# Numerical Analysis of anH1-Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation

Authors: Jinfeng Wang; Meng Zhao; Min Zhang; Yang Liu; Hong Li
Journal: The Scientific World Journal
 (2014)

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## Abstract

We discuss and analyze anH1-Galerkin mixed finite element (H1-GMFE) method to look for the numerical solution of time fractional telegraph equation. We introduce an auxiliary variable to reduce the original equation into lower-order coupled equations and then formulate an H1-GMFE scheme with two important variables. We discretize the Caputo time fractional derivatives using the finite difference methods and approximate the spatial direction by applying the H1-GMFE method. Based on the discussion on the theoretical error analysis in L2-norm for the scalar unknown and its gradient in one dimensional case, we obtain the optimal order of convergence in space-time direction. Further, we also derive the optimal error results for the scalar unknown in H1-norm. Moreover, we derive and analyze the stability of H1-GMFE scheme and give the results of a priori error estimates in two- or three-dimensional cases. In order to verify our theoretical analysis, we give some results of numerical calculation by using the Matlab procedure.


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## Body

## 1. Introduction

In this paper, our purpose is to present and discuss a mixed finite element method for the time fractional telegraph equation(1)∂0,t2αu(x,t)+2κ∂0,tαu(x,t)-Δu(x,t)+βu(x,t)=f(x,t),(x,t)∈Ω×J,
with boundary condition
(2)u(x,t)=0,(x,t)∈∂Ω×J¯,
and initial conditions
(3)u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,
where Ω(⊂Rd,d=1,2,3) is a bounded domain with boundary ∂Ω and J=(0,T] is the time interval with 0<T<∞. The coefficients κ>0 and β≥0 are two constants, f(x,t) is a given source function, u0(x) and u1(x) are two given initial functions, and the time Caputo fractional-order derivatives ∂0,tαu(x,t) and ∂0,t2αu(x,t) are defined, respectively, by
(4)∂0,tαu(x,t)=1Γ(1-α)∫0t‍∂u(x,τ)∂τdτ(t-τ)α,∂0,t2αu(x,t)=1Γ(2-2α)∫0t‍∂2u(x,τ)∂τ2dτ(t-τ)2α-1,
where 1/2<α<1.In the current literatures, we can see that some numerical methods for solving fractional partial differential equations (PDEs), which include finite element methods [1–8], mixed finite element methods [9], finite difference methods [10–24], finite volume methods [25, 26], spectral methods [27], and discontinuous Galerkin methods [28–31], have been considered and analyzed. In 2014, Liu et al. [9] gave some theoretical error analysis for a class of fractional PDE based on a nonstandard mixed method in spatial direction and a finite difference scheme in time direction. In [32], Zhao and Li discussed finite element method for the fractional telegraph equation. In 2014, Wei et al. [31] studied the numerical solution for time fractional telegraph equation based on the LDG method. But, we have not seen any related studies on mixed finite element methods for solving the fractional telegraph equation.Recently, some people have made use of the method to obtain the numerical solution for some partial differential equations since Pani (in 1998) [33] proposed an H1-GMFE method. This method includes some advantages, such as avoiding the LBB consistency condition, allowing different polynomial degrees of the finite element spaces, and obtaining the optimal a priori estimates in both H1 and L2-norms. In [34], Pani and Fairweather discussed some detailed a priori error results of two numerical schemes based on the H1-GMFE method for linear parabolic integrodifferential equations. In [35], Pani et al. gave some error analysis based on H1-GMFE scheme for the partial differential equation of hyperbolic type. At the same time, a modified H1-GMFE procedure was also proposed and analyzed for the case of two or three dimensions. Guo and Chen [36, 37] obtained some theoretical error analysis and numerical results of H1-GMFE method for RLW equation and Sobolev equation. In 2013, Liu et al. [38] introduced another auxiliary variable, which is different from the one in [36], and then proposed and studied an explicit multistep H1-GMFE scheme for a RLW equation. Liu and Li [39] and Zhou [40] gave some different discussions on H1-GMFE method for pseudohyperbolic equations (heat transport equation), respectively. In [41], Che et al. studied the H1-GMFE method for a nonlinear integrodifferential equation. Recently, Shi et al. [42, 43] proposed some nonconforming mixed scheme based on the H1-GMFE method.Based on the above review onH1-GMFE method, we easily see that the method was studied based on the integer-order partial differential equations. However, the theoretical results of H1-GMFE method for fractional telegraph equation have not been presented and analyzed.In this paper, our aim is to give some detailed a priori error analysis and numerical results on theH1-GMFE method for time fractional telegraph equation. We apply the difference schemes to approximate the time fractional derivatives and use the H1-GMFE method to discretize spatial direction. We obtain some optimal a priori error results of one dimension for the scalar unknown in L2 and H1-norms. Moreover, we also get an a priori error result of the optimal L2-norm for the auxiliary variable. At the same time, we use the H1-GMFE method to deal with the cases in several dimensions and analyze the stable results for the H1-GMFE scheme. We calculate some numerical results to verify the theoretical analysis of H1-GMFE method for fractional telegraph equation.The layout of the paper is as follows. In Section2, we formulate an H1-GMFE scheme for time fractional telegraph equation (5). In Section 3, we introduce some lemmas of two important projections and two difference approximations for time fractional derivatives and then analyze some a priori error results. In Section 4, some theoretical results are given in the cases of two and three dimensions. In Section 5, we choose a numerical example to verify the theoretical analysis of our method. In Section 6, we give some remarks and extensions about the H1-GMFE method for fractional PDEs.For the need of study, we denote the natural inner product as(·,·) in (L2(Ω))d, d=1,2,3. Further, we write the classicalSobolev spaces Wm,2(Ω) as Hm with norm ∥z∥m=[∑0≤|β|≤m∫Ω‍|Dβz|2dx]1/2. When m=0, we simply write the norm ∥·∥0 as ∥·∥.


## 2. AnH1-GMFE Scheme in One Space Dimension

In this section, we first consider theH1-GMFE method for the following time fractional telegraph equation in 1D case:
(5)∂0,t2αu(x,t)+2κ∂0,tαu(x,t)-∂2u(x,t)∂x2+βu(x,t)=f(x,t),(x,t)∈Ω×J,
with boundary condition
(6)u(xL,t)=u(xR,t)=0,t∈J¯,
and initial conditions
(7)u(x,0)=u0(x),ut(x,0)=u1(x),x∈Ω,
where Ω=[xL,xR]⊂R.In order to get theH1-GMFE formulation, we first introduce an auxiliary variable σ=∂u(x,t)/∂x and split (5) into the following first-order system by
(8)∂0,t2αu(x,t)+2κ∂0,tαu(x,t)-∂σ(x,t)∂x+βu(x,t)=f(x,t),(9)σ-∂u(x,t)∂x=0.Multiplying (8) by -∂w/∂x, w∈H1, integrating with respect to space from xL to xR, and using an integration by parts with ∂u(xL,t)/∂t=∂u(xR,t)/∂t=0 and ∂2u(xL,t)/∂t2=∂2u(xR,t)/∂t2=0, we easily get
(10)(∂0,t2ασ,w)+2κ(∂0,tασ,w)+(∂σ∂x,∂w∂x)=β(u,∂w∂x)-(f,∂w∂x).
Multiply (9) by ∂v/∂x, v∈H01, and integrate with respect to space from xL to xR to obtain
(11)(∂u∂x,∂v∂x)=(σ,∂v∂x),∀v∈H01.
For formulating finite element scheme, we now choose the finite element spaces Vh⊂H01 and Wh⊂H1, which satisfy the following approximation properties: for 1≤p≤∞ and k, r positive integers [33],
(12)inf⁡vh∈Vh{∥v-vh∥Lp+h∥v-vh∥W1,p}≤Chk+1∥v∥Wk+1,p,v∈H01∩Wk+1,p,inf⁡wh∈Wh{∥w-wh∥Lp+h∥w-wh∥W1,p}≤Chr+1∥w∥Wr+1,p,w∈Wr+1,p.
Based on the chosen finite element spaces, the semidiscrete H1-GMFE scheme is described by
(13)(∂uh∂x,∂vh∂x)=(σh,∂vh∂x),∀vh∈Vh,(∂0,t2ασh,wh)+2κ(∂0,tασh,wh)+(∂σh∂x,∂wh∂x)=β(uh,∂wh∂x)-(f,∂wh∂x),∀wh∈Wh.


