ) ) }, \end{aligned}$$

indicates

$$\begin{aligned} {}&\left\| U \right\| _{ H^{1+\gamma _0}( 0,T; L^2(\Omega ) ) } + \left\| U(T) \right\| _{ H_0^1(\Omega ) } \\&\quad \lesssim \left\| U(0) \right\| _{ H_0^1(\Omega ) } + \left\| u_1 \right\| _{L^2(\Omega )} + \left\| f \right\| _{ L^2(\Omega _T) }. \end{aligned}$$

As the definition of $R_h$ and the fact $U(0) = R_hu_0$ imply

$$\begin{aligned} \left\| U(0) \right\| _{H_0^1(\Omega )} \leqslant \left\| u_0 \right\| _{H_0^1(\Omega )}, \end{aligned}$$

this proves (3) and thus concludes the proof of Theorem 4.1. $\square$

Proof of Theorem 4.2

Set $\rho := (I-Q_MR_h)u$ and $\theta := U - Q_MR_hu$. By Lemma 5.5 and integration by parts, using (1) gives

$$\begin{aligned} \left( D_{0+}^{\gamma _0} (u' - u_1), D_{T-}^{\gamma _0} \theta ' \right) _{L^2(\Omega _T)} + (\nabla u, \theta ')_{L^2(\Omega _T)} = (f, \theta ')_{L^2(\Omega _T)}, \end{aligned}$$

which, together with (2), yields

$$\begin{aligned} \left( D_{0+}^{\gamma _0} \theta ', D_{T-}^{\gamma _0} \theta ' \right) _{L^2(\Omega _T)} + ( \nabla \theta ,\nabla \theta ' )_{L^2(\Omega _T)} = \mathbb I_1 + \mathbb I_2 + \mathbb I_3, \end{aligned}$$

where

$$\begin{aligned} \mathbb I_1&:= ( \nabla \rho ,\nabla \theta ' )_{L^2(\Omega _T)}, \\ \mathbb I_2&:= \left( D_{0+}^{\gamma _0} \rho ', D_{T-}^{\gamma _0} \theta ' \right) _{ L^2(\Omega _T) }, \\ \mathbb I_3&:= -\left( D_{0+}^{\gamma _0} ( u_1 - u_{h,1} ), D_{T-}^{\gamma _0} \theta ' \right) _{ L^2(\Omega _T) }. \end{aligned}$$

Moreover, the fact $\theta (0) = 0$ gives

$$\begin{aligned} ( \nabla \theta , \nabla \theta ' )_{ L^2(\Omega _T) } = \frac{1}{2} \left\| \theta (T) \right\| _{ H_0^1(\Omega ) }^2 \end{aligned}$$

by integration by parts, and Lemma 5.4 implies

$$\begin{aligned} \left( D_{0+}^{\gamma _0} \theta ', D_{T-}^{\gamma _0} \theta ' \right) _{L^2(\Omega _T)} \sim \left\| \theta ' \right\| _{ H^{\gamma _0}( 0,T; L^2(\Omega ) ) }^2. \end{aligned}$$

Therefore, it follows

$$\begin{aligned} \left\| \theta ' \right\| _{ H^{\gamma _0}( 0,T; L^2(\Omega ) ) }^2 + \left\| \theta (T) \right\| _{H_0^1(\Omega )}^2 \lesssim \mathbb I_1 + \mathbb I_2 + \mathbb I_3. \end{aligned}$$

(14)

Let us first estimate $\mathbb I_1$. Since $R_h: H_0^1(\Omega ) \rightarrow \mathring{V}_h$ and $-\Delta : H^2(\Omega ) \rightarrow L^2(\Omega )$ are two bounded linear operators, Lemma 5.6 implies

$$\begin{aligned} Q_M R_h u = R_h Q_M u \quad \text { and } \quad Q_M(-\Delta u) = -\Delta Q_M u, \end{aligned}$$

so that, by integration by parts and the definition of $R_h$, a straightforward calculation gives

$$\begin{aligned} \mathbb I_1&= \int _0^T \big ( \nabla (I-R_hQ_M) u , \nabla \theta ' \big )_{L^2(\Omega )} \, \mathrm {d}t \\&= \int _0^T \big ( \nabla (I-Q_M) u , \nabla \theta ' \big )_{L^2(\Omega )} \, \mathrm {d}t \\&= \int _0^T \big ( -\Delta (I-Q_M) u, \theta ' \big )_{L^2(\Omega )} \\&= \int _0^T \big ( (I-Q_M)(-\Delta u) , \theta ' \big )_{L^2(\Omega )} \, \mathrm {d}t, \end{aligned}$$

