Finite Element Solutions for the Space Fractional Diffusion Equation with a Nonlinear Source Term

and Applied Analysis 3 Baeumer et al. 8, 13 have proved existence and uniqueness of a strong solution for 1.2 using the semigroup theory when f x, t, u is globally Lipschitz continuous. Furthermore, when f x, t, u is locally Lipschitz continuous, existence of a unique strong solution has also been shown by introducing the cut-off function. Finite difference methods have been studied in 14–16 for linear space fractional diffusion problems. They used the right-shifted Grüwald-Letnikov approximate for the fractional derivative since the standard Grüwald-Letnikov approximate gives the unconditional instability even for the implicit method. Using the right-shifted Grüwald-Letnikov approximation, the method of lines has been applied in 12 for numerical approximate solutions. For the space fractional diffusion problems with a nonlinear source term, Lynch et al. 17 used the so-called L2 and L2C methods in 6 and compared computational accuracy of them. Baeumer et al. 8 give existence of the solution and computational results using finite difference methods. Choi et al. 18 have shown existence and stability of numerical solutions of an implicit finite difference equation obtained by using the right-shifted Grüwald-Letnikov approximation. For the time fractional diffusion equations, explicit and implicit finite difference methods have been used in 11, 19–23 . Compared to finite difference methods on the fractional diffusion equation, finite element methods have been rarely discussed. Ervin and Roop 24 have considered finite element analysis for stationary linear advection dispersion equations, and Roop 25 has studied finite element analysis for nonstationary linear advection dispersion equations. The finite element numerical approximations have been discussed for the time and space fractional Fokker-Planck equation in Deng 9 and for the space general fractional diffusion equations with a nonlocal quadratic nonlinearity but a linear source term in Ervin et al. 26 . As far as we know, finite element methods have not been considered for the space fractional diffusion equation with nonlinear source terms. In this paper, we will discuss finite element solutions for the problem 1.2 – 1.4 under the assumption of existence of a sufficiently regular solution u of the equation. Finite element numerical analysis of the semidiscrete and fully discrete methods for 1.2 – 1.4 will be considered using the backward Euler method in time and Galerkin finite element method in space as well as the semidiscrete method.Wewill discuss existence, uniqueness, and stability of the numerical solutions for the problem 1.2 – 1.4 . Also, L2-error estimate will be considered for the problem 1.2 – 1.4 . The outline of the paper is as follows. We introduce some properties of the space fractional derivatives in Section 2, which will be used in later discussion. In Section 3, the semidiscrete variational formulation for 1.2 based on Galerkin method is given. Existence, stability and L2-error estimate of the semidiscrete solution are analyzed. In Section 4, existence and unconditional stability of approximate solutions for the fully discrete backward Euler method are shown following the idea of the semidiscrete method. Further, L2-error estimates are obtained, whose convergence is of O k h , where γ̃ μ if μ/ 3/2 and γ̃ μ − , 0 < < 1/2, if μ 3/2. Finally, numerical examples are given in order to see the theoretical convergence order discussed in Section 5. We will see that numerical solutions of fractional diffusion equations diffuse more slowly than that of the classical diffusion problem and diffusivity depends on the order of fractional derivatives. 2. The Variational Form In this sectionwewill consider the variational form of problem 1.2 – 1.4 and show existence and stability of the weak solution. We first recall some basic properties of Riemann-Liouville fractional calculus 9, 24 . 4 Abstract and Applied Analysis For any given positive number μ > 0, define the seminorm |u|Jμ L R ‖D u‖L2 R 2.1 and the norm ‖u‖Jμ L R ( ‖u‖L2 R |u|2Jμ L R 1/2 , 2.2 where the left fractional derivative space J L R denotes the closure of C ∞ 0 R with respect to the norm ‖ · ‖Jμ L R . Similarly, we may define the right fractional derivative space J R R as the closure of C∞ 0 R with respect to the norm ‖ · ‖Jμ R R , where ‖u‖Jμ R R ( ‖u‖L2 R |u|2Jμ R R 1/2 2.3 and the seminorm |u|Jμ R R ‖D u‖L2 R . 2.4 Furthermore, with the help of Fourier transform we define a seminorm |u|Hμ R ∥ ∥|ω|û∥∥L2 R 2.5 and the norm ‖u‖Hμ R ( ‖u‖L2 R |u|Hμ R 1/2 . 2.6 Here H R denotes the closure of C∞ 0 R with respect to ‖ · ‖Hμ R . It is known in 24 that the spaces J L R , J μ R R , andH μ R are all equal with equivalent seminorms and norms. Analogously, when the domain Ω is a bounded interval, the spaces J L,0 Ω , J μ R,0 Ω , and H μ 0 Ω are equal with equivalent seminorms and norms 24, 27 . The following lemma on the Riemann-Liouville fractional integral operators will be used in our analysis, which can be proved by using the property of Fourier transform 24 . Lemma 2.1. For a given μ > 0 and a real valued function u Dμu,Dμ∗u cos ( πμ )‖Du‖L2 R . 2.7 Remark 2.2. It follows from 2.7 that we may use the following norm: ‖u‖ H μ/2 0 R ‖u‖L2 R κμ ∣ ∣ ∣cos ( π · μ 2 )∣ ∣ ∣|u| H μ/2 0 R 2.8 instead of the norm ‖u‖Hμ R . Abstract and Applied Analysis 5 For the seminorm on H 0 Ω with Ω a, b , the following fractional PoincaréFriedrich’s inequality holds. For the proof, we refer to 9, 24 . Lemma 2.3. For u ∈ H 0 Ω , there is a positive constant C such that ‖u‖L2 Ω ≤ C|u|Hμ 0 Ω 2.9 and for 0 < s < μ, s / n − 1/2, n − 1 ≤ μ < n, n ∈ N, |u|Hs 0 Ω ≤ C|u|Hμ 0 Ω . 2.10 Hereafter, a positive numberCwill denote a generic constant. Also the semigroup property and the adjoint property hold for the Riemann-Liouville fractional integral operators 9, 24 : for all μ, ν > 0, if u ∈ L Ω , p ≥ 1, then aD −μ x aD −ν x u x aD −μ−ν x u x , ∀x ∈ Ω, xD −μ b x D−ν b u x xD −μ−ν b u x , ∀x ∈ Ω, 2.11and Applied Analysis 5 For the seminorm on H 0 Ω with Ω a, b , the following fractional PoincaréFriedrich’s inequality holds. For the proof, we refer to 9, 24 . Lemma 2.3. For u ∈ H 0 Ω , there is a positive constant C such that ‖u‖L2 Ω ≤ C|u|Hμ 0 Ω 2.9 and for 0 < s < μ, s / n − 1/2, n − 1 ≤ μ < n, n ∈ N, |u|Hs 0 Ω ≤ C|u|Hμ 0 Ω . 2.10 Hereafter, a positive numberCwill denote a generic constant. Also the semigroup property and the adjoint property hold for the Riemann-Liouville fractional integral operators 9, 24 : for all μ, ν > 0, if u ∈ L Ω , p ≥ 1, then aD −μ x aD −ν x u x aD −μ−ν x u x , ∀x ∈ Ω, xD −μ b x D−ν b u x xD −μ−ν b u x , ∀x ∈ Ω, 2.11