## 3. Full Discrete Scheme and A Priori Error Estimates

### 3.1. Two Projection Lemmas

For a priori error estimates for fully discrete scheme, we introduce two projection operators [33, 44] in Lemmas 1 and 2.Lemma 1.
One defines a Ritz projectionPhu∈Vh for the variable u by
(14)(ux-Phux,vhx)=0,vh∈Vh.
Then the following estimates hold, for j=0,1:
(15)∥u-Phu∥j≤C⋆hk+1-j∥u∥k+1.Lemma 2.
Further, one also defines an elliptic projectionRhσ∈Wh of σ as the solution of
(16)B(σ-Rhσ,wh)=0,wh∈Wh,
where B(σ,w)=(σx,wx)+λ(σ,w). Here λ>0 is chosen to satisfy
(17)B(w,w)≥μ0∥w∥12,w∈H1,μ0>0.
Then the following estimates are found: for j=0,1,
(18)∥σ-Rhσ∥j≤C*hr+1-j∥σ∥r+1.


### 3.2. Approximation of Time-Fractional Derivative

For formulating fully discrete scheme, let0=t0<t1<t2<⋯<tM=T be a given partition of the time interval [0,T] with step length Δt=T/M and nodes tn=nΔt, for some positive integer M. For a smooth function ϕ on [0,T], define ϕn=ϕ(tn). In the following analysis, for deriving the convenience of theoretical process, we now denote
(19)Bn-kjα=(n-k+1)j-jα-(n-k)j-jα,j=1,2Dtσk=σk-σk-1Δt,Dttσk-1=σk-2σk-1+σk-2Δt2.
Now, we will introduce two lemmas on the approximations of time fractional derivatives.Lemma 3 (see [27]).
The time fractional derivative∂0,tασ(x,t) at t=tn is approximated by, for 0<α<1,
(20)∂0,tασ(x,tn)=Δt1-αΓ(2-α)∑k=1n‍Bn-kαDtσk+Eαn,
and then holds
(21)|Eαn|≤C0Δt2-α.Lemma 4 (see [1]).
The time fractional order derivative∂0,t2ασ(x,t) at t=tn is estimated by, for 1<2α<2,
(22)∂0,t2ασ(x,tn)=Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2αDttσk-1+E2αn,
and then holds
(23)|E2αn|≤C0Δt.In the next analysis, we will derive and prove some a priori error results foru and σ.