Therefore, Lemma 5.8 leads to

$$\begin{aligned} \mathbb I_1 \lesssim C M^{-1-2\gamma _0} \left\| (I-Q_M)\Delta u \right\| _{ H^{1+\gamma _0}(0,T;L^2(\Omega )) } \left\| \theta ' \right\| _{ H^{\gamma _0}( 0,T;L^2(\Omega ) ) }. \end{aligned}$$

(15)

Next let us estimate $\mathbb I_2$ and $\mathbb I_3$. The definition of $Q_M$ gives

$$\begin{aligned} \mathbb I_2 = \left( D_{0+}^{\gamma _0} (u - Q_MR_hu)', D_{T-}^{\gamma _0} \theta ' \right) _{L^2(\Omega _T)} = \left( D_{0+}^{\gamma _0} ( u-R_hu )', D_{T-}^{\gamma _0} \theta ' \right) _{L^2(\Omega _T)}, \end{aligned}$$

so that the Cauchy–Schwarz inequality and Lemma 5.4 indicate

$$\begin{aligned} \mathbb I_2 \lesssim \left\| (I-R_h)u \right\| _{H^{1+\gamma _0}(0,T;L^2(\Omega ))} \left\| \theta ' \right\| _{H^{\gamma _0}(0,T;L^2(\Omega ))}. \end{aligned}$$

(16)

By the evident estimate

$$\begin{aligned} \left\| u_1 - u_{h,1} \right\| _{ H^{\gamma _0}(0,T;\Omega _T) } \sim \left\| u_1 - u_{h,1} \right\| _{L^2(\Omega )}, \end{aligned}$$

the Cauchy–Schwarz inequality and Lemma 5.4 also yield

$$\begin{aligned} \mathbb I_3 \lesssim \left\| u_1 - u_{h,1} \right\| _{L^2(\Omega )} \left\| \theta ' \right\| _{ H^{\gamma _0}( 0,T; L^2(\Omega ) ) }. \end{aligned}$$

(17)

Finally, by the Young’s inequality with $\epsilon$, combining (14), (15), (16), (17) gives

$$\begin{aligned} \left\| \theta ' \right\| _{ H^{\gamma _0} ( 0,T; L^2(\Omega ) ) } + \left\| \theta (T) \right\| _{ H_0^1(\Omega ) } \lesssim \eta _1 + \eta _2 + \eta _3. \end{aligned}$$

Since $\theta (0) = 0$ implies

$$\begin{aligned} \left\| \theta \right\| _{ H^{1+\gamma _0}( 0,T; L^2(\Omega ) ) } \sim \left\| \theta ' \right\| _{ H^{\gamma _0} 0,T; L^2(\Omega ) ) }, \end{aligned}$$

it follows

$$\begin{aligned} \left\| \theta \right\| _{ H^{1+\gamma _0} ( 0,T; L^2(\Omega ) ) } + \left\| \theta (T) \right\| _{ H_0^1(\Omega ) } \lesssim \eta _1 + \eta _2 + \eta _3. \end{aligned}$$

As (4), (5) are evident from the above estimate, this concludes the proof of Theorem 4.2.$\square$

Proof of Corollary 4.1

It suffices to prove $\eta _i \lesssim \xi _i$ for all $1 \leqslant i \leqslant 5$, where $\{ \eta _i \}_{i=1}^5$ are defined in Theorem 4.2. Observing that $\eta _1 \lesssim \xi _1$ is a standard result [4], that $\eta _2 \lesssim \xi _2$ follows from Lemma 5.9, and that $\eta _3 \lesssim \xi _3$ follows from Lemma 5.1, we only need to prove $\eta _4 \lesssim \xi _4$ and $\eta _5 \lesssim \xi _5$.