Introduction
Fractional calculus is an old mathematical topic but it has not been attracted enough for almost three hundred years.However, it has been recently proven that fractional calculus is a significant tool in the modeling of many phenomena in various fields such as engineering, physics, porous media, economics, and biological sciences.One can see related references in 1-7 .
In the classical diffusion model, it is assumed that particles are distributed in a normal bell-shaped pattern based on the Brownian motion.In general, the nature of diffusion is characterized by the mean squared displacement where d is the spatial dimension and κ μ is the diffusion constant.The classical normal diffusion case arises when the exponent μ 1.When μ / 1, anomalous diffusions arise.

Abstract and Applied Analysis
The anomalous diffusion is classified as the process is subdiffusive diffusive slowly when μ < 1 or superdiffusive diffusive fast when μ > 1.
As mentioned before, in many real problems, it is more adequate to use anomalous diffusion described by fractional derivatives than the classical normal diffusion 4, 5, 8-12 .One typical model for anomalous diffusion is the fractional superdiffusion equation arising in chaotic and turbulent processes, where the usual second derivative in space is replaced by a fractional derivative of order 1 < μ < 2.
In this paper we discuss Galerkin approximate solutions for the space fractional diffusion equation with a nonlinear source term.The equation is described as ∂u x, t ∂t κ μ ∇ μ u x, t f x, t, u 1.2 with an initial condition and boundary conditions u x, t 0, x ∈ ∂Ω, 0 ≤ t ≤ T, 1.4 where κ μ denotes the anomalous diffusion coefficient and ∂Ω is the boundary of the domain Ω.And the differential operator ∇ μ is