### 3.3. Error Estimates for Fully Discrete Scheme

Based on the approximation formulas (20) and (22) of time-fractional derivatives, we obtain the time semidiscrete scheme of (10) and (11):
(24)(∂un∂x,∂v∂x)=(σn,∂v∂x),∀v∈H01,Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttσk-1,w)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtσk,w)+(∂σn∂x,∂w∂x)=β(un,∂w∂x)-(fn,∂w∂x)+(E0n,w),∀w∈H1,
where E0n=Eαn+E2αn.Now, we formulate a fully discrete procedure: find(uhn,σhn)∈Vh×Wh(n=1,…,M-1) such that
(25)(∂uhn∂x,∂vh∂x)=(σhn,∂vh∂x),∀vh∈Vh,Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttσhk-1,wh)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtσhk,wh)+(∂σhn∂x,∂wh∂x)=β(uhn,∂wh∂x)-(fn,∂wh∂x),∀wh∈Wh.Making a combination of (24)-(25) with two projections (14) and (16), we get the following error equations:
(26)(∂ϑn∂x,∂vh∂x)=(δn,∂vh∂x)+(ϱn,∂vh∂x),∀vh∈Vh,(27)Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttδk-1,wh)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtδk,wh)+B(δn,wh)=-Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttϱk-1,wh)-2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtϱk,wh)+λ(ϱn,wh)+(E0n,wh)+β(ϕn+ϑn,∂wh∂x),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∀wh∈Wh,
where
(28)ϕn+ϑn≐(u(tn)-Phun)+(Phun-uhn)=u(tn)-uhn;ϱn+δn≐(σ(tn)-Rhσn)+(Rhσn-σhn)=σ(tn)-σhn.In the following discussion, we will derive the proof for the fully discrete a priori error estimates.Theorem 5.
Withσ0=u0x and σt(0)=u1x, suppose that un∈H01∩Hk+1, σn∈Hr+1, σh0=Rhσ(0), and σht(0)=Lhσt(0), where Lh is the L2 projection defined by (w-Lhw,wh)=0, wh∈Wh. Then there exists a positive constant C(u,σ,α,β) free of space-time discrete parameters h and Δt such that, for 1/2<α<1,
(29)∥σn-σhn∥≤C(u,σ,α,β)T2α-12-2α(Δt1-2αhmin⁡⁡{k+1,r+1}+Δt3-2α)+C(σ)hr+1,(30)∥un-uhn∥j≤C(u,σ,α,β)T2α-12-2α(Δt1-2αhmin⁡⁡{k+1,r+1}+Δt3-2α)+C(σ)hr+1+C(u)hk+1-j,j=0,1,
and for α→1(31)∥σn-σhn∥≤C(u,σ,β)(Δt-1hmin⁡⁡{k+1,r+1}+Δt)+C(σ)hr+1,(32)∥un-uhn∥j≤C(u,σ,β)(Δt-1hmin⁡⁡{k+1,r+1}+Δt)+C(σ)hr+1+C(u)hk+1-j,j=0,1.Proof.
For the need of error analysis, we first consider theL2-norm ∥ϑn∥ and the H1-norm ∥ϑn∥1. Taking vh=ϑn in (26) and using Poincaré inequality based on the H01-space and Cauchy-Schwarz inequality, we easily get
(33)∥ϑn∥≤∥ϑn∥1≤C∥∂ϑn∂x∥≤C(∥ϱn∥+∥δn∥).
In the next analysis, we will give the estimates of ∥δn∥ and ∥ϑn∥ in L2-norm. Noting that
(34)∑k=1n‍Bn-kαDtδk=∑k=0n-1‍BkαDtδn-k,∑k=1n‍Bn-k2αDttδk-1=∑k=0n-1‍Bk2αDttδn-k-1,
then (27) may be rewritten as
(35)Δt2-2αΓ(3-2α)∑k=0n-1‍Bk2α(Dttδn-k-1,wh)+2κΔt1-αΓ(2-α)∑k=0n-1‍Bkα(Dtδn-k,wh)+B(δn,wh)=-Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttϱk-1,wh)-2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtϱk,wh)+λ(ϱn,wh)+(E0n,wh)+β(ϕn+ϑn,∂wh∂x).
We take wh=δn in (35) and multiply by
(36)Γα≐Γ(3-2α)Γ(2-α)Δt2α,
to arrive at
(37)Γ(2-α)∑k=0n-1‍Bk2α(δn-k-2δn-k-1+δn-k-2,δn)+2κΓ(3-2α)Δtα∑k=0n-1‍Bkα(δn-k-δn-k-1,δn)+ΓαB(δn,δn)=-Γ(2-α)∑k=0n-1‍Bk2α(ϱn-k-2ϱn-k-1+ϱn-k-2,δn)-2κΓ(3-2α)Δtα∑k=0n-1‍Bkα(ϱn-k-ϱn-k-1,δn)+λΓα(ϱn,δn)+Γα(E0n,δn)+βΓα(ϕn+ϑn,∂ϑn∂x).
By the simple calculation, we get the following equality:
(38)Γ(2-α)∑k=0n-1‍Bk2α(δn-k-2δn-k-1+δn-k-2,δn)+2κΓ(3-2α)Δtα∑k=0n-1‍Bkα(δn-k-δn-k-1,δn)=(Γ(2-α)+2κΓ(3-2α)Δtα)∥δn∥2+Γ(2-α)×[(B12α-2B02α)(δn-1,δn)(∑k=1n-1‍)iiiiiiiiiiiii-Bn2α(δ0,δn)+Bn-12α(δ-1,δn)iiiiiiiiiiiii+(∑k=1n-1‍(Bk+12α-2Bk2α+Bk-12α)δn-k-1,δn)]+2κΓ(3-2α)Δtα×[(∑k=1n-1‍(Bkα-Bk-1α)δn-k,δn)-Bn-1α(δ0,δn)].
By applying the similar process of calculation to (38), we use Cauchy-Schwarz inequality to get
(39)-Γ(2-α)∑k=0n-1‍Bk2α(ϱn-k-2ϱn-k-1+ϱn-k-2,δn)-2κΓ(3-2α)Δtα∑k=0n-1‍Bkα(ϱn-k-ϱn-k-1,δn)≤(Γ(2-α)+2κΓ(3-2α)Δtα)∥ϱn∥+Γ(2-α)×[(2B02α-B12α)∥ϱn-1∥+Bn2α∥ϱ0∥+Bn-12α∥ϱ-1∥∑k=1n-1‍iiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|∥ϱn-k-1∥]∥δn∥+2κΓ(3-2α)Δtα×[∑k=1n-1‍|Bkα-Bk-1α|∥ϱn-k∥+Bn-1α∥ϱ0∥]∥δn∥.
Noting (33), we use Cauchy-Schwarz inequality and Young inequality to have
(40)βΓα(ϕn+ϑn,∂ϑn∂x)≤βΓα∥ϕn∥∥∂ϑn∂x∥+βΓα∥ϑn∥∥∂ϑn∂x∥≤C(β)Γα∥ϕn∥(∥ϱn∥+∥δn∥)+C(β)Γα(∥ϱn∥+∥δn∥)2≤C(β)Γα(∥ϕn∥2+∥ϱn∥2)+C(β)Γα∥δn∥2.
Substitute (38), (39), and (40) into (37) and use Cauchy-Schwarz inequality to arrive at
(41)(Γ(2-α)+2κΓ(3-2α)Δtα-C(β)Γα)∥δn∥2+ΓαB(δn,δn)≤((Γ(2-α)+2κΓ(3-2α)Δtα)∥ϱn∥+Γ(2-α)[∑k=1n-1‍]iiiiiiiiii×[(2B02α-B12α)(∥δn-1∥+∥ϱn-1∥)∑k=1n-1‍iiiiiiiiiiiiiii+Bn2α(∥δ0∥+∥ϱ0∥)+Bn-12α(∥δ-1∥+∥ϱ-1∥)iiiiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|iiiiiiiiiiiiiiiiiiiiii∑k=1n-1‍×(∥δn-k-1∥+∥ϱn-k-1∥)]iiiiiiiiii[∑k=1n-1‍]+λΓα∥ϱn∥+Γα∥E0n∥)∥δn∥+2κΓ(3-2α)Δtα×[∑k=1n-1‍|Bkα-Bk-1α|(∥δn-k∥+∥ϱn-k∥)iiiiiiiiiiii∑k=1n-1‍+Bn-1α(∥δ0∥+∥ϱ0∥)]∥δn∥+C(β)Γα(∥ϕn∥2+∥ϱn∥2).
By an application of Young inequality, we get
(42)Γ(2-α)∥δn∥+ΓαB(δn,δn)≤1Γ(2-α)×((Γ(2-α)+2κΓ(3-2α)Δtα)∥ϱn∥+Γ(2-α)[∑k=1n-1‍]iiiiiiiiiiiii×[(2B02α-B12α)(∥δn-1∥+∥ϱn-1∥)∑k=1n-1‍iiiiiiiiiiiiiiiiii+Bn2α(∥δ0∥+∥ϱ0∥)+Bn-12α(∥δ-1∥+∥ϱ-1∥)iiiiiiiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|iiiiiiiiiiiiiiiiiiiiiiiiii∑k=1n-1‍×(∥δn-k-1∥+∥ϱn-k-1∥)]iiiiiiiiiiiii[∑k=1n-1‍]+λΓα∥ϱn∥+Γα∥E0n∥)+2κΓ(3-2α)ΔtαΓ(2-α)×[∑k=1n-1‍|Bkα-Bk-1α|(∥δn-k∥+∥ϱn-k∥)iiiiiiiiiiii∑k=1n-1‍+Bn-1α(∥δ0∥+∥ϱ0∥)]+C(β)Γα(∥ϕn∥+∥ϱn∥).
Noting thatBk2α-Bk+12α>0, we arrive at
(43)∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|≤∑k=1n-1‍(Bk2α-Bk+12α)+∑k=1n-1‍(Bk-12α-Bk2α)=B12α-Bn2α+B02α-Bn-12α.
At the same time, noting that Bkα-Bk+1α>0, we use the similar method to have
(44)∑k=1n-1‍|Bkα-Bk-1α|=B0α-Bn-1α.
By a combination of (43) and (44) with (18) and noting that B0α=B02α=1, we arrive at
(45)1Γ(2-α)[((Γ(2-α)+2κΓ(3-2α)Δtα)∥ϱn∥[∑k=1n-1‍]iiiiiiiiiiiiiiiiiiiii+Γ(2-α)iiiiiiiiiiiiiiiiiiiii×[(2B02α-B12α)∥ϱn-1∥∑k=1n-1‍iiiiiiiiiiiiiiiiiiiiiiiiii+Bn2α∥ϱ0∥+Bn-12α∥ϱ-1∥iiiiiiiiiiiiiiiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|∥ϱn-k-1∥])iiiiiiiiiiiiiiiiii+2κΓ(3-2α)Δtαiiiiiiiiiiiiiiiiii×[∑k=1n-1‍|Bkα-Bk-1α|∥ϱn-k∥+Bn-1α∥ϱ0∥]iiiiiiiiiiiiiiiiii([∑k=1n-1‍])+λΓα∥ϱn∥+Γα∥E0n∥]+C(β)Γα(∥ϕn∥+∥ϱn∥)≤CΓ(2-α)(Γ(2-α)+2κΓ(3-2α)Δtα+Γ(2-α)[(B02α)]iiiiiiiiiiiiiiiiiiiiiiiiii×[(2B02α-B12α)+Bn2α+Bn-12α+B12αiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii-Bn2α+B02α-Bn-12α]iiiiiiiiiiiiiiiiiiiiiiiiii+2κΓ(3-2α)Δtαiiiiiiiiiiiiiiiiiiiiiiiii[(B02α)]×[B0α-Bn-1α+Bn-1α]+λΓα)hr+1+CΓ(3-2α)Γ(2-α)Δt1+2α=CΓ(2-α)(Γ(2-α)+2κΓ(3-2α)Δtα)hr+1+C(u,σ,β)Γα(hr+1+hk+1)+CΓ(3-2α)Γ(2-α)Δt1+2α.
Substitute (45) into (42) and note that μ0∥δn∥1≤B(δn,δn)/∥δn∥1≤B(δn,δn)/∥δn∥ to get
(46)Γ(2-α)∥δn∥+μ0Γα∥δn∥1≤Γ(2-α)[(2B02α-B12α)∥δn-1∥∑k=1n-1‍iiiiiiiiiiiiiiiiiiiiiiiiii+Bn2α∥δ0∥+Bn-12α∥δ-1∥iiiiiiiiiiiiiiiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|∥δn-k-1∥]+2κΓ(3-2α)ΔtαΓ(2-α)×[∑k=1n-1‍|Bkα-Bk-1α|∥δn-k∥+Bn-1α∥δ0∥]+CΓ(2-α)(Γ(2-α)+2κΓ(3-2α)Δtα)hr+1+C(u,σ,β)Γα(hr+1+hk+1)+CΓ(3-2α)Γ(2-α)Δt1+2α.
In the following discussion, we apply mathematical induction to obtain the error result
(47)∥δn∥≤C(u,σ,α,β)Bn-12α(hk+1+hr+1+Δt2+Δt1+2α).
Take n=1 in (46) and note that B02α=1; it is easy to find that the following inequality holds:
(48)∥δ1∥+Δtα∥δ1∥1≤C(u,σ,α,β)(hk+1+hr+1+Δt2+Δt1+2α)≤C(u,σ,α,β)B02α(hk+1+hr+1+Δt2+Δt1+2α).
Assuming that the inequalities ∥δj∥≤(C(u,σ,α,β)/Bj-12α)(hk+1+hr+1+Δt2+Δt1+2α) hold for j=1,2,…,n-1, we now prove that the inequality (47) holds. Noting that 1/Bkjα<1/Bk+1jα, j=1,2, and combining (43) and (44) with (46), we have
(49)∥δn∥+Δtα∥δn∥1≤(Γ(2-α)[(2B02α-B12α)C(u,σ,α,β)Bn-12α∑k=1n-1‍iiiiiiiiiiiiiiiiiiiiiiiiiiii×(hr+1+hk+1+Δt2+Δt1+2α)iiiiiiiiiiiiiiiiiiiiiiiiiiii+Bn2α∥δ0∥+Bn-12α∥δ-1∥iiiiiiiiiiiiiiiiiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii×C(u,σ,α,β)Bn-k-22αiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∑k=1n-1‍×(hr+1+hk+1+Δt2+Δt1+2α)])+2κΓ(3-2α)ΔtαΓ(2-α)×[∑k=1n-1‍|Bkα-Bk-1α|C(u,σ,α,β)Bn-k-12αiiiiiiiiiiiiiiiii∑k=1n-1‍×(hr+1+hk+1+Δt2+Δt1+2α)+Bn-1α∥δ0∥]+CΓ(2-α)(Γ(2-α)+2κΓ(3-2α)Δtα)hr+1+C(u,σ,β)Γα(hr+1+hk+1)+CΓ(3-2α)Γ(2-α)Δt1+2α≤C(u,σ,α,β)Bn-12α(hr+1+hk+1+Δt2+Δt1+2α).
Based on the process of mathematical induction, we claim that (47) holds.
For the need of the next proof, we have to estimate the termn1-2α/Bn-12α. We now use Taylor formula to arrive at
(50)n1-2αBn-12α=n1-2αn2-2α-(n-1)2-2α=1n-(n-1)(1+1/(n-1))2α-1=(n-(n-1)[(1n-1)2]iiiiii×[1+(2α-1)1n-1+(2α-1)(2α-2)2!(1n-1)2iiiiiiiiiiii×(1+θ1n-1)2α-3(1n-1)2])-1=(2-2α+(2α-1)(2-2α)2!(1+θ1n-1)2α-3iiiiiii×(1+θ1n-1)2α-3(1n-1))-1<12-2α,
where 0<θ<1 and 1/2<α<1. Noting that (nΔt)2α-1≤T2α-1 and (50), we have
(51)∥δn∥+Δtα∥δn∥1≤C(u,σ,α,β)Bn-12α(hr+1+hk+1+Δt2+Δt1+2α)=n2α-1Δt2α-1Δt1-2αn1-2αC(u,σ,α,β)Bn-12α×(hr+1+hk+1+Δt2+Δt1+2α)≤T2α-1Δt1-2α2-2αC(u,σ,α,β)Bn-12α×(hr+1+hk+1+Δt2+Δt1+2α)=C(u,σ,α,β)T2α-12-2α×(Δt1-2α(hr+1+hk+1)+Δt2+Δt3-2α).
Substitute (51) into (33) and use (18) to get
(52)∥ϑn∥+∥ϑn∥1≤C(∥ϱn∥+∥δn∥)≤C(u,σ,α,β)T2α-12-2α×(Δt1-2α(hr+1+hk+1)+Δt2+Δt3-2α)+C(σ)hr+1.
By combining (51) and (18) with triangle inequality, the L2-norm estimate ∥σn-σhn∥ is got. Similarly, the estimate ∥un-uhn∥ in the L2-norm and the estimate ∥un-uhn∥1 in the H1-norm also are obtained by a combination of (52) and (15) with triangle inequality.
Now we mainly analyze the caseα→1. We can find that the error inequality (29) in this case has no meaning since the coefficient C(u,σ,α,β)/(2-2α)→∞ as α→1. So, we have to look for another error estimate’s process. Noting the fact that nΔt≤T (n=1,2,…,M), we can obtain the following error inequality:
(53)∥δn∥≤C(u,σ,α,β)nΔt(Δt1-2α(hr+1+hk+1)+Δt3-2α).
Now we can use induction to prove the inequality (53). The detailed proof is similar to the above process of analysis, so we do not give the detailed proof again. Making a combination of (53) and (18) with triangle inequality, we can get the estimate (31). A similar discussion for (32) can also be made.


