A New Mixed Element Method for a Class of Time-Fractional Partial Differential Equations
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 .
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  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.  discussed the detailed error estimate theories of finite element methods for nonlinear space-time-fractional differential equations with subdiffusion and superdiffusion. Jiang and Ma  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  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 , fourth-order parabolic equation , and extended Fisher-Kolmogorov equation .
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 , 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 .
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 :
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.
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
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  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.
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
The authors declare that there is no conflict of interests regarding the publication of this paper.
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).
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