1.6
Here n is the smallest integer such that n − 1 ≤ μ < n.
Throughout this paper, we will assume that the nonlinear source term f x, t, u is locally Lipschitz continuous with constants C l and C f such that For any given positive number μ > 0, define the seminorm and the norm where the left fractional derivative space . Similarly, we may define the right fractional derivative space , where and the seminorm Furthermore, with the help of Fourier transform we define a seminorm and the norm

2.6
Here The following lemma on the Riemann-Liouville fractional integral operators will be used in our analysis, which can be proved by using the property of Fourier transform 24 .
Lemma 2.1.For a given μ > 0 and a real valued function u Remark 2.2.It follows from 2.7 that we may use the following norm: For the seminorm on H μ 0 Ω with Ω a, b , the following fractional Poincaré-Friedrich's inequality holds.For the proof, we refer to 9, 24 .
Hereafter, a positive number C will denote a generic constant.Also the semigroup property and the adjoint property hold for the Riemann-Liouville fractional integral operators 9, 24 : for all μ, ν > 0 and specially In the rest of this section, we will consider a weak problem for 1.2 -1.4 with Since there is a weak solution of 2.13 when f is locally Lipschitz continuous as in 8, 13 , we here only discuss the stability of the weak solution, to show that we need the following lemma.Lemma 2.4.For all v ∈ H μ/2 0 Ω , the following inequality holds:

2.14
Proof.Following the ideas in 9, 26 , we obtain the following inequality by using the properties 2.11 -2.12 and Lemmas 2.1-2.3:

2.15
This completes the proof.
We consider the stability of a weak solution u for 2.13 .
Theorem 2.5.Let u be a solution of 2.13 .Then there is a constant C such that Proof.Taking v u t in 2.13 , we obtain

2.17
Since the second term on the left hand side is nonnegative from Lemma 2.4, we have 1 2

2.18
Integrating both sides with respect to t, we obtain An application of Gronwall's inequality gives that there is a constant C such that This completes the proof.

The Semidiscrete Variational Form
In this section, we will analyze the stability and error estimates of Galerkin finite element solutions for the semidiscrete variational formulation for 1.2 .Let S h be a partition of Ω with a grid parameter h such that Ω {∪K | K ∈ S h } and h max K∈S h h K , where h K is the width of the subinterval K. Associated with the partition S h , we may define a finite-dimensional subspace

3.3
Since u h can be represented as we may rewrite 3.1 in a matrix form: where N × N matrices A and B and vectors u and F are

3.6
It follows from N i,j 1 α i α j ϕ i , ϕ j N i 1 α i ϕ i , N j 1 α j ϕ j ≥ 0 and Lemma 2.4 that matrices A and B are nonnegative definite and nonsingular.Thus this system 3.5 of ordinary differential equations has a unique solution since f is locally Lipschitz continuous.
The stability for the semidiscrete variational problem 3.1 can be obtained by following the proof of Theorem 2.5, which is 3.7 Now we will consider estimates of error between the weak solution of 2.13 and the one of semidiscrete form 3.1 .The finite dimensional subspace Ω is chosen so that the interpolation I h u of u satisfies an approximation property 9, 28 : for u ∈ H γ Ω , 0 < γ ≤ n, and 0 ≤ s ≤ γ, there exists a constant C depending only on Ω such that 3.8 Since the norm • H s Ω is equivalent to the seminorm | • | H s Ω , we may replace 3.8 by the relation 3.9 Further we need an adjoint problem to find w 0, on ∂Ω.

3.10
Bai and L ü 29 have proved existence of a solution to the problem 3.10 .We assume as in Ervin and Roop 24 that the solution w satisfies the regularity

3.12
Let u h P h u be the elliptic projection

3.13
Let θ u h − u h and ρ u h − u.Then the error is expressed as First, we consider the following estimates on ρ. 15 Proof.It follows from the fractional Poincaré-Friedrich's inequality and the adjoint property 2.12 that for ψ, χ

3.16
Similarly we obtain

10
Abstract and Applied Analysis Using the equivalence of seminorms and norms, we obtain

3.19
In case of μ / 3/2 and v ∈ V h , by taking g ρ in 3.10 and using 3.13 , 3.16 -3.17 and the adjoint property 2.12 , we have