## 3.1. Two Projection Lemmas

For a priori error estimates for fully discrete scheme, we introduce two projection operators [33, 44] in Lemmas 1 and 2.Lemma 1.
One defines a Ritz projectionPhu∈Vh for the variable u by
(14)(ux-Phux,vhx)=0,vh∈Vh.
Then the following estimates hold, for j=0,1:
(15)∥u-Phu∥j≤C⋆hk+1-j∥u∥k+1.Lemma 2.
Further, one also defines an elliptic projectionRhσ∈Wh of σ as the solution of
(16)B(σ-Rhσ,wh)=0,wh∈Wh,
where B(σ,w)=(σx,wx)+λ(σ,w). Here λ>0 is chosen to satisfy
(17)B(w,w)≥μ0∥w∥12,w∈H1,μ0>0.
Then the following estimates are found: for j=0,1,
(18)∥σ-Rhσ∥j≤C*hr+1-j∥σ∥r+1.


## 3.2. Approximation of Time-Fractional Derivative

For formulating fully discrete scheme, let0=t0<t1<t2<⋯<tM=T be a given partition of the time interval [0,T] with step length Δt=T/M and nodes tn=nΔt, for some positive integer M. For a smooth function ϕ on [0,T], define ϕn=ϕ(tn). In the following analysis, for deriving the convenience of theoretical process, we now denote
(19)Bn-kjα=(n-k+1)j-jα-(n-k)j-jα,j=1,2Dtσk=σk-σk-1Δt,Dttσk-1=σk-2σk-1+σk-2Δt2.
Now, we will introduce two lemmas on the approximations of time fractional derivatives.Lemma 3 (see [27]).
The time fractional derivative∂0,tασ(x,t) at t=tn is approximated by, for 0<α<1,
(20)∂0,tασ(x,tn)=Δt1-αΓ(2-α)∑k=1n‍Bn-kαDtσk+Eαn,
and then holds
(21)|Eαn|≤C0Δt2-α.Lemma 4 (see [1]).
The time fractional order derivative∂0,t2ασ(x,t) at t=tn is estimated by, for 1<2α<2,
(22)∂0,t2ασ(x,tn)=Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2αDttσk-1+E2αn,
and then holds
(23)|E2αn|≤C0Δt.In the next analysis, we will derive and prove some a priori error results foru and σ.