Let us first consider $\eta _4 \lesssim \xi _4$. By Lemma 5.4, the definition of $Q_M$ implies

$$\begin{aligned} \left\| Q_M(I-R_h)u \right\| _{ H^{1+\gamma _0}( 0,T; L^2(\Omega ) ) } \lesssim \left\| (I-R_h)u \right\| _{ H^{1+\gamma _0}( 0,T; L^2(\Omega ) ) }, \end{aligned}$$

so that Lemma 5.1 and [18, Lemma 22.3] yield

$$\begin{aligned} \left\| Q_M(I-R_h)u \right\| _{ H^{1+\gamma _0}( 0,T; L^2(\Omega ) ) } \lesssim h^{m+1} \left\| u \right\| _{ H^{1+\gamma _0}( 0,T; H^{m+1}(\Omega ) ) }. \end{aligned}$$

Moreover, Lemma 5.9 gives

$$\begin{aligned} \left\| (I - Q_M)u \right\| _{ H^{1+\gamma _0}(0,T; L^2(\Omega )) } \lesssim C M^{\gamma _0 -1 - r } \left\| u'' \right\| _{ B^r(0,T; L^2(\Omega )) }. \end{aligned}$$

Consequently, $\eta _4 \lesssim \xi _4$ is a direct consequence of the inequality

$$\begin{aligned}&\left\| (I-Q_MR_h)u \right\| _{ H^{1+\gamma _0}( 0,T; L^2(\Omega ) ) } \\&\quad \leqslant \left\| (I-Q_M)u \right\| _{ H^{1+\gamma _0}( 0,T; L^2(\Omega ) ) } + \left\| Q_M(I-R_h)u \right\| _{ H^{1+\gamma _0}( 0,T; L^2(\Omega ) ) }. \end{aligned}$$

Then let us consider $\eta _5 \lesssim \xi _5$. Since Lemma 5.6 gives $R_hQ_M u = Q_M R_h u$, the definition of $R_h$ yields

$$\begin{aligned} \left\| (R_hu - Q_MR_hu)(T) \right\| _{ H_0^1(\Omega ) } \leqslant \left\| (u-Q_Mu)(T) \right\| _{ H_0^1(\Omega ) }, \end{aligned}$$

and hence Lemma 5.9 indicates

$$\begin{aligned} \left\| (R_hu - Q_MR_hu)(T) \right\| _{H_0^1(\Omega )} \lesssim C M^{-1.5-r} \left\| u'' \right\| _{B^r(0,T;H_0^1(\Omega ))}. \end{aligned}$$

Therefore, as Lemma 5.1 implies

$$\begin{aligned} \left\| (I-R_h)u(T) \right\| _{H_0^1(\Omega )} \lesssim h^m \left\| u(T) \right\| _{H^{m+1}(\Omega )}, \end{aligned}$$

the estimate $\eta _5 \lesssim \xi _5$ follows from the inequality

$$\begin{aligned}&\left\| (u - Q_MR_hu)(T) \right\| _{H_0^1(\Omega )} \\&\quad \leqslant \left\| (I - R_h) u(T) \right\| _{H_0^1(\Omega )} + \left\| (R_h u - Q_MR_hu)(T) \right\| _{H_0^1(\Omega )}. \end{aligned}$$

This concludes the proof of Corollary 4.1. $\square$

6 Numerical Experiments

This section performs some numerical experiments to demonstrate the high order accuracy of the proposed algorithm in two dimensional case. Throughout this section we set $\gamma := 1.5$, $T := 1$ and $\Omega := (0,1)^2$.

Example 1

In this example the solution to problem (1) is

$$\begin{aligned} u(x,t) := t^{20} x_1x_2(1-x_1)(1-x_2), \quad (x,t) \in \Omega _T, \end{aligned}$$

where $x = (x_1,x_2)$. Let us first consider the spatial discretization errors of the proposed algorithm, and, to this end, we set $M := 20$ to ensure that the temporal discretization errors are negligible compared with the former. The corresponding numerical results, presented in Table 1, illustrate that the convergence orders of

$$\begin{aligned} \left\| (u-U)(T) \right\| _{H_0^1(\Omega )} \quad \text { and } \quad \left\| u-U \right\| _{H^{1+\gamma _0}(0,T;L^2(\Omega ))} \end{aligned}$$

are m and $m + 1$ respectively, which agrees well with Corollary 4.1. Then let us consider the temporal discretization errors and hence set $m := 4$ and $h := 1/32$ to ensure that the temporal discretization error is dominant. We plot the log-linear relationship between the errors and the polynomial degree M in Fig. 1. As indicated by Corollary 4.1, these numerical results demonstrate that the errors reduce exponentially as M increases.

