Leapfrog/Finite Element Method for Fractional Diffusion Equation

We analyze a fully discrete leapfrog/Galerkin finite element method for the numerical solution of the space fractional order (fractional for simplicity) diffusion equation. The generalized fractional derivative spaces are defined in a bounded interval. And some related properties are further discussed for the following finite element analysis. Then the fractional diffusion equation is discretized in space by the finite element method and in time by the explicit leapfrog scheme. For the resulting fully discrete, conditionally stable scheme, we prove an L 2-error bound of finite element accuracy and of second order in time. Numerical examples are included to confirm our theoretical analysis.


Introduction
Fractional calculus and fractional partial differential equations (FPDEs) have many applications in various aspects such as in viscoelastic mechanics, power-law phenomenon in fluid and complex network, allometric scaling laws in biology and ecology, colored noise, electrode-electrolyte polarization, dielectric polarization, boundary layer effects in ducts, electromagnetic waves, quantitative finance, quantum evolution of complex systems, and fractional kinetics [1]. And a lot of attention has recently been paid to the problem of the numerical approximation of FPDEs.
Generally speaking, the finite difference method and the finite element method are the two main means to solve FPDEs. Recently, some typical fractional difference methods have been utilized to solve FPDEs numerically [2][3][4]. On the other hand, the finite element method has also been used to find the variational solution of FPDEs [5][6][7][8][9][10][11][12][13][14]. But there are still some interesting schemes that can be constructed to enhance the convergence order by using the finite difference/finite element mixed method.
In this paper, we use the explicit leapfrog difference/ Galerkin finite element mixed method to numerically solve the space fractional diffusion equation in order to get a higher convergence order.
The fractional diffusion equation as a typical kind of fractional partial differential equation [15] is a generalization of the classical diffusion equation, which can be used to better characterize anomalous diffusion phenomena. Besides, the spatial fractional diffusion equation usually describes the Lévy flights. The operator RL 2 , ( RL 2 , ) is commonly referred to the left (right) sided Lévy stable distribution, where the underlying stochastic process is Lévy -stable flights; see [16][17][18]. And a more general form 1 ⋅ RL 2 , + 2 ⋅ RL 2 , is widely used for mathematical modelling and numerical computation.
Here, we mainly focus on constructing and analyzing a kind of efficient numerical schemes for approximately solving space fractional diffusion equation. The considered problem reads as follows: for 1/2 < < 1, Here the spatial fractional differential operator Δ is denoted by 1 where 0 ≤ 1 , 2 ≤ 1, and 1 + 2 = 1. When = 1, the problem models a Brownian diffusion process. And is a source term, is a positive constant.
The rest of this paper is constructed as follows. In Section 2, the preliminary knowledge of fractional derivative and the generalized fractional derivative spaces are defined. And some related properties are further discussed. The approximate system of the equation, existence and uniqueness of the weak solution, and the error estimates of the fully discrete scheme for (1) are studied in Section 3. In Section 4, numerical examples are presented to demonstrate the efficiency of the theoretical results derived in Section 3.

Generalized Fractional Derivative Spaces
In this section, we first give the definition of fractional derivatives. There are several definitions for the fractional derivatives, but Riemann-Liouville derivative is one of the most often used fractional derivatives, which is a reasonable generalization of the classical derivative [1,[19][20][21][22]. Then we define the generalized fractional derivative spaces by using Riemann-Liouville derivative, which is extended from the 2 sense to the sense.
in which − 1 < < ∈ + . Obviously, they are the integer derivatives of the left and right fractional integrals, respectively. Now, we give some lemmas and corollaries which are necessary to define the generalized fractional derivative spaces.
Corollary 7 (see [13]). Consider under the assumption that ( ) ∈ (Ω), such that ∈ (Ω), which is obtained by Lemma 3. And RL , ( ) ∈ (Ω) naturally holds. So, by the above idea, we define the following fractional derivative spaces from the 2 sense to the sense, which will be proved to be equivalent with the fractional Sobolev spaces under some certain conditions. Definition 8. Define the following norms of the left (with symbol , ) fractional derivative space and the right (with symbol , ) fractional derivative space in a bounded interval Ω = [ , ] as follows correspondingly, where 1 < < +∞: and norm and norm Definition 9. Define the symmetric fractional derivative space (with symbol ) in a bounded interval Ω = [ , ] in the 2 sense equipped with seminorm and norm From [6], we can get the following lemma, which is true in the 2 sense.
,0 (Ω), and 0 (Ω) are equal to equivalent seminorms and norms, where (Ω) is the fractional Sobolev space in terms of the Fourier transform.
Therefore, in this paper we always use 0 when = 2, to denote the fractional derivative space equipped with the norm ‖ ⋅ ‖ which can be any one of (12), (15), and (18), and Moreover, we can present some new properties about norms for the above left and right fractional derivative spaces in the sense. Proof. Properties (1) and (2) Property (4) follows similarly.