## 3.3. Error Estimates for Fully Discrete Scheme

Based on the approximation formulas (20) and (22) of time-fractional derivatives, we obtain the time semidiscrete scheme of (10) and (11):
(24)(∂un∂x,∂v∂x)=(σn,∂v∂x),∀v∈H01,Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttσk-1,w)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtσk,w)+(∂σn∂x,∂w∂x)=β(un,∂w∂x)-(fn,∂w∂x)+(E0n,w),∀w∈H1,
where E0n=Eαn+E2αn.Now, we formulate a fully discrete procedure: find(uhn,σhn)∈Vh×Wh(n=1,…,M-1) such that
(25)(∂uhn∂x,∂vh∂x)=(σhn,∂vh∂x),∀vh∈Vh,Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttσhk-1,wh)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtσhk,wh)+(∂σhn∂x,∂wh∂x)=β(uhn,∂wh∂x)-(fn,∂wh∂x),∀wh∈Wh.Making a combination of (24)-(25) with two projections (14) and (16), we get the following error equations:
(26)(∂ϑn∂x,∂vh∂x)=(δn,∂vh∂x)+(ϱn,∂vh∂x),∀vh∈Vh,(27)Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttδk-1,wh)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtδk,wh)+B(δn,wh)=-Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttϱk-1,wh)-2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtϱk,wh)+λ(ϱn,wh)+(E0n,wh)+β(ϕn+ϑn,∂wh∂x),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∀wh∈Wh,
where
(28)ϕn+ϑn≐(u(tn)-Phun)+(Phun-uhn)=u(tn)-uhn;ϱn+δn≐(σ(tn)-Rhσn)+(Rhσn-σhn)=σ(tn)-σhn.In the following discussion, we will derive the proof for the fully discrete a priori error estimates.Theorem 5.
Withσ0=u0x and σt(0)=u1x, suppose that un∈H01∩Hk+1, σn∈Hr+1, σh0=Rhσ(0), and σht(0)=Lhσt(0), where Lh is the L2 projection defined by (w-Lhw,wh)=0, wh∈Wh. Then there exists a positive constant C(u,σ,α,β) free of space-time discrete parameters h and Δt such that, for 1/2<α<1,
(29)∥σn-σhn∥≤C(u,σ,α,β)T2α-12-2α(Δt1-2αhmin⁡⁡{k+1,r+1}+Δt3-2α)+C(σ)hr+1,(30)∥un-uhn∥j≤C(u,σ,α,β)T2α-12-2α(Δt1-2αhmin⁡⁡{k+1,r+1}+Δt3-2α)+C(σ)hr+1+C(u)hk+1-j,j=0,1,
and for α→1(31)∥σn-σhn∥≤C(u,σ,β)(Δt-1hmin⁡⁡{k+1,r+1}+Δt)+C(σ)hr+1,(32)∥un-uhn∥j≤C(u,σ,β)(Δt-1hmin⁡⁡{k+1,r+1}+Δt)+C(σ)hr+1+C(u)hk+1-j,j=0,1.Proof.
For the need of error analysis, we first consider theL2-norm ∥ϑn∥ and the H1-norm ∥ϑn∥1. Taking vh=ϑn in (26) and using Poincaré inequality based on the H01-space and Cauchy-Schwarz inequality, we easily get
(33)∥ϑn∥≤∥ϑn∥1≤C∥∂ϑn∂x∥≤C(∥ϱn∥+∥δn∥).
In the next analysis, we will give the estimates of ∥δn∥ and ∥ϑn∥ in L2-norm. Noting that
(34)∑k=1n‍Bn-kαDtδk=∑k=0n-1‍BkαDtδn-k,∑k=1n‍Bn-k2αDttδk-1=∑k=0n-1‍Bk2αDttδn-k-1,
then (27) may be rewritten as
(35)Δt2-2αΓ(3-2α)∑k=0n-1‍Bk2α(Dttδn-k-1,wh)+2κΔt1-αΓ(2-α)∑k=0n-1‍Bkα(Dtδn-k,wh)+B(δn,wh)=-Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttϱk-1,wh)-2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtϱk,wh)+λ(ϱn,wh)+(E0n,wh)+β(ϕn+ϑn,∂wh∂x).
We take wh=δn in (35) and multiply by
(36)Γα≐Γ(3-2α)Γ(2-α)Δt2α,
to arrive at
(37)Γ(2-α)∑k=0n-1‍Bk2α(δn-k-2δn-k-1+δn-k-2,δn)+2κΓ(3-2α)Δtα∑k=0n-1‍Bkα(δn-k-δn-k-1,δn)+ΓαB(δn,δn)=-Γ(2-α)∑k=0n-1‍Bk2α(ϱn-k-2ϱn-k-1+ϱn-k-2,δn)-2κΓ(3-2α)Δtα∑k=0n-1‍Bkα(ϱn-k-ϱn-k-1,δn)+λΓα(ϱn,δn)+Γα(E0n,δn)+βΓα(ϕn+ϑn,∂ϑn∂x).
By the simple calculation, we get the following equality:
(38)Γ(2-α)∑k=0n-1‍Bk2α(δn-k-2δn-k-1+δn-k-2,δn)+2κΓ(3-2α)Δtα∑k=0n-1‍Bkα(δn-k-δn-k-1,δn)=(Γ(2-α)+2κΓ(3-2α)Δtα)∥δn∥2+Γ(2-α)×[(B12α-2B02α)(δn-1,δn)(∑k=1n-1‍)iiiiiiiiiiiii-Bn2α(δ0,δn)+Bn-12α(δ-1,δn)iiiiiiiiiiiii+(∑k=1n-1‍(Bk+12α-2Bk2α+Bk-12α)δn-k-1,δn)]+2κΓ(3-2α)Δtα×[(∑k=1n-1‍(Bkα-Bk-1α)δn-k,δn)-Bn-1α(δ0,δn)].
By applying the similar process of calculation to (38), we use Cauchy-Schwarz inequality to get
(39)-Γ(2-α)∑k=0n-1‍Bk2α(ϱn-k-2ϱn-k-1+ϱn-k-2,δn)-2κΓ(3-2α)Δtα∑k=0n-1‍Bkα(ϱn-k-ϱn-k-1,δn)≤(Γ(2-α)+2κΓ(3-2α)Δtα)∥ϱn∥+Γ(2-α)×[(2B02α-B12α)∥ϱn-1∥+Bn2α∥ϱ0∥+Bn-12α∥ϱ-1∥∑k=1n-1‍iiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|∥ϱn-k-1∥]∥δn∥+2κΓ(3-2α)Δtα×[∑k=1n-1‍|Bkα-Bk-1α|∥ϱn-k∥+Bn-1α∥ϱ0∥]∥δn∥.
Noting (33), we use Cauchy-Schwarz inequality and Young inequality to have
(40)βΓα(ϕn+ϑn,∂ϑn∂x)≤βΓα∥ϕn∥∥∂ϑn∂x∥+βΓα∥ϑn∥∥∂ϑn∂x∥≤C(β)Γα∥ϕn∥(∥ϱn∥+∥δn∥)+C(β)Γα(∥ϱn∥+∥δn∥)2≤C(β)Γα(∥ϕn∥2+∥ϱn∥2)+C(β)Γα∥δn∥2.
Substitute (38), (39), and (40) into (37) and use Cauchy-Schwarz inequality to arrive at
(41)(Γ(2-α)+2κΓ(3-2α)Δtα-C(β)Γα)∥δn∥2+ΓαB(δn,δn)≤((Γ(2-α)+2κΓ(3-2α)Δtα)∥ϱn∥+Γ(2-α)[∑k=1n-1‍]iiiiiiiiii×[(2B02α-B12α)(∥δn-1∥+∥ϱn-1∥)∑k=1n-1‍iiiiiiiiiiiiiii+Bn2α(∥δ0∥+∥ϱ0∥)+Bn-12α(∥δ-1∥+∥ϱ-1∥)iiiiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|iiiiiiiiiiiiiiiiiiiiii∑k=1n-1‍×(∥δn-k-1∥+∥ϱn-k-1∥)]iiiiiiiiii[∑k=1n-1‍]+λΓα∥ϱn∥+Γα∥E0n∥)∥δn∥+2κΓ(3-2α)Δtα×[∑k=1n-1‍|Bkα-Bk-1α|(∥δn-k∥+∥ϱn-k∥)iiiiiiiiiiii∑k=1n-1‍+Bn-1α(∥δ0∥+∥ϱ0∥)]∥δn∥+C(β)Γα(∥ϕn∥2+∥ϱn∥2).
By an application of Young inequality, we get
(42)Γ(2-α)∥δn∥+ΓαB(δn,δn)≤1Γ(2-α)×((Γ(2-α)+2κΓ(3-2α)Δtα)∥ϱn∥+Γ(2-α)[∑k=1n-1‍]iiiiiiiiiiiii×[(2B02α-B12α)(∥δn-1∥+∥ϱn-1∥)∑k=1n-1‍iiiiiiiiiiiiiiiiii+Bn2α(∥δ0∥+∥ϱ0∥)+Bn-12α(∥δ-1∥+∥ϱ-1∥)iiiiiiiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|iiiiiiiiiiiiiiiiiiiiiiiiii∑k=1n-1‍×(∥δn-k-1∥+∥ϱn-k-1∥)]iiiiiiiiiiiii[∑k=1n-1‍]+λΓα∥ϱn∥+Γα∥E0n∥)+2κΓ(3-2α)ΔtαΓ(2-α)×[∑k=1n-1‍|Bkα-Bk-1α|(∥δn-k∥+∥ϱn-k∥)iiiiiiiiiiii∑k=1n-1‍+Bn-1α(∥δ0∥+∥ϱ0∥)]+C(β)Γα(∥ϕn∥+∥ϱn∥).
Noting thatBk2α-Bk+12α>0, we arrive at
(43)∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|≤∑k=1n-1‍(Bk2α-Bk+12α)+∑k=1n-1‍(Bk-12α-Bk2α)=B12α-Bn2α+B02α-Bn-12α.
At the same time, noting that Bkα-Bk+1α>0, we use the similar method to have
(44)∑k=1n-1‍|Bkα-Bk-1α|=B0α-Bn-1α.
By a combination of (43) and (44) with (18) and noting that B0α=B02α=1, we arrive at
(45)1Γ(2-α)[((Γ(2-α)+2κΓ(3-2α)Δtα)∥ϱn∥[∑k=1n-1‍]iiiiiiiiiiiiiiiiiiiii+Γ(2-α)iiiiiiiiiiiiiiiiiiiii×[(2B02α-B12α)∥ϱn-1∥∑k=1n-1‍iiiiiiiiiiiiiiiiiiiiiiiiii+Bn2α∥ϱ0∥+Bn-12α∥ϱ-1∥iiiiiiiiiiiiiiiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|∥ϱn-k-1∥])iiiiiiiiiiiiiiiiii+2κΓ(3-2α)Δtαiiiiiiiiiiiiiiiiii×[∑k=1n-1‍|Bkα-Bk-1α|∥ϱn-k∥+Bn-1α∥ϱ0∥]iiiiiiiiiiiiiiiiii([∑k=1n-1‍])+λΓα∥ϱn∥+Γα∥E0n∥]+C(β)Γα(∥ϕn∥+∥ϱn∥)≤CΓ(2-α)(Γ(2-α)+2κΓ(3-2α)Δtα+Γ(2-α)[(B02α)]iiiiiiiiiiiiiiiiiiiiiiiiii×[(2B02α-B12α)+Bn2α+Bn-12α+B12αiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii-Bn2α+B02α-Bn-12α]iiiiiiiiiiiiiiiiiiiiiiiiii+2κΓ(3-2α)Δtαiiiiiiiiiiiiiiiiiiiiiiiii[(B02α)]×[B0α-Bn-1α+Bn-1α]+λΓα)hr+1+CΓ(3-2α)Γ(2-α)Δt1+2α=CΓ(2-α)(Γ(2-α)+2κΓ(3-2α)Δtα)hr+1+C(u,σ,β)Γα(hr+1+hk+1)+CΓ(3-2α)Γ(2-α)Δt1+2α.
Substitute (45) into (42) and note that μ0∥δn∥1≤B(δn,δn)/∥δn∥1≤B(δn,δn)/∥δn∥ to get
(46)Γ(2-α)∥δn∥+μ0Γα∥δn∥1≤Γ(2-α)[(2B02α-B12α)∥δn-1∥∑k=1n-1‍iiiiiiiiiiiiiiiiiiiiiiiiii+Bn2α∥δ0∥+Bn-12α∥δ-1∥iiiiiiiiiiiiiiiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|∥δn-k-1∥]+2κΓ(3-2α)ΔtαΓ(2-α)×[∑k=1n-1‍|Bkα-Bk-1α|∥δn-k∥+Bn-1α∥δ0∥]+CΓ(2-α)(Γ(2-α)+2κΓ(3-2α)Δtα)hr+1+C(u,σ,β)Γα(hr+1+hk+1)+CΓ(3-2α)Γ(2-α)Δt1+2α.
In the following discussion, we apply mathematical induction to obtain the error result
(47)∥δn∥≤C(u,σ,α,β)Bn-12α(hk+1+hr+1+Δt2+Δt1+2α).
Take n=1 in (46) and note that B02α=1; it is easy to find that the following inequality holds:
(48)∥δ1∥+Δtα∥δ1∥1≤C(u,σ,α,β)(hk+1+hr+1+Δt2+Δt1+2α)≤C(u,σ,α,β)B02α(hk+1+hr+1+Δt2+Δt1+2α).
Assuming that the inequalities ∥δj∥≤(C(u,σ,α,β)/Bj-12α)(hk+1+hr+1+Δt2+Δt1+2α) hold for j=1,2,…,n-1, we now prove that the inequality (47) holds. Noting that 1/Bkjα<1/Bk+1jα, j=1,2, and combining (43) and (44) with (46), we have
(49)∥δn∥+Δtα∥δn∥1≤(Γ(2-α)[(2B02α-B12α)C(u,σ,α,β)Bn-12α∑k=1n-1‍iiiiiiiiiiiiiiiiiiiiiiiiiiii×(hr+1+hk+1+Δt2+Δt1+2α)iiiiiiiiiiiiiiiiiiiiiiiiiiii+Bn2α∥δ0∥+Bn-12α∥δ-1∥iiiiiiiiiiiiiiiiiiiiiiiiiiii+∑k=1n-1‍|Bk+12α-2Bk2α+Bk-12α|iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii×C(u,σ,α,β)Bn-k-22αiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii∑k=1n-1‍×(hr+1+hk+1+Δt2+Δt1+2α)])+2κΓ(3-2α)ΔtαΓ(2-α)×[∑k=1n-1‍|Bkα-Bk-1α|C(u,σ,α,β)Bn-k-12αiiiiiiiiiiiiiiiii∑k=1n-1‍×(hr+1+hk+1+Δt2+Δt1+2α)+Bn-1α∥δ0∥]+CΓ(2-α)(Γ(2-α)+2κΓ(3-2α)Δtα)hr+1+C(u,σ,β)Γα(hr+1+hk+1)+CΓ(3-2α)Γ(2-α)Δt1+2α≤C(u,σ,α,β)Bn-12α(hr+1+hk+1+Δt2+Δt1+2α).
Based on the process of mathematical induction, we claim that (47) holds.
For the need of the next proof, we have to estimate the termn1-2α/Bn-12α. We now use Taylor formula to arrive at
(50)n1-2αBn-12α=n1-2αn2-2α-(n-1)2-2α=1n-(n-1)(1+1/(n-1))2α-1=(n-(n-1)[(1n-1)2]iiiiii×[1+(2α-1)1n-1+(2α-1)(2α-2)2!(1n-1)2iiiiiiiiiiii×(1+θ1n-1)2α-3(1n-1)2])-1=(2-2α+(2α-1)(2-2α)2!(1+θ1n-1)2α-3iiiiiii×(1+θ1n-1)2α-3(1n-1))-1<12-2α,
where 0<θ<1 and 1/2<α<1. Noting that (nΔt)2α-1≤T2α-1 and (50), we have
(51)∥δn∥+Δtα∥δn∥1≤C(u,σ,α,β)Bn-12α(hr+1+hk+1+Δt2+Δt1+2α)=n2α-1Δt2α-1Δt1-2αn1-2αC(u,σ,α,β)Bn-12α×(hr+1+hk+1+Δt2+Δt1+2α)≤T2α-1Δt1-2α2-2αC(u,σ,α,β)Bn-12α×(hr+1+hk+1+Δt2+Δt1+2α)=C(u,σ,α,β)T2α-12-2α×(Δt1-2α(hr+1+hk+1)+Δt2+Δt3-2α).
Substitute (51) into (33) and use (18) to get
(52)∥ϑn∥+∥ϑn∥1≤C(∥ϱn∥+∥δn∥)≤C(u,σ,α,β)T2α-12-2α×(Δt1-2α(hr+1+hk+1)+Δt2+Δt3-2α)+C(σ)hr+1.
By combining (51) and (18) with triangle inequality, the L2-norm estimate ∥σn-σhn∥ is got. Similarly, the estimate ∥un-uhn∥ in the L2-norm and the estimate ∥un-uhn∥1 in the H1-norm also are obtained by a combination of (52) and (15) with triangle inequality.
Now we mainly analyze the caseα→1. We can find that the error inequality (29) in this case has no meaning since the coefficient C(u,σ,α,β)/(2-2α)→∞ as α→1. So, we have to look for another error estimate’s process. Noting the fact that nΔt≤T (n=1,2,…,M), we can obtain the following error inequality:
(53)∥δn∥≤C(u,σ,α,β)nΔt(Δt1-2α(hr+1+hk+1)+Δt3-2α).
Now we can use induction to prove the inequality (53). The detailed proof is similar to the above process of analysis, so we do not give the detailed proof again. Making a combination of (53) and (18) with triangle inequality, we can get the estimate (31). A similar discussion for (32) can also be made.


