The Scientific World Journal

Volume 2014 (2014), Article ID 141467, 8 pages

http://dx.doi.org/10.1155/2014/141467

## A New Mixed Element Method for a Class of Time-Fractional Partial Differential Equations

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

Received 4 December 2013; Accepted 16 January 2014; Published 9 March 2014

Academic Editors: Q. Liu, F. Soleymani, and G. Tsiatas

Copyright © 2014 Yang Liu et al. 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.

#### Abstract

A kind of new mixed element method for time-fractional partial differential equations is studied. The Caputo-fractional derivative of time direction is approximated by two-step difference method and the spatial direction is discretized by a new mixed element method, whose gradient belongs to the simple space replacing the complex space. Some a priori error estimates in -norm for the scalar unknown and in -norm for its gradient . Moreover, we also discuss a priori error estimates in -norm for the scalar unknown .

#### 1. Introduction

In this paper, we consider the following time-fractional partial differential equation with initial and boundary conditions In (1), is a bounded convex polygonal domain in , with continuous boundary , is the time interval with . and are given functions and is Caputo fractional derivative defined by

Fractional partial differential equations (PDEs) mainly include three types: PDEs with space fractional derivative, PDEs with time-fractional derivative, and PDEs with space-time-fractional derivative. So far, more and more people have started to pay attention to looking for the analytical and numerical solutions of fractional PDEs. In [1–14], authors proposed a lot of finite difference methods for time, space, and space-time-fractional PDEs. Lin and Xu [15] proposed and analyzed the spectral methods for solving time-fractional diffusion equation. In [16, 17], authors presented local discontinuous Galerkin methods for fractional PDEs. Li et al. [18] discussed the detailed error estimate theories of finite element methods for nonlinear space-time-fractional differential equations with subdiffusion and superdiffusion. Jiang and Ma [19] developed high-order finite element methods for one-dimensional time fractional PDE (1). In [20, 21], the finite element methods were analyzed for space fractional PDEs. In [22–24], some time-fractional PDEs were solved by the finite element methods. Zhao and Li [25] presented the fractional difference/finite element approximations for the space-time-fractional telegraph equation.

Based on the summary of the above numerical methods for solving fractional PDEs, we can see that many numerical methods, such as finite difference methods, LDG methods, finite element methods, and spectral methods, have been studied and developed. However mixed finite element methods for solving fractional PDEs have not been proposed in the current literatures.

In recent years, a lot of mixed finite element methods have been proposed by many mathematical scholars. In [26, 27], authors presented a new mixed finite element method based on the linear elliptic equations. Compared to classical mixed methods, this method has several distinct characteristics: the gradient of the new one belongs to the simple space avoiding space, the optimal a priori error estimates in -norm for the scalar unknown can be obtained, the number of total degrees of freedom for this method is less than that for classical mixed methods, and the regularity requirements on the solution are reduced. In view of the method's characteristics, the new mixed method has been developed to solve some integer-order partial differential equations, such as parabolic equation [28–31], Sobolev equation [32], fourth-order parabolic equation [33], and extended Fisher-Kolmogorov equation [34].

In this paper, our aim is to study the new numerical method based on the new mixed finite element method [26, 27] for solving a class of time-fractional PDEs. We derive a new discrete method for time-fractional derivative, formulate a fully discrete mixed finite element scheme, and prove some a priori error estimates in for the scalar unknown and in -norm for its gradient . What is more, we derive an a priori error estimate in -norm for the scalar unknown .

The layout of the paper is as follows. In Section 2, we introduce a new discrete method for the Caputo time-fractional derivative and give the ‘‘proof’’ of the truncation error’s boundedness. In Section 3, we formulate a new mixed scheme for time-fractional PDE (1) and give the detailed proof for the a priori error estimates for two important variables based on fully discrete scheme. In Section 4, we give some remarks and extensions about the new mixed method and fractional PDEs. Throughout this paper, will denote a generic constant independent of the space-time discretization parameter and . At the same time, we denote the natural inner product in or by with the corresponding norm . The other notations and definitions of Sobolev spaces as in [35, 36] are used.

#### 2. Approximation of Time-Fractional Derivative

For the discretization for time-fractional derivative, let be a given partition of the time interval with step length and nodes , for some positive integer . For a smooth function on , define .

Lemma 1. *The time-fractional derivative at is approximated by the following: for **
where
*

*Proof. *As [37], using Taylor’s expansion at time , we can arrive at
By (5), Taylor’s expansion, and some simple calculations of definite integral, we have
So, the conclusion of Lemma 1 has been arrived at by the above calculations.

*Remark 2. *In a number of studies [18, 19, 22], the time-fractional derivative with order is discretized by
However, the study on the discrete formulation (3) for the Caputo fractional derivative with order is fairly limited.

