Numerical Analysis of an H1 -Galerkin Mixed Finite Element Method for Time Fractional Telegraph Equation

Abstract

We discuss and analyze an [superscript]H1[/superscript] -Galerkin mixed finite element ([superscript]H1[/superscript] -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 [superscript]H1[/superscript] -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 [superscript]H1[/superscript] -GMFE method. Based on the discussion on the theoretical error analysis in [superscript]L2[/superscript] -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 [superscript]H1[/superscript] -norm. Moreover, we derive and analyze the stability of [superscript]H1[/superscript] -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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Jinfeng Wang 1 and Meng Zhao 2 and Min Zhang 2 and Yang Liu 2 and Hong Li 2

Academic Editor:Yui Chuin Shiah

1, School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China
2, School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

Received 7 April 2014; Revised 10 July 2014; Accepted 13 July 2014; 24 July 2014

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper, our purpose is to present and discuss a mixed finite element method for the time fractional telegraph equation [figure omitted; refer to PDF] with boundary condition [figure omitted; refer to PDF] and initial conditions [figure omitted; refer to PDF] 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...5;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,tα u(x,t) and ^∂0,t2α u(x,t) are defined, respectively, by [figure omitted; refer to PDF] 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 on ^H1 -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 the ^H1 -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 Section 2, 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 classical Sobolev spaces ^Wm,2 (Ω) as ^Hm with norm _||z||m =[_{∑0...4;|β|...4;m∫Ω} ...|^Dβ z^|2 dx^]1/2 . When m=0 , we simply write the norm _||·||0 as ||·|| .

2. An ^H1 -GMFE Scheme in One Space Dimension

In this section, we first consider the ^H1 -GMFE method for the following time fractional telegraph equation in 1D case: [figure omitted; refer to PDF] with boundary condition [figure omitted; refer to PDF] and initial conditions [figure omitted; refer to PDF] where Ω=[_xL ,_xR ]⊂R .

In order to get the ^H1 -GMFE formulation, we first introduce an auxiliary variable σ=∂u(x,t)/∂x and split (5) into the following first-order system by [figure omitted; refer to PDF]

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 ^∂2 u(_xL ,t)/∂^t2 =^∂2 u(_xR ,t)/∂^t2 =0 , we easily get [figure omitted; refer to PDF] Multiply (9) by ∂v/∂x , v∈^H01 , and integrate with respect to space from _xL to _xR to obtain [figure omitted; refer to PDF] 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...4;p...4;∞ and k , r positive integers [33], [figure omitted; refer to PDF] Based on the chosen finite element spaces, the semidiscrete ^H1 -GMFE scheme is described by [figure omitted; refer to PDF]

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 projection _Ph u∈_Vh for the variable u by [figure omitted; refer to PDF] Then the following estimates hold, for j=0,1 : [figure omitted; refer to PDF]

Lemma 2.

Further, one also defines an elliptic projection _Rh σ∈_Wh of σ as the solution of [figure omitted; refer to PDF] where B(σ,w)=(_σx ,_wx )+λ(σ,w) . Here λ>0 is chosen to satisfy [figure omitted; refer to PDF] Then the following estimates are found: for j=0,1 , [figure omitted; refer to PDF]

3.2. Approximation of Time-Fractional Derivative

For formulating fully discrete scheme, let 0=_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 [varphi] on [0,T] , define ^[varphi]n =[varphi](_tn ) . In the following analysis, for deriving the convenience of theoretical process, we now denote [figure omitted; refer to PDF] Now, we will introduce two lemmas on the approximations of time fractional derivatives.

Lemma 3 (see [27]).

The time fractional derivative ^∂0,tα σ(x,t) at t=_tn is approximated by, for 0<α<1 , [figure omitted; refer to PDF] and then holds [figure omitted; refer to PDF]

Lemma 4 (see [1]).

The time fractional order derivative ^∂0,t2α σ(x,t) at t=_tn is estimated by, for 1<2α<2 , [figure omitted; refer to PDF] and then holds [figure omitted; refer to PDF]

In the next analysis, we will derive and prove some a priori error results for u 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): [figure omitted; refer to PDF] where ^E0n =^Eαn +^E2αn .