## 4. AnH1-GMFE Scheme for Several Spaces Variables

In this section, we consider (1) with two and three spaces variables.LetL2(Ω)=(L2(Ω))d, (d=2 or 3) with inner product and norm
(54)(q,w)=∑i=1d‍(qi,wi),∥w∥=(∑i=1d‍∥wi∥2)1/2.
Further, let
(55)W=H(div;Ω)={w∈L2(Ω)∣∇·w∈L2(Ω)}
with norm
(56)∥w∥H(div;Ω)=(∥w∥2+∥∇·w∥2)1/2.
Taking q=∇u, we use a similar process to the system (10) and (11) to get the H1-Galerkin mixed weak formulation {u,q}:[0,T]↦H01×W by
(57)(∇u,∇v)=(q,∇v),∀v∈H01,(∂0,t2αq,w)+2κ(∂0,tαq,w)+(∇·q,∇·w)=β(u,∇·w)-(f,∇·w),∀w∈W.
The corresponding time semidiscrete system is defined by
(58)(∇un,∇v)=(qn,∇v),∀v∈H01,Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttqk-1,w)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtqk,w)+(∇·qn,∇·w)=β(un,∇·w)-(fn,∇·w)+(R0n,w),∀w∈W,
where R0n=O(Δt), which can be estimated by a similar process to E0n.In order to get fully discrete mixed finite element scheme, we now choose the finite element spacesVh⊂H01 and Wh⊂W, which satisfy the following approximation properties: for 1≤p≤∞ and k, r positive integers [33],
(59)inf⁡vh∈Vh{∥v-vh∥+h∥v-vh∥1}≤Chk+1∥v∥k+1,v∈H01∩Hk+1,inf⁡wh∈Wh{∥w-wh∥+h∥w-wh∥H(div;Ω)}≤Chr+1∥w∥r+1,w∈Hr+1.
The fully discrete H1-GMFE scheme is to find {uhn,qhn}:[0,T]↦Vh×Wh such that
(60)(∇uhn,∇vh)=(qhn,∇vh),∀vh∈Vh,(61)Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttqhk-1,wh)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtqhk,wh)+(∇·qhn,∇·wh)=β(uhn,∇·wh)-(fn,∇·wh),∀w∈Wh.