*Lemma 3. The truncation error is bounded by
where .*

*Proof. *By the simple calculations, we arrive at
From (9), we can see easily that the conclusion for Lemma 3 is obtained.

*3. New Mixed Finite Element Method*

*3.1. Mixed Formulation and Projections*

*In order to get the mixed scheme, we first split (1) into the following coupled system of two lower-order equations by introducing an auxiliary variable :
*

*Based on the new mixed method in [26, 27], using Green's formula, the new mixed weak formulation of (10) is to determine such that *

*
In order to formulate a new mixed finite element scheme, we first define the mixed finite element spaces. As shown in the literatures [26, 27], we choose the mixed space with finite element pair as
As discussed in [26, 27], we know that satisfies the so-called discrete Ladyzhenskaya-Babuska-Brezzi condition.*

*In view of the definition of the above mixed space, the corresponding semidiscrete mixed scheme of (11a) and (11b) is to find such that *

*Remark 4. *(i) If the standard mixed method is considered, the mixed weak formulation for problem (1) is to find such that where , .

(ii) Compared with the classical mixed weak formulation (14a) and (14b), the gradient in our scheme (11a) and (11b) belongs to the simple square integrable space avoiding the use of the complex space. Obviously, the regularity requirements on the solution is reduced.

(iii) So far, we have not seen any related reports on the study of mixed finite element methods for solving Fractional PDEs. Here, we will give some detailed theoretical analysis on a kind of new mixed element method for solving the fractional PDE (1).

*In order to analyze the convergence of the method, we first introduce two mixed elliptic projection associated with our equations.*

*Lemma 5. There exists a linear operator such that
*

*Lemma 6. There exists a linear operator such that
*

*From [26–28], we can obtain the proof for Lemmas 5 and 6.*

*3.2. A Priori Error Estimates for Fully Discrete Scheme*

*3.2. A Priori Error Estimates for Fully Discrete Scheme**In the following discussion, we will analyze some a priori error estimates for fully discrete schemes based on the case . For the convenience of theoretical analysis, we now denote
By the discrete formula (3) for time-fractional derivative, (11a) and (11b) have the following equivalent formulation:*

*
Now, we formulate a completely discrete procedure: find , such that *

*
For the convenience of the analysis, we now decompose the errors as
*

*Subtracting (21a) and (21b) from (20a) and (20b) and using two projections (15) and (17), we get the error equationsIn the following discussion, we will derive the detailed process of proof for the fully discrete a priori error estimates.*

*Theorem 7. Supposing that , ; then the error estimates hold with a parameter : *

*Proof. *Noting that
Then (23a) may be rewritten as
We add (26) to (23b), take , and multiply by to arrive at
Now we consider the first term on the left-hand side of (27). Noting that and denoting , we have
Substitute (28) into (27) to arrive at
For (29), we use Cauchy-Schwarz inequality to get
By (30), we can arrive at
Now, we use induction to prove the conclusion (31).*Step 1*. Setting in (30) and noting that and , we can arrive easily at
So, when , (31) holds. *Step 2*. Supposing that (31) holds, for ,
Now, we consider the case for . By (30) and the supposition (33), we have
Noting that in inequality (34); then we have
In order to obtain the estimate for (35), we have to discuss the boundedness for . Noting that , we have
Combining (36) with (35), we arrive at
Making use of induction based on (32) and (37), we claim that (31) holds.

Note that the relationship [15] holds; then we have
By a substitution (38) into (31), we get
By a combination of (16) and (18) with triangle inequality, we get the error results of theorem.

*Theorem 8. With the same condition, one has the following a priori error estimates:
*

*Proof. *By (21a) and (21b), we easily get
We take in (41) and in (41) and use Cauchy-Schwarz inequality and Young’s inequality to get
By (42), we have
By a similar discussion to Theorem 7, we get
Taking in (23b) and using (31), we arrive at
Combining (44), (45), (16), and (18) with triangle inequality, we complete the proof.

*Remark 9. *It is easy to find that a priori error estimate in -norm for the variable , which cannot be derived based on the classical mixed scheme (14a) and (14b), is gotten.