Now, we formulate a fully discrete procedure: find (^uhn ,^σhn )∈_Vh ×_Wh (n=1,...,M-1) such that [figure omitted; refer to PDF]

Making a combination of (24)-(25) with two projections (14) and (16), we get the following error equations: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

In the following discussion, we will derive the proof for the fully discrete a priori error estimates.

Theorem 5.

With _σ0 =_u0x and _σt (0)=_u1x , suppose that ^un ∈^H01 ∩^Hk+1 , ^σn ∈^Hr+1 , ^σh0 =_Rh σ(0) , and _σht (0)=_Lhσt (0) , where _Lh is the ^L2 projection defined by (w-_Lh w,_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 , [figure omitted; refer to PDF] and for α[arrow right]1 [figure omitted; refer to PDF]

Proof.

For the need of error analysis, we first consider the ^L2 -norm ||^[vartheta]n || and the ^H1 -norm _{||^[vartheta]n ||1} . Taking _vh =^[vartheta]n in (26) and using Poincaré inequality based on the ^H01 -space and Cauchy-Schwarz inequality, we easily get [figure omitted; refer to PDF] In the next analysis, we will give the estimates of ||^δn || and ||^[vartheta]n || in ^L2 -norm. Noting that [figure omitted; refer to PDF] then (27) may be rewritten as [figure omitted; refer to PDF] We take _wh =^δn in (35) and multiply by [figure omitted; refer to PDF] to arrive at [figure omitted; refer to PDF] By the simple calculation, we get the following equality: [figure omitted; refer to PDF] By applying the similar process of calculation to (38), we use Cauchy-Schwarz inequality to get [figure omitted; refer to PDF] Noting (33), we use Cauchy-Schwarz inequality and Young inequality to have [figure omitted; refer to PDF] Substitute (38), (39), and (40) into (37) and use Cauchy-Schwarz inequality to arrive at [figure omitted; refer to PDF] By an application of Young inequality, we get [figure omitted; refer to PDF]

Noting that ^Bk2α -^Bk+12α >0 , we arrive at [figure omitted; refer to PDF] At the same time, noting that ^Bkα -^Bk+1α >0 , we use the similar method to have [figure omitted; refer to PDF] By a combination of (43) and (44) with (18) and noting that ^B0α =^B02α =1 , we arrive at [figure omitted; refer to PDF] Substitute (45) into (42) and note that _{μ0||^δn ||1} ...4;B(^δn ,^δn )/_{||^δn ||1} ...4;B(^δn ,^δn )/||^δn || to get [figure omitted; refer to PDF] In the following discussion, we apply mathematical induction to obtain the error result [figure omitted; refer to PDF] Take n=1 in (46) and note that ^B02α =1 ; it is easy to find that the following inequality holds: [figure omitted; refer to PDF] Assuming that the inequalities ||^δj ||...4;(C(u,σ,α,β)/^Bj-12α )(^hk+1 +^hr+1 +Δ^t2 +Δ^t1+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 [figure omitted; refer to PDF] Based on the process of mathematical induction, we claim that (47) holds.

For the need of the next proof, we have to estimate the term ^n1-2α /^Bn-12α . We now use Taylor formula to arrive at [figure omitted; refer to PDF] where 0<θ<1 and 1/2<α<1 . Noting that (nΔt^)2α-1 ...4;^T2α-1 and (50), we have [figure omitted; refer to PDF] Substitute (51) into (33) and use (18) to get [figure omitted; refer to PDF] 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 case α[arrow right]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)[arrow right]∞ as α[arrow right]1 . So, we have to look for another error estimate's process. Noting the fact that nΔt...4;T (n=1,2,...,M ), we can obtain the following error inequality: [figure omitted; refer to PDF]

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. An ^H1 -GMFE Scheme for Several Spaces Variables

In this section, we consider (1) with two and three spaces variables.

Let ^L2 (Ω)=(^L2 (Ω)^)d , (d=2 or 3) with inner product and norm [figure omitted; refer to PDF] Further, let [figure omitted; refer to PDF] with norm [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] The corresponding time semidiscrete system is defined by [figure omitted; refer to PDF] 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 spaces _Vh ⊂^H01 and _Wh ⊂W , which satisfy the following approximation properties: for 1...4;p...4;∞ and k , r positive integers [33], [figure omitted; refer to PDF] The fully discrete ^H1 -GMFE scheme is to find {^uhn ,^qhn }:[0,T]..._Vh ×_Wh such that [figure omitted; refer to PDF]

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.