### 4.1. The Analysis of Stability

In the following discussion, we will give the stability for the system (60) and (61). First, we need to obtain an important lemma on two initial value conditions.Lemma 6.
Withq0=∇u0 and qt(0)=∇u1, the following inequality holds:
(62)∥qh-1∥≤C(∥u0∥1+Δt∥u1∥1).Proof.
By a similar discussion as in [32], we have
(63)∥qh-1∥≤C(∥q0∥+Δt∥qt(0)∥).
Based on the given initial value conditions and (63), we arrive at
(64)∥qh-1∥≤C(∥∇u0∥+Δt∥∇u1∥)≤C(∥u0∥1+Δt∥u1∥1).Theorem 7.
The following stable inequality for the system (60) and (61) holds:
(65)∥uhn∥1+∥qhn∥+κΓα∥∇·qhn∥≤C(α)(Γαmax⁡0≤j≤n∥fj∥+∥u0∥1+Δt∥u1∥1).Proof.
In (60), we take vh=∇uhn and use Cauchy-Schwarz inequality and Poincaré inequality to arrive at
(66)∥uhn∥≤∥uhn∥1≤C∥∇uhn∥≤C∥qhn∥.
In (61), we choose wh=qhn and use Cauchy-Schwarz inequality, Young inequality, and (66) to get
(67)Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttqhk-1,qhn)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtqhk,qhn)+∥∇·qhn∥2≤β∥uhn∥∥∇·qhn∥+∥fn∥∥∇·qhn∥≤C∥qhn∥∥∇·qhn∥+∥fn∥∥∇·qhn∥≤C(∥qhn∥2+∥fn∥2)+12∥∇·qhn∥2.
By the simple simplification for (67), we easily get
(68)Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttqhk-1,qhn)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtqhk,qhn)+12∥∇·qhn∥2≤C(∥qhn∥2+∥fn∥2).
For the case n=1 in (68), we multiply (36) to get easily
(69)Γ(2-α)B02α(qh1-2qh0+qh-1,qh1)+2κΔtαΓ(3-2α)B0α(qh1-qh0,qh1)+κΓα∥∇·qh1∥2≤CΓα(∥qh1∥2+∥f1∥2).
Noting that B0α=1=B02α and Lemma 6, we use Cauchy-Schwarz inequality to get
(70)(12Γ(2-α)+κΔtαΓ(3-2α)-CΓα)∥qh1∥2+κΓα∥∇·qh1∥2≤CΓα∥f1∥2+C(α)(∥qh-1∥2+∥qh0∥2)≤Γα∥f1∥2+C(α)(∥u0∥12+Δt2∥u1∥12)≤C(α)(Γαmax⁡0≤j≤n∥fj∥+∥u0∥1+Δt∥u1∥1)2.
By the simple calculation for (70), we arrive at
(71)∥qh1∥+κΓα∥∇·qh1∥≤C(α)(Γαmax⁡0≤j≤n∥fj∥+∥u0∥1+Δt∥u1∥1).
For the case n≥1, using a similar process to the proof of Theorem 5 based on the mathematical induction, we can get
(72)∥qhn∥+κΓα∥∇·qhn∥≤C(α)(Γαmax⁡0≤j≤n∥fj∥+∥u0∥1+Δt∥u1∥1).
By a combination of (72) and (66), we get the conclusion of Theorem 7.


### 4.2. A Priori Error Results

For deriving the a priori error analysis, we define the Ritz projectionu~h∈Vh by
(73)(∇(u-u~h),∇vh)=0,vh∈Vh.
Further, let q~h∈Wh be the standard finite element interpolant of q.Letρ=q-q~h, η=u-u~h; see [33, 44]; we obtain
(74)∥η∥j≤Chk+1-j∥u∥k+1,j=0,1,∥ρ∥+h∥ρ∥H(div;Ω)≤Chr+1∥q∥r+1.
Now, we can get the following theorem of a priori error estimates based on the above contents.Theorem 8.
Withq0=∇u0, qt(0)=∇u1, un∈H01∩Hk+1, and qn∈Hr+1, there exists a positive constant C(u,q,α,β) free of space-time meshes h and Δt such that, for 1/2<α<1,
(75)∥qn-qhn∥≤C(u,q,α,β)T2α-12-2α(Δt1-2αhmin⁡⁡{k+1,r}+Δt3-2α)+C(q)hr+1,∥un-uhn∥j≤C(u,q,α,β)T2α-12-2α(Δt1-2αhmin⁡⁡{k+1,r}+Δt3-2α)+C(q)hr+1+C(u)hk+1-j,j=0,1,
and for α→1(76)∥qn-qhn∥≤C(u,q,β)(Δt-1hmin⁡⁡{k+1,r}+Δt)+C(q)hr+1,∥un-uhn∥j≤C(u,q,β)(Δt-1hmin⁡⁡{k+1,r}+Δt)+C(q)hr+1+C(u)hk+1-j,j=0,1.Proof.
We can use a similar proof as in Theorem5 to get the conclusion of Theorem 8, so we do not discuss that again.


## 4.1. The Analysis of Stability

In the following discussion, we will give the stability for the system (60) and (61). First, we need to obtain an important lemma on two initial value conditions.Lemma 6.
Withq0=∇u0 and qt(0)=∇u1, the following inequality holds:
(62)∥qh-1∥≤C(∥u0∥1+Δt∥u1∥1).Proof.
By a similar discussion as in [32], we have
(63)∥qh-1∥≤C(∥q0∥+Δt∥qt(0)∥).
Based on the given initial value conditions and (63), we arrive at
(64)∥qh-1∥≤C(∥∇u0∥+Δt∥∇u1∥)≤C(∥u0∥1+Δt∥u1∥1).Theorem 7.
The following stable inequality for the system (60) and (61) holds:
(65)∥uhn∥1+∥qhn∥+κΓα∥∇·qhn∥≤C(α)(Γαmax⁡0≤j≤n∥fj∥+∥u0∥1+Δt∥u1∥1).Proof.
In (60), we take vh=∇uhn and use Cauchy-Schwarz inequality and Poincaré inequality to arrive at
(66)∥uhn∥≤∥uhn∥1≤C∥∇uhn∥≤C∥qhn∥.
In (61), we choose wh=qhn and use Cauchy-Schwarz inequality, Young inequality, and (66) to get
(67)Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttqhk-1,qhn)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtqhk,qhn)+∥∇·qhn∥2≤β∥uhn∥∥∇·qhn∥+∥fn∥∥∇·qhn∥≤C∥qhn∥∥∇·qhn∥+∥fn∥∥∇·qhn∥≤C(∥qhn∥2+∥fn∥2)+12∥∇·qhn∥2.
By the simple simplification for (67), we easily get
(68)Δt2-2αΓ(3-2α)∑k=1n‍Bn-k2α(Dttqhk-1,qhn)+2κΔt1-αΓ(2-α)∑k=1n‍Bn-kα(Dtqhk,qhn)+12∥∇·qhn∥2≤C(∥qhn∥2+∥fn∥2).
For the case n=1 in (68), we multiply (36) to get easily
(69)Γ(2-α)B02α(qh1-2qh0+qh-1,qh1)+2κΔtαΓ(3-2α)B0α(qh1-qh0,qh1)+κΓα∥∇·qh1∥2≤CΓα(∥qh1∥2+∥f1∥2).
Noting that B0α=1=B02α and Lemma 6, we use Cauchy-Schwarz inequality to get
(70)(12Γ(2-α)+κΔtαΓ(3-2α)-CΓα)∥qh1∥2+κΓα∥∇·qh1∥2≤CΓα∥f1∥2+C(α)(∥qh-1∥2+∥qh0∥2)≤Γα∥f1∥2+C(α)(∥u0∥12+Δt2∥u1∥12)≤C(α)(Γαmax⁡0≤j≤n∥fj∥+∥u0∥1+Δt∥u1∥1)2.
By the simple calculation for (70), we arrive at
(71)∥qh1∥+κΓα∥∇·qh1∥≤C(α)(Γαmax⁡0≤j≤n∥fj∥+∥u0∥1+Δt∥u1∥1).
For the case n≥1, using a similar process to the proof of Theorem 5 based on the mathematical induction, we can get
(72)∥qhn∥+κΓα∥∇·qhn∥≤C(α)(Γαmax⁡0≤j≤n∥fj∥+∥u0∥1+Δt∥u1∥1).
By a combination of (72) and (66), we get the conclusion of Theorem 7.


## 4.2. A Priori Error Results

For deriving the a priori error analysis, we define the Ritz projectionu~h∈Vh by
(73)(∇(u-u~h),∇vh)=0,vh∈Vh.
Further, let q~h∈Wh be the standard finite element interpolant of q.Letρ=q-q~h, η=u-u~h; see [33, 44]; we obtain
(74)∥η∥j≤Chk+1-j∥u∥k+1,j=0,1,∥ρ∥+h∥ρ∥H(div;Ω)≤Chr+1∥q∥r+1.
Now, we can get the following theorem of a priori error estimates based on the above contents.Theorem 8.
Withq0=∇u0, qt(0)=∇u1, un∈H01∩Hk+1, and qn∈Hr+1, there exists a positive constant C(u,q,α,β) free of space-time meshes h and Δt such that, for 1/2<α<1,
(75)∥qn-qhn∥≤C(u,q,α,β)T2α-12-2α(Δt1-2αhmin⁡⁡{k+1,r}+Δt3-2α)+C(q)hr+1,∥un-uhn∥j≤C(u,q,α,β)T2α-12-2α(Δt1-2αhmin⁡⁡{k+1,r}+Δt3-2α)+C(q)hr+1+C(u)hk+1-j,j=0,1,
and for α→1(76)∥qn-qhn∥≤C(u,q,β)(Δt-1hmin⁡⁡{k+1,r}+Δt)+C(q)hr+1,∥un-uhn∥j≤C(u,q,β)(Δt-1hmin⁡⁡{k+1,r}+Δt)+C(q)hr+1+C(u)hk+1-j,j=0,1.Proof.
We can use a similar proof as in Theorem5 to get the conclusion of Theorem 8, so we do not discuss that again.