Table 1 The errors for Example 1 with $M = 20$

Fig. 1

The log-linear relationship between the errors and the polynomial degree M for Example 1 with $m=4$ and $h=1/32$

Example 2

This example adopts

$$\begin{aligned} u(x,t) := t^2 \left| 1-2t \right| ^\beta x_1 (1-x_1) \sin (\pi x_2), \quad (x,t) \in \Omega _T \end{aligned}$$

as the solution to problem (1), where $\beta$ is a positive constant. Here we only consider the temporal discretization errors and hence set $m := 6$ and $h := 2^{-4}$ to ensure that the temporal discretization errors are dominant. The corresponding numerical results are presented in Tables 2 and 3. Observing that

$$\begin{aligned} \left| 1-2t \right| ^{\beta } \in H^{\beta +0.5-\epsilon }(0,T) \quad \text {for all }\epsilon > 0 , \end{aligned}$$

by Corollary 4.1 and [18, Lemma 22.3] we have

$$\begin{aligned} \begin{aligned} \left\| (u-U)(T) \right\| _{ H_0^1(\Omega ) }&\lesssim C(\epsilon )M^{-\beta +\epsilon } ,\\ \left\| u-U \right\| _{ H^{1+\gamma _0}( 0,T;L^2(\Omega ) ) }&\lesssim C(\epsilon )M^{0.75-\beta +\epsilon }, \end{aligned} \end{aligned}$$

where $C(\epsilon )$ is a constant that depends on $\epsilon$. Evidently, for the convergence order of $\left\| u-U \right\| _{ H^{1+\gamma _0}( 0,T;L^2(\Omega ) ) }$, the numerical results are in agreement with Corollary 4.1. However, in this case, $\left\| (u-U)(T) \right\| _{H_0^1(\Omega )}$ reduces significantly faster than that predicted by Corollary 4.1.

Table 2 The errors for Example 2 with $\beta =2.5$

Table 3 The errors for Example 2 with $\beta =2.1$

Example 3

This example investigates the temporal accuracy of the algorithm in the case that the underlying solution has singularity at $t = 0$. The solution to problem (1) is

$$\begin{aligned} u(x,t) = t^\beta x_1 x_2(1-x_1)(1-x_2), \quad (x,t)\in \Omega _T, \end{aligned}$$

where $\beta = 1.2$, 1.5 or 1.8. We set $m := 4$ and $h := 2^{-5}$, and display the corresponding numerical results in Tables 4, 5 and 6. These numerical results illustrate that both $\left\| (u-U)(T) \right\| _{H_0^1(\Omega )}$ and $\left\| u-U \right\| _{H^{1+\gamma _0}(0,T;L^2(\Omega ))}$ converge significantly faster than that implied by Corollary 4.1.

Table 4 The errors for Example 3 with $\beta = 1.2$

Table 5 The errors for Example 3 with $\beta = 1.5$

Table 6 The errors for Example 3 with $\beta = 1.8$

7 Conclusions

In this paper, a high accuracy algorithm for time fractional wave problems is developed, which adopts a spectral method to approximate the fractional derivative and uses a finite element method in the spatial discretization. Stability and a priori error estimates of this algorithm are derived, and numerical experiments are also performed to verify its high accuracy.

In future work, we shall consider the following issues. Firstly, the optimal error estimates of $\left\| (u-U)(T) \right\| _{L^\infty (\Omega )}$ and $\left\| (u-U)(T) \right\| _{L^2(\Omega )}$ are not established. Secondly, it is worth applying the idea of approximating fractional differential operators of order $\gamma$ ($1< \gamma < 2$) by spectral methods to other fractional differential equations, such as nonlinear fractional ordinary differential equations and nonlinear time fractional wave equations.

Appendix A: Weak Solution

We call

$$\begin{aligned} u \in H^{(\gamma +1)/2}(0,T; L^2(\Omega )) \cap L^2(0,T; H_0^1(\Omega )) \end{aligned}$$

a weak solution to problem (1) if $u(0) = u_0$ and

(18)

for all $v \in H^{(\gamma -1)/2}(0,T; L^2(\Omega )) \cap L^2(0,T; H_0^1(\Omega ))$.