Corollary 13. Consider
It is obviously true by using the norms of fractional derivative spaces and imbedding theorems for (Ω).

Error Estimates of the Leapfrog/Finite Element Scheme
In this section, we firstly give a fully discrete scheme, where we use the leapfrog difference method in the temporal direction and the finite element method in the spatial direction and then analyze the error estimate. Let ℎ denote a uniform partition on Ω, with grid parameter ℎ. For ∈ , let (Ω) denote the space of polynomials on Ω with degree not greater than . Then we define ℎ as the finite element space on ℎ with the basis of the piecewise polynomials of order ∈ + ; that is, in which is the unit of ℎ .
The following property of finite element spaces is necessary for our subsequent analysis [23]: for ∈ +1 (Ω), 0 ≤ ≤ + 1, there exists V ∈ ℎ such that The Gronwall's lemma is also needed for the error analysis.
In the following, we give the fully discrete scheme of (1). Let Δ denote the step size for so that = Δ , = 1, 2, . . . , − 1. For notational convenience, we denote := (⋅, ) and The Scientific World Journal 5 Let ℎ of (1) be the finite element solution at time = of the following fully discrete scheme: that is, where (⋅, ⋅) is denoted by an 2 inner product and (Δ ( ⋅ . For brevity, we always use (Δ ( ⋅ ℎ ), V) instead of the right hand side of this equation.
Obviously the true solution of this problem (1) also satisfies Therefore, subtracting (36) from (43) gives that is, (45) After adding ‖ ‖ 2 to both sides of (46), we obtain the identity Define now the quantity +1 , for 1 ≤ ≤ − 1, by We can rewrite (47) as The Scientific World Journal Denoting then (49) can be abbreviated as We now estimate each term in ( −1 , , +1 ). For the second term of the right hand side, one has where For the third term of the right hand side, one has in which For the fourth term of the right hand side, one has And for the term 2Δ ( − , +1 + −1 ), by using the Cauchy-Schwarz inequality, we obtain where by Taylor's theorem Hence, summing from = 1 to − 1, one has that is, The Scientific World Journal 7 We now show that, under our stability assumption (40), +1 is positive and comparable to ‖ ‖ 2 + ‖ +1 ‖ 2 . To this end, we use the inverse inequality ‖V‖ ≤ 12 ℎ − ‖V‖, V ∈ ℎ , and this yields Hence, if Δ ⋅ ℎ −2 is sufficiently small such that 12 Δ ⋅ ℎ −2 ≤ 13 ≤ 1, we get So we have that Therefore, we obtain Hence, where denoting Now denoting (Δ , ℎ) = ℎ 2 +2 ‖ ‖ 2 0, +1 +ℎ 2 +2−2 ‖ ‖ 2 0, +1 +(Δ ) 4 ‖ ‖ 2 0,0 and using the condition (39), we get that By using the interpolation property and the following result which yields estimate (42).

Numerical Examples for Piecewise Linear Polynomials
Let ℎ denote a uniform partition on Ω = [ , ] and ℎ the space of continuous piecewise linear functions on ℎ ; that is, = 1. Then we use the Galerkin finite element method for the spatial variables. After the spatial discretization, we get classical ODEs systems with variables ℎ , = 1, 2, . . . , /Δ . In order to satisfy the condition (39) in Theorem 17, we use the two-order Runge-Kutta method to compute the variable 1 ℎ .
In this section, we present numerical calculations which support the error estimates in Theorem 17. If we suppose Δ = ℎ 2 , then we have the convergence rate The Scientific World Journal   ) . (73) The experiential error results and convergence rates are presented in Table 1.
The experiential error results and convergence rates are displayed in Table 2.

Conclusion
In this paper, we study the finite element method for fractional diffusion equation. We use the simple, second order accurate explicit scheme, leapfrog difference method in time, and the finite element method in space. Under the suitably accurate initial conditions and the stability requirement that Δ ⋅ ℎ −2 be sufficiently small, the error analysis for the fully discrete scheme is discussed, which is an 2 -error bound of finite element accuracy and of second order in time. Numerical examples are given to demonstrate the efficiency of the theoretical results.