3.20
Taking v I h w in the previously mentioned inequalities, we have

3.21
Thus we obtain

3.22
We now differentiate 3.13 .Then we obtain −κ μ ∇ μ ρ t , v 0 for all v ∈ V h .Using the previous duality arguments again, we have In case of μ 3/2, we can similarly prove 3.15 by applying the assumption 3.12 .This completes the proof.
We now consider the estimates on θ.Lemma 3.2.Let u h and u h be the solutions of 3.1 -3.3 and 3.13 , respectively.Let θ t u h t − u h t .Then there is a constant C such that 24

Abstract and Applied Analysis 11
Proof.It follows from 3.1 and 3.13 that for v ∈ V h ,

3.27
Integration on time t gives
Combining Lemmas 3.1 and 3.2, we obtain the following error estimates.
Theorem 3.3.Let u h and u be the solutions of 3.1 -3.3 and 1.2 -1.4 , respectively.Then there is a constant C u such that 3.32

The Fully Discrete Variational Form
In this section, we consider a fully discrete variational formulation of 1.2 .Existence and uniqueness of numerical solutions for the fully discrete variational formulation are discussed.
The corresponding error estimates are also analyzed.
For the temporal discretization let k T/M for a positive integer M and t m mk.Let u m be the solution of the backward Euler method defined by with an initial condition and boundary conditions Then we get the fully discrete variational formulation of 1.2 to find Thus a finite Galerkin solution Ω is a solution of the equation with an initial condition and boundary conditions Now we prove the existence and uniqueness of solutions for 4.5 using the Brouwer fixed-point theorem.

Theorem 4.1. There exists a unique solution u
Then G v is obviously a continuous function from V h to V h .In order to show the existence of solution for G v 0, we adopt the mathematical induction.Assume that u 0 h , u 1 h , . . ., u m h exist for m < M. It follows from 1.8 , Lemma 2.4, and Young's inequality that 4.9 If we take sufficiently small k so that k < 1/2C f and v L 2 Ω > u m h L 2 Ω / 1 − 2C f k , then the Brouwer's fixed-point theorem implies the existence of a solution.
For the proof of the uniqueness of solutions, we assume that u and v are two solutions of 4.5 .Then we obtain Replacing ψ u − v in the above equation and applying Lemma 2.4, we obtain
The following theorem presents the unconditional stability for 4.4 .
Theorem 4.2.The fully discrete scheme 4.4 is unconditionally stable.In fact, for any m Proof.It follows from 1.8 , Lemma 2.4, and Young's inequality that by taking v u m 1 in 4.4 , we obtain

4.14
Adding the above inequality from m 0 to m, we obtain

4.15
Applying the discrete Gronwall's inequality with sufficiently small k such that k < 1/2C f , we obtain the desired result.
The following theorem is an error estimate for the fully discrete problem 4.4 .

4.18
Applying the locally Lipschitz continuity of f and Young's inequality, we obtain e m 1 2 ≤ kC l e m 1 2

4.19
That is, ≤ kC l e m 1 2

4.21
Applying the discrete Gronwall's inequality with sufficiently small k such that As in the previous section, denote θ m 1 u m 1 and when μ 3/2, 0 < < 1/2, Proof.Since we know the estimates on ρ from Lemma 3.1, we have only to show boundedness of θ m 1 .Using the property 3.13 , we obtain for where r O k .Taking v θ m 1 and applying Lemma 2.4, the locally Lipschitz continuity of f, Young's inequality, and the triangle inequality, we obtain

4.26
Abstract and Applied Analysis 17

4.28
Applying the discrete Gronwall's inequality with sufficiently small k such that

4.29
Also, using Lemma 3.1 and the initial conditions 1.3 and 4.6 , we obtain

Numerical Experiments
In this section, we present numerical results for the Galerkin approximations which supports the theoretical analysis discussed in the previous section.
Let S h denote a uniform partition of Ω and let V h denote the space of continuous piecewise linear functions defined on S h .In order to implement the Galerkin finite element approximation, we adapt finite element discretization on the spatial axis and the backward Euler finite difference scheme along the temporal axis.We associate shape functions of space V h with the standard basis of the functions on the uniform interval with length h.Example 5.1.We first consider a space fractional linear diffusion equation: and boundary conditions u 0, t u 1, t 0.