With _q0 =∇_u0 and _qt (0)=∇_u1 , the following inequality holds: [figure omitted; refer to PDF]

Proof.

By a similar discussion as in [32], we have [figure omitted; refer to PDF] Based on the given initial value conditions and (63), we arrive at [figure omitted; refer to PDF]

Theorem 7.

The following stable inequality for the system (60) and (61) holds: [figure omitted; refer to PDF]

Proof.

In (60), we take _vh =∇^uhn and use Cauchy-Schwarz inequality and Poincaré inequality to arrive at [figure omitted; refer to PDF] In (61), we choose _wh =^qhn and use Cauchy-Schwarz inequality, Young inequality, and (66) to get [figure omitted; refer to PDF] By the simple simplification for (67), we easily get [figure omitted; refer to PDF] For the case n=1 in (68), we multiply (36) to get easily [figure omitted; refer to PDF] Noting that ^B0α =1=^B02α and Lemma 6, we use Cauchy-Schwarz inequality to get [figure omitted; refer to PDF] By the simple calculation for (70), we arrive at [figure omitted; refer to PDF] For the case n...5;1 , using a similar process to the proof of Theorem 5 based on the mathematical induction, we can get [figure omitted; refer to PDF] 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 projection _u~h ∈_Vh by [figure omitted; refer to PDF] Further, let _q~h ∈_Wh be the standard finite element interpolant of q .

Let ρ=q-_q~h , η=u-_u~h ; see [33, 44]; we obtain [figure omitted; refer to PDF] Now, we can get the following theorem of a priori error estimates based on the above contents.

Theorem 8.

With _q0 =∇_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 , [figure omitted; refer to PDF] and for α[arrow right]1 [figure omitted; refer to PDF]

Proof.

We can use a similar proof as in Theorem 5 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 choose κ=1/2 and β=0 , the exact solution u(x,t)=^t2+α sin(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...4;i...4;N-1 . Now we choose the piecewise linear spaces _Vh =span...{_{[straight phi]1} ,_{[straight phi]2} ,...,_{[straight phi]N-1} } and _Wh =span...{_ψ0 ,_ψ1 ,...,_ψN } with index k=r=1 . Let [figure omitted; refer to PDF] Based on the above expressions (77), we can get the following equivalent algebraic equation of (25): [figure omitted; refer to PDF] where a=Δ^t-2α /Γ(3-2α) , b=Δ^t-α /Γ(2-α) , ^{u[arrow right]n} =(^u1n ,^u2n ,...,^uN-1n)T , ^{σ[arrow right]n} =(^σ0n ,^σ1n ,...,^σN-1n ,^σNn)T , ^{f[arrow right]n} =((^fn ,_ψ0 ),...,(^fn ,_ψN )^)T , [figure omitted; refer to PDF]

Based on the algebraic equation (78), we calculate and get the detailed numerical results listed in Tables 1-4.

Table 1: Spatial convergence results in ^L2 -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 )ll l l l	Order (h2h3 )gg g g g
|| 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	1.769798

Table 2: Spatial convergence results in ^H1 -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 )ll l l l	Order (h2h3 )gg g g g
	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	0.999406

Table 3: Space-time convergence results in ^L2 -norm with h=5Δt=1/M and changed α .

L 2 -norm	α	M 1 = 20	M 2 = 40	M 3 = 80	Order (M1M2 )ll l l l	Order (M2M3 )gg g g g
|| 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	1.373505

Table 4: Space-time convergence results in ^H1 -norm with h=5Δt=1/M and changed α .

H 1 -norm	α	M 1 = 20	M 2 = 40	M 3 = 80	Order (M1M2 )ll l l l	Order (M2M3 )gg g g g
|| 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	0.999579

In Table 1, 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 Table 3, 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(Δ^t3-2α +^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 the ^H1 -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 of ^H1 -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 an ^H1 -Galerkin moving mixed finite element method, which is based on a combination of ^H1 -GMFE methods and moving finite element methods.

Acknowledgments

The authors thank the anonymous referees and editor for their valuable comments and suggestions, which greatly help them improve this work. This work is supported by the National Natural Science Fund (11301258, 11361035), the Natural Science Fund of Inner Mongolia Autonomous Region (2012MS0108, 2012MS0106), and the Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011, NJZY13199).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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