To prove that problem (1) admits a unique weak solution, we first consider the following problem: given $c_0$, $c_1 \in \mathbb R$ and $g \in L^2(0,T)$, seek $y \in H^\gamma (0,T)$ such that

$$\begin{aligned} D_{0+}^\gamma (y - c_0 - c_1t) + \lambda y = g, \end{aligned}$$

(19)

where $\lambda$ is a positive constant such that $\lambda \geqslant 1$.

Lemma A.1

Suppose that $v\in H^{(\gamma +1)/2}(0,T)$ and $D_{0+}^{\gamma }v\in L^2(0,T)$, then

$$\begin{aligned} \left\| v \right\| _{H^\gamma (0,T)} \lesssim \left\| D_{0+}^\gamma v \right\| _{L^2(0,T)}. \end{aligned}$$

(20)

Proof

Since $D_{0+}^\gamma v\in L^2(0,T)$, by [9, Lemmas A.4] we conclude that $I_{0+}^\gamma D_{0+}^\gamma v\in H^{\gamma }(0,T)$ with

$$\begin{aligned} \left\| I_{0+}^\gamma D_{0+}^\gamma v \right\| _{H^\gamma (0,T)} \lesssim \left\| D_{0+}^\gamma v \right\| _{L^2(0,T)}. \end{aligned}$$

(21)

A simple calculation yields

$$\begin{aligned} v = c_0t^{\gamma -2} + c_1t^{\gamma -1} +I_{0+}^\gamma D_{0+}^\gamma v, \end{aligned}$$

which indicates that $c_0=c_1=0$ by the fact $v\in H^{(\gamma +1)/2}(0,T)$. Then (20) follows from (21). This completes the proof. $\square$

Lemma A.2

Suppose that $v\in H^{(\gamma +1)/2}(0,T)$ with $v(0)=0$, then we have the following properties.

• (a) It holds that

  $$\begin{aligned} \left( D_{0+}^\frac{\gamma +1}{2} v, D_{T-}^\frac{\gamma -1}{2} v' \right) _{L^2(0,T)} \sim \left\| v \right\| _{H^{(\gamma +1)/2}(0,T)}^2. \end{aligned}$$

  (22)

• (b) For any $w\in H^{(\gamma -1)/2}(0,T)$, it holds that

  $$\begin{aligned} \left( D_{0+}^\frac{\gamma +1}{2} v, D_{T-}^\frac{\gamma -1}{2} w \right) _{L^2(0,T)}\lesssim \left\| v \right\| _{H^{(\gamma +1)/2}(0,T)} \left\| w \right\| _{H^{(\gamma -1)/2}(0,T)}. \end{aligned}$$

  (23)

• (c) For any $\varphi \in C_0^{\infty }(0,T)$, it holds that

  $$\begin{aligned} \left\langle {D_{0+}^\gamma v, \varphi } \right\rangle = \left( D_{0+}^\frac{\gamma +1}{2} v, D_{T-}^\frac{\gamma -1}{2} \varphi \right) _{L^2(0,T)}. \end{aligned}$$

  (24)

Proof

Let us first prove (a). Since $v\in H^{(\gamma +1)/2}(0,T)$ and $v(0)=0$, we have

$$\begin{aligned} \left\| v' \right\| _{H^{(\gamma -1)/2}(0,T)}\sim \left\| v \right\| _{H^{(\gamma +1)/2}(0,T)}. \end{aligned}$$

(25)

In addition, a straightforward calculation gives

$$\begin{aligned} D_{0+}^\frac{\gamma +1}{2} v = D^{2}I_{0+}^{\frac{3-\gamma }{2}}I_{0+}v' = D^{2}I_{0+}^{\frac{5-\gamma }{2}}v' = D_{0+}^{\frac{\gamma -1}{2}}v'. \end{aligned}$$

(26)

So (22) follows from (25), (26) and Lemma 5.4.

Then let us prove (b). In view of (25), (26), using Lemma 5.4 yields (23).