## 5. Some Numerical Results

Here, in order to show the numerical performances on the rate of convergence and a priori error estimates, we now chooseκ=1/2 and β=0, the exact solution u(x,t)=t2+αsin(2πx), for all (x,t)∈[0,1]×[0,1], and the determined source term f(x,t) by the exact solution in (5) and then calculate some numerical results by using Matlab procedure. In the numerical calculation, the time direction is approximated by finite difference schemes and the spatial direction is discretized by the H1-GMFE method. Now, we divide interval [0,1] into equal-length intervals ei=[xi,xi+1], 0≤i≤N-1. Now we choose the piecewise linear spaces Vh=span⁡{φ1,φ2,…,φN-1} and Wh=span⁡{ψ0,ψ1,…,ψN} with index k=r=1. Let
(77)uhn=∑k=1N-1‍ukφk,σhn=∑k=0N‍σkψk.
Based on the above expressions (77), we can get the following equivalent algebraic equation of (25):
(78)[(a+b)E+F]σ→n=a∑k=1n-1‍(-Bn-k+12α+2Bn-k2α-Bn-k-12α)Eσ→k-1+a(2B02α-B12α)Eσ→n-1+b∑k=1n‍(-Bn-k+1α+Bn-kα)Eσ→k-1-aBn-12αEσ→-1-f→n,Gu→n=Hσ→n,
where a=Δt-2α/Γ(3-2α), b=Δt-α/Γ(2-α), u→n=(u1n,u2n,…,uN-1n)T, σ→n=(σ0n,σ1n,…,σN-1n,σNn)T, f→n=((fn,ψ0),…,(fn,ψN))T,
(79)E=(h3h6h62h3h6⋱⋱⋱h62h3h6h6h3)(n+1)×(n+1),F=(1h-1h-1h2h-1h⋱⋱⋱-1h2h-1h-1h1h)(n+1)×(n+1),G=(2h-1h-1h2h-1h⋱⋱⋱-1h2h-1h-1h2h)(n-1)×(n-1),H=(120-12⋱⋱⋱120-12)(n-1)×(n+1).Based on the algebraic equation (78), we calculate and get the detailed numerical results listed in Tables 1–4.Table 1
Spatial convergence results inL2-norm with fixed Δt=1/4000 and changed α.
L
2-norm
α
h
1
=
1
20
h
2
=
1
40
h
3
=
1
80
Order(h1h2)lllll
Order(h2h3)ggggg
∥
u
-
u
h
∥
0.6
5.163644E − 03
1.280561E − 03
3.129308E − 04
2.013469
2.039022
0.7
5.051593E − 03
1.251162E − 03
3.044436E − 04
2.013469
2.039022
0.8
4.916797E − 03
1.215362E − 03
2.936933E − 04
2.016333
2.049004
0.9
4.758728E − 03
1.172117E − 03
2.795694E − 04
2.021459
2.067839
∥
σ
-
σ
h
∥
0.6
4.396500E − 03
1.122373E − 03
3.219747E − 04
1.969804
1.801533
0.7
5.141035E − 03
1.312040E − 03
3.759918E − 04
1.970247
1.803038
0.8
6.036700E − 03
1.543010E − 03
4.444122E − 04
1.968013
1.795777
0.9
7.087004E − 03
1.822010E − 03
5.343041E − 04
1.959645
1.769798Table 2
Spatial convergence results inH1-norm with fixed Δt=1/4000 and changed α.
H
1-norm
α
h
1
=
1
20
h
2
=
1
40
h
3
=
1
80
Order(h1h2)lllll
Order(h2h3)ggggg
0
.
6
6.933461E − 01
3.484812E − 01
1.744206E − 01
0.992495
0.998510
∥
u
-
u
h
∥
1
0.7
6.933489E − 01
3.484816E − 01
1.744207E − 01
0.992499
0.998511
0.8
6.933531E − 01
3.484823E − 01
1.744208E − 01
0.992505
0.998513
0.9
6.933591E − 01
3.484833E − 01
1.744210E − 01
0.992513
0.998516
0.6
4.374396E + 00
2.191164E + 00
1.096078E + 00
0.997386
0.999347
∥
σ
-
σ
h
∥
1
0.7
4.374658E + 00
2.191198E + 00
1.096083E + 00
0.997450
0.999363
0.8
4.374979E + 00
2.191241E + 00
1.096089E + 00
0.997528
0.999383
0.9
4.375365E + 00
2.191294E + 00
1.096098E + 00
0.997620
0.999406Table 3
Space-time convergence results inL2-norm with h=5Δt=1/M and changed α.
L
2-norm
α
M
1
=
20
M
2
=
40
M
3
=
80
Order(M1M2)lllll
Order(M2M3)ggggg
∥
u
-
u
h
∥
0.6
4.781196E − 03
1.095478E − 03
2.262375E − 04
2.125811
2.275651
0.7
4.587916E − 03
1.028885E − 03
2.011905E − 04
2.156757
2.354448
0.8
4.319590E − 03
9.315674E − 04
1.630376E − 04
2.213162
2.514455
0.9
3.941100E − 03
7.841911E − 04
2.154685E − 04
2.329321
1.863729
∥
σ
-
σ
h
∥
0.6
6.937717E − 03
2.316455E − 03
8.737355E − 04
1.582542
1.406650
0.7
8.221980E − 03
2.746089E − 03
1.033148E − 03
1.582108
1.410332
0.8
1.000490E − 02
3.373945E − 03
1.275972E − 03
1.568199
1.402840
0.9
1.251982E − 02
4.324761E − 03
1.669151E − 03
1.533521
1.373505Table 4
Space-time convergence results inH1-norm with h=5Δt=1/M and changed α.
H
1-norm
α
M
1
=
20
M
2
=
40
M
3
=
80
Order(M1M2)lllll
Order(M2M3)ggggg
∥
u
-
u
h
∥
1
0.6
6.933582E − 01
3.484856E − 01
1.744222E − 01
0.992502
0.998515
0.7
6.933669E − 01
3.484880E − 01
1.744230E − 01
0.992510
0.998519
0.8
6.933819E − 01
3.484924E − 01
1.744244E − 01
0.992523
0.998525
0.9
6.934089E − 01
3.485010E − 01
1.744275E − 01
0.992543
0.998536
∥
σ
-
σ
h
∥
1
0.6
4.375309E + 00
2.191391E + 00
1.096134E + 00
0.997538
0.999422
0.7
4.375792E + 00
2.191479E + 00
1.096153E + 00
0.997639
0.999456
0.8
4.376484E + 00
2.191614E + 00
1.096183E + 00
0.997779
0.999505
0.9
4.377505E + 00
2.191830E + 00
1.096235E + 00
0.997973
0.999579In Table1, we calculate the numerical results of a priori error results and orders of convergence in L2-norm for both u and σ with a fixed time step length Δt=1/4000 and the changed spatial meshes h1=1/20, h2=1/40, and h3=1/80. From the calculated data, we can clearly see that the orders of convergence in L2-norm keep unchanged with the changed α=0.6,0.7,0.8,0.9, which confirm the optimal second-order convergence results of H1-Galerkin mixed finite element method. Similarly, we obtain optimal first-order rates of convergence of H1-norm for u and σ in Table 2.In Table3, we calculate the numerical results of a priori error results and orders of convergence in L2-norm for both u and σ with different space-time step length h=5Δt=1/M(M=20,40,80). From the calculated data, we can find that the rates of convergence, which are higher than the results O(Δt3-2α+h2) of theory, gradually decrease with the increased α (which is taken from 0.6 to 0.9 with interval 0.1). In Table 4, some first-order convergence results for both u and σ in H1-norm, which are unchanged with the changed α=0.6,0.7,0.8,0.9, are given.In view of the above analysis on the numerical results, we now announce that the time fractional telegraph equation can be well solved by theH1-GMFE method.


## 6. Some Concluding Remarks and Extensions

As far as we know, more and more people have proposed and analyzed a lot of numerical methods for fractional partial differential equations. However, the discussions on mixed finite element methods for solving fractional partial differential equations are fairly limited. The a priori error analysis ofH1-GMFE method for time fractional telegraph equation, especially, has not been made and discussed. In this paper, we give the detailed proof’s process of the error analysis on H1-GMFE method for time fractional telegraph equation. Further, we provide a numerical procedure to verify the theoretical results of the studied method.In the near future, our aim is to study anH1-Galerkin moving mixed finite element method, which is based on a combination of H1-GMFE methods and moving finite element methods.



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*Source: 371413-2014-07-24.xml*