*4. Some Concluding Remarks and Extensions*

*4. Some Concluding Remarks and Extensions**As far as I know, the mixed finite element methods for fractional partial differential equations have not been proposed and studied. In this paper, our purpose is to present and analyze a kind of novel mixed finite element method for seeking the numerical solution of time-fractional partial differential equation with () order derivative. We discuss two-step difference method in time direction (the approximations of the time-fractional derivative) and a class of new mixed finite element methods proposed by [26, 27] in spatial direction. We obtain some a priori error estimates in for the scalar unknown and in -norm for its gradient . What is more, an a priori error estimate in -norm for the scalar unknown is derived, too.*

*In the near future, we will develop the new mixed finite element method to solve two-dimensional time-fractional Tricomi-type equations, fractional telegraph equation, and so on. At the same time, we will try to find some new approximation method for fractional derivatives and to study some other mixed finite element methods for seeking the numerical solutions of the fractional PDEs.*

*Conflict of Interests*

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

*Acknowledgments*

*Acknowledgments**The authors thank the editors and reviewers for their very valuable comments and suggestions, which greatly improved the paper. This work is supported by the National Natural Science Fund (11301258, 11361035), the Key Project of Chinese Ministry of Education (12024), the Natural Science Fund of Inner Mongolia Autonomous Region (2012MS0108, 2012MS0106, and 2011BS0102), the Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011, NJZY13199), the Program of Higher-Level Talents of Inner Mongolia University (125119, 30105-125132).*