Finally we prove (c). Observe that (26) implies $I_{0+}^{\frac{3-\gamma }{2}}v'\in H^1(0,T)$, and a simple computation implies

$$\begin{aligned} \left( I_{0+}^{\frac{3-\gamma }{2}}v'\right) (t)\leqslant \frac{t^{\frac{1-\gamma }{2}}\left\| v' \right\| _{L^2(0,t)}}{\Gamma (\frac{3-\gamma }{2})\sqrt{2-\gamma }},\quad 0<t\leqslant T. \end{aligned}$$

Thus,

$$\begin{aligned} \left( I_{0+}^{\frac{3-\gamma }{2}}v'\right) (0) = 0. \end{aligned}$$

Using integration by parts gives

$$\begin{aligned} \begin{aligned} {}&\left\langle {D_{0+}^\gamma v, \varphi } \right\rangle = \left\langle {D^2I_{0+}^{2-\gamma } v, \varphi } \right\rangle \\&\quad = (I_{0+}^{2-\gamma } v, \varphi '')_{L^2(0,T)} = (I_{0+}^{3-\gamma } v', \varphi '')_{L^2(0,T)} \\&\quad = (I_{0+}^{\frac{3-\gamma }{2}}v' , I_{T-}^{\frac{3-\gamma }{2}}\varphi '')_{L^2(0,T)} =-(DI_{0+}^{\frac{3-\gamma }{2}}v' , I_{T-}^{\frac{3-\gamma }{2}}\varphi ')_{L^2(0,T)} \\&\quad = (D_{0+}^{\frac{\gamma -1}{2}}v' , D_{T-}^{\frac{\gamma -1}{2}}\varphi )_{L^2(0,T)} =(D_{0+}^{\frac{\gamma +1}{2}}v , D_{T-}^{\frac{\gamma -1}{2}}\varphi )_{L^2(0,T)} \end{aligned} \end{aligned}$$

for all $\varphi \in C_0^\infty (0,T)$. This shows (24) and completes the proof of this lemma. $\square$

Lemma A.3

Problem (19) has a unique solution $y \in H^\gamma (0,T)$, and y satisfies that $y(0) = c_0$ and

$$\begin{aligned} \left( D_{0+}^\frac{\gamma +1}{2} ( y - c_0 - c_1t ), \ D_{T-}^\frac{\gamma -1}{2} z \right) _{L^2(0,T)} + \lambda (y,z)_{L^2(0,T)} = (g,z)_{L^2(0,T)} \end{aligned}$$

(27)

for all $z \in H^\frac{\gamma -1}{2} (0,T)$. Moreover,

$$\begin{aligned} \left\| y \right\| _{ H^\frac{\gamma +1}{2}(0,T) } + \lambda ^\frac{1}{2} \left\| y \right\| _{L^2(0,T)} \lesssim \left\| g \right\| _{L^2(0,T)} + \lambda ^\frac{1}{2} \left| c_0 \right| + \left| c_1 \right| . \end{aligned}$$

(28)

Proof

Set

$$\begin{aligned} b(z) := \left( g,z \right) _{L^2(0,T)} + \left( D_{0+}^\frac{\gamma +1}{2} (c_1t), D_{T-}^\frac{\gamma -1}{2} z \right) _{L^2(0,T)} - \lambda \left( c_0, z \right) _{L^2(0,T)} \end{aligned}$$

for all $z \in H^\frac{\gamma -1}{2}(0,T)$. Since Lemma 5.4 implies $b \in H^\frac{1-\gamma }{2}(0,T)$, Lemma A.2 and the well-known Lax-Milgram Theorem guarantee the unique existence of $w \in H^\frac{\gamma +1}{2}(0,T)$ with $w(0) = 0$ such that

$$\begin{aligned} \left( D_{0+}^\frac{\gamma +1}{2} w, D_{T-}^\frac{\gamma -1}{2} z \right) _{L^2(0,T)} + \lambda (w, z)_{L^2(0,T)} = b(z) \end{aligned}$$

(29)

for all $z \in H^\frac{\gamma -1}{2}(0,T)$. Using Lemma A.2 gives

$$\begin{aligned} \left\langle {D_{0+}^\gamma w, \varphi } \right\rangle&= \left( D_{0+}^\frac{\gamma +1}{2} w, D_{T-}^\frac{\gamma -1}{2} \varphi \right) _{L^2(0,T)}, \\ \left\langle {D_{0+}^\gamma (c_1t), \varphi } \right\rangle&= \left( D_{0+}^\frac{\gamma +1}{2} (c_1t), D_{T-}^\frac{\gamma -1}{2} \varphi \right) _{L^2(0,T)} \end{aligned}$$

for all $\varphi \in C_0^\infty (0,T)$, so that from (29) it follows that

$$\begin{aligned} D_{0+}^\gamma ( w - c_1t ) = g - \lambda ( w+ c_0 ). \end{aligned}$$

Putting $y := w + c_0$ gives

$$\begin{aligned} D_{0+}^\gamma ( y - c_0 - c_1t ) + \lambda y = g, \end{aligned}$$

and then by Lemma A.1 and A.2 it is evident that y is the unique $H^\gamma (0,T)$-solution to problem (19). Also, $y(0) = c_0$ is obvious, and (27) follows directly from (29).

Now let us prove (28). Firstly, substituting $z := y'$ into (27) and using integration by parts yield

$$\begin{aligned} \left( D_{0+}^\frac{\gamma +1}{2} ( y-c_0-c_1t ), \ D_{T-}^\frac{\gamma -1}{2} y' \right) _{L^2(0,T)} + \frac{\lambda }{2} y^2(T) = \frac{\lambda }{2} c_0^2+ (g,y')_{L^2(0,T)}. \end{aligned}$$

Therefore, Lemma A.2, the Cauchy–Schwarz inequality and the Young’s inequality with $\epsilon$ imply

$$\begin{aligned} \left\| y - c_0 \right\| _{ H^\frac{\gamma +1}{2} (0,T) }^2 + \lambda y^2(T) \lesssim \left\| g \right\| _{L^2(0,T)}^2 + \lambda c_0^2 + c_1^2, \end{aligned}$$

and so

$$\begin{aligned} \left\| y \right\| _{ H^\frac{\gamma +1}{2}(0,T) } \lesssim \left\| g \right\| _{L^2(0,T)} + \lambda ^\frac{1}{2} \left| c_0 \right| + \left| c_1 \right| . \end{aligned}$$

(30)

Secondly, substituting $z := y$ into (27) yields

$$\begin{aligned} \lambda \left\| y \right\| _{L^2(0,T)}^2 = \left( g,y \right) _{L^2(0,T)} - \left( D_{0+}^\frac{\gamma +1}{2} (y - c_0 - c_1t), \ D_{T-}^\frac{\gamma -1}{2} y \right) _{L^2(0,T)}, \end{aligned}$$

so that using Lemmas 5.4 and A.2, the Cauchy–Schwarz inequality and the Young’s inequality with $\epsilon$ gives

$$\begin{aligned} \lambda \left\| y \right\| _{L^2(0,T)}^2 \lesssim \left\| y - c_0 - c_1t \right\| _{ H^\frac{\gamma +1}{2}(0,T) } \left\| y \right\| _{H^\frac{\gamma -1}{2}(0,T)} + \lambda ^{-1} \left\| g \right\| _{L^2(0,T)}^2, \end{aligned}$$

which, together with (30), yields

$$\begin{aligned} \lambda ^\frac{1}{2} \left\| y \right\| _{L^2(0,T)} \lesssim \left\| g \right\| _{L^2(0,T)} + \lambda ^\frac{1}{2} \left| c_0 \right| + \left| c_1 \right| . \end{aligned}$$

(31)

Finally, collecting (30), (31) proves (28), and thus proves this lemma. $\square$

Finally, by the above lemma and the Galerkin method, we readily conclude that problem (1) admits a unique weak solution indeed. We summarize the result as follows.

Theorem A.1

The weak solution u of problem (1) satisfies that $u(0) = u_0$ and that

$$\begin{aligned} \left( D_{0+}^\frac{\gamma +1}{2} ( u - u_0 - tu_1 ), \ D_{T-}^\frac{\gamma -1}{2} v \right) _{L^2(\Omega _T)} + (\nabla u,\nabla v)_{L^2(\Omega _T)} = (f,v)_{L^2(\Omega _T)} \end{aligned}$$

(32)

for all $v \in H^\frac{\gamma -1}{2} (0,T;H_0^1(\Omega ))$. Furthermore, we have