*References*

*References*

- D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, Series on Complexity, Nonlinearity and Chaos, World Scientific, 2012. - A. Atangana and D. Baleanu, “Numerical solution of a kind of fractional parabolic equations via two difference schemes,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 828764, 8 pages, 2013. View at Publisher · View at Google Scholar - P. Zhuang, F. Liu, V. Anh, and I. Turner, “Numerical methods for the variable-order fractional advection diffusion equation with a nonlinear source term,”
*SIAM Journal on Numerical Analysis*, vol. 47, no. 3, pp. 1760–1781, 2009. View at Google Scholar - M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for fractional advection-dispersion flow equations,”
*Journal of Computational and Applied Mathematics*, vol. 172, no. 1, pp. 65–77, 2004. View at Publisher · View at Google Scholar · View at Scopus - C.-M. Chen, F. Liu, and K. Burrage, “Finite difference methods and a fourier analysis for the fractional reaction-subdiffusion equation,”
*Applied Mathematics and Computation*, vol. 198, no. 2, pp. 754–769, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. Shen, F. Liu, and V. Anh, “Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation,”
*Numerical Algorithms*, vol. 56, no. 3, pp. 383–403, 2011. View at Publisher · View at Google Scholar · View at Scopus - M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for two-sided space-fractional partial differential equations,”
*Applied Numerical Mathematics*, vol. 56, no. 1, pp. 80–90, 2006. View at Publisher · View at Google Scholar · View at Scopus - F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, “Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation,”
*Applied Mathematics and Computation*, vol. 191, no. 1, pp. 12–20, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. Shen, F. Liu, V. Anh, I. Turner, and J. Chen, “A characteristic difference method for the variable-order fractional advection-diffusion equation,”
*Journal of Applied Mathematics and Computing*, vol. 42, no. 1-2, pp. 371–386, 2013. View at Google Scholar - K. Wang and H. Wang, “A fast characteristic finite difference method for fractional advection-diffusion equations,”
*Advances in Water Resources*, vol. 34, no. 7, pp. 810–816, 2011. View at Publisher · View at Google Scholar · View at Scopus - R. Lin, F. Liu, V. Anh, and I. Turner, “Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation,”
*Applied Mathematics and Computation*, vol. 212, no. 2, pp. 435–445, 2009. View at Publisher · View at Google Scholar · View at Scopus - A. Ashyralyev and F. Dal, “Finite difference and iteration methods for fractional hyperbolic partial differential equations with the Neumann condition,”
*Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 434976, 15 pages, 2012. View at Publisher · View at Google Scholar - S. B. Yuste, “Weighted average finite difference methods for fractional diffusion equations,”
*Journal of Computational Physics*, vol. 216, no. 1, pp. 264–274, 2006. View at Publisher · View at Google Scholar · View at Scopus - E. Sousa, “A second order explicit finite difference method for the fractional advection diffusion equation,”
*Computers and Mathematics with Applications*, vol. 64, no. 10, pp. 3141–3152, 2012. View at Publisher · View at Google Scholar · View at Scopus - Y. Lin and C. Xu, “Finite difference/spectral approximations for the time-fractional diffusion equation,”
*Journal of Computational Physics*, vol. 225, no. 2, pp. 1533–1552, 2007. View at Publisher · View at Google Scholar · View at Scopus - L. L. Wei, Y. N. He, X. D. Zhang, and S. L. Wang, “Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation,”
*Finite Elements in Analysis & Design*, vol. 59, pp. 28–34, 2012. View at Google Scholar - L. L. Wei, Y. N. He, and Y. Zhang, “Numerical analysis of the fractional seventh-order KdV equation using an implicit fully discrete local discontinuous Galerkin method,”
*International Journal of Numerical Analysis and Modeling*, vol. 10, no. 2, pp. 430–444, 2013. View at Google Scholar - C. Li, Z. Zhao, and Y. Chen, “Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion,”
*Computers and Mathematics with Applications*, vol. 62, no. 3, pp. 855–875, 2011. View at Publisher · View at Google Scholar · View at Scopus - Y. Jiang and J. Ma, “High-order finite element methods for time-fractional partial differential equations,”
*Journal of Computational and Applied Mathematics*, vol. 235, no. 11, pp. 3285–3290, 2011. View at Publisher · View at Google Scholar · View at Scopus - Y. Zheng, C. Li, and Z. Zhao, “A note on the finite element method for the space-fractional advection diffusion equation,”
*Computers and Mathematics with Applications*, vol. 59, no. 5, pp. 1718–1726, 2010. View at Publisher · View at Google Scholar · View at Scopus - H. Zhang, F. Liu, and V. Anh, “Galerkin finite element approximation of symmetric space-fractional partial differential equations,”
*Applied Mathematics and Computation*, vol. 217, no. 6, pp. 2534–2545, 2010. View at Publisher · View at Google Scholar · View at Scopus - Y. J. Jiang and J. T. Ma, “Moving finite element methods for time fractional partial differential equations,”
*Science China Mathematics*, vol. 56, no. 6, pp. 1287–1300, 2013. View at Google Scholar - N. J. Ford, J. Xiao, and Y. Yan, “A finite element method for time fractional partial differential equations,”
*Fractional Calculus and Applied Analysis*, vol. 14, no. 3, pp. 454–474, 2011. View at Publisher · View at Google Scholar · View at Scopus - X. D. Zhang, P. Z. Huang, X. L. Feng, and L. L. Wei, “Finite element method for twodimensional time-fractional Tricomi-type equations,”
*Numerical Methods for Partial Differential Equations*, vol. 29, no. 4, pp. 1081–1096, 2013. View at Google Scholar - Z. G. Zhao and C. P. Li, “Fractional difference/finite element approximations for the timespace fractional telegraph equation,”
*Applied Mathematics and Computation*, vol. 219, no. 6, pp. 2975–2988, 2012. View at Google Scholar - S. C. Chen and H. R. Chen, “New mixed element schemes for a second-order elliptic problem,”
*Mathematica Numerica Sinica*, vol. 32, no. 2, pp. 213–218, 2010. View at Google Scholar - F. Shi, J. Yu, and K. Li, “A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair,”
*International Journal of Computer Mathematics*, vol. 88, no. 11, pp. 2293–2305, 2011. View at Publisher · View at Google Scholar · View at Scopus - L. Li, P. Sun, and Z. D. Luo, “A new mixed finite element scheme and error estiamtes for parabolic equations,”
*Acta Mathematica Scientia*, vol. 32, no. 6, pp. 1158–1165, 2012. View at Google Scholar - Y. Liu, H. Li, W. Gao, S. He, and Z. C. Fang, “A novel characteristic expanded mixed method for reaction-convection-diffusion problems,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 683205, 11 pages, 2013. View at Publisher · View at Google Scholar - D. Y. Shi and Q. L. Tang, “A new characteristic nonconforming mixed finite element scheme for convection-dominated diffusion problem,”
*Journal of Applied Mathematics*, vol. 2013, Article ID 951692, 10 pages, 2013. View at Publisher · View at Google Scholar - Z. F. Weng, X. L. Feng, and D. M. Liu, “A fully discrete stabilized mixed finite element method for parabolic problems,”
*Numerical Heat Transfer A*, vol. 63, no. 10, pp. 755–775, 2013. View at Google Scholar - D. Shi and Y. Zhang, “High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equations,”
*Applied Mathematics and Computation*, vol. 218, no. 7, pp. 3176–3186, 2011. View at Publisher · View at Google Scholar · View at Scopus - Y. Liu, H. Li, Z. C. Fang, S. He, and J. Wang, “A coupling method of new EMFE and FE for fourth-order partial differential equation of parabolic type,”
*Advances in Mathematical Physics*, vol. 2013, Article ID 787891, 14 pages, 2013. View at Publisher · View at Google Scholar - J. F. Wang, H. Li, S. He, W. Gao, and Y. Liu, “A new linearized Crank-Nicolson mixed element scheme for the extended Fisher-Kolmogorov equation,”
*The Scientific World Journal*, vol. 2013, Article ID 756281, 11 pages, 2013. View at Publisher · View at Google Scholar - P. G. Ciarlet,
*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam, The Netherlands, 1978. - Z. D. Luo,
*Mixed Finite Element Methods and Applications*, Chinese Science Press, Beijing, China, 2006. - Y. Liu, H. Li, Y. W. Du, and J. F. Wang, “Explicit multistep mixed finite element method for RLW equation,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 768976, 12 pages, 2013. View at Publisher · View at Google Scholar

*
*