5.3
In this case, the exact solution is Tables 1, 2, and 4 show the order of convergence and L 2 -error between the exact solution and the Galerkin approximate solution of the fully discrete backward Euler method for 5.1 when μ 1.6, μ 1.8 and μ 1.5, respectively.For numerical computation, the temporal step size k 0.001 is used in all three cases.Table 3 shows L 2 -errors and orders of convergence for the Galerkin approximate solution when μ 1.8 and the spatial step size h 0.0625.According to Tables 1-3, we may find the order of convergence of O k h μ for this linear fractional diffusion problem 5.1 -5.3 when μ / 3/2.Furthermore, Table 4 shows orders of numerical convergence for the problem when μ 3/2, where we may see that the order of convergence is of O k h μ− , 0 < < 1/2.It follows from Tables 1-4 that numerical computations confirm the theoretical results.
We plot the exact solution and approximate solutions obtained by the backward Euler Galerkin method using h 1/32 and k 1/1000 for 5.1 with μ 1.6 and μ 1.8. Figure 1 shows the contour plots of an exact solution and numerical solutions at t 1, and Figure 2 shows log-log graph for the order of convergence.
Example 5.2.We consider a space fractional diffusion equation with a nonlinear Fisher type source term which is described as ∂u x, t ∂t κ μ ∇ μ u x, t λu x, t 1 − βu x, t 5.5   In fact, we will consider the case of κ μ 0.1, β 1 in 5.5 with an initial condition u 0 x e −10x , x ≥ 0, e 10x , x < 0. 5.8 For numerical computations, we have to take care of the nonlinear term f u λu 1 − βu .This gives a complicated nonlinear matrix.In order to avoid the difficulty of solving nonlinear system, we adopted a linearized method replacing λu n 1 1 − βu n 1 by λu n 1 1 − βu n .Figure 3 shows contour plots of numerical solutions at t 1 for 5.5 -5.8 with λ 0.25.For numerical computations, step sizes h 0.01 and k 0.005 are used.From the numerical results we may find that numerical solutions converge to the solution of classical diffusion equation as μ approaches to 2.
Example 5.3.We now consider 5.5 with κ μ 0.1, β 1 and boundary conditions lim |x| → ∞ u x, t 0. 5.9 We will consider an initial condition with a sharp peak in the middle as u 0 x sech 2 10x 5.10 and an initial condition with a flat roof in the middle as

5.11
Tang and Weber 30 have obtained computational solutions for 5.5 with initial conditions 5.10 and 5.11 using a Petrov-Galerkin method when 5.5 is a classical diffusion problem.We obtain computational results using the method as in Example 5.2. Figure 4 shows contour plots of numerical solutions at t 1 for 5.5 with an initial condition 5.10 when Ω −2, 2 and λ 0.25.Figure 5 shows also contour plots of numerical solutions at t 4 for 5.5 and 5.10 when Ω −4, 4 and λ 1.In both cases, step sizes h 0.01 and k 0.005 are used for computation.According to Figures 4 and 5, we may see that the diffusivity depends on μ but it is far less than that of the classical solution.That is, the fractional diffusion problem keeps the peak in the middle for longer time than the classical one does.
Figure 6 shows contour plots of numerical solutions for 5.5 with an initial condition 5.10 when μ 1.8, Ω −2, 2 and λ 1.In this case, step sizes h 0.01 and k 0.005 are also used for computation.But the period of time is from t 0 to t 5.According to Figure 6, we may see that the peak goes down rapidly for a short time, and it begins to go up after the contour arrives at the lowest level.
Figure 7 shows contour plots of numerical solutions at t 1 for 5.5 with an initial condition 5.11 when Ω −4, 4 and λ 0.25.In this case, step sizes h 0.01 and k 0.005 are also used for computation.According to Figure 7, we may find that the fractional diffusion problem keeps the flat roof in the middle for longer time than the classical one does.

Concluding Remarks
Galerkin finite element methods are considered for the space fractional diffusion equation with a nonlinear source term.We have derived the variational formula of the semidiscrete scheme by using the Galerkin finite element method in space.We showed existence and stability of solutions for the semidiscrete scheme.Furthermore, we derived the fully time-space discrete variational formulation using the backward Euler method.Existence and uniqueness of solutions for the fully discrete Galerkin method have been discussed.Also we proved that the scheme is unconditionally stable, and it has the order of convergence of O k h γ , where γ is a constant depending on the order of fractional derivative.Numerical computations confirm the theoretical results discussed in the previous section for the problem with a linear source term.For the fractional diffusion problem with a nonlinear source term, we may find that the diffusivity depends on the order of fractional derivative, and numerical solutions of fractional order problems are less diffusive than the solution of a classical diffusion problem.

b
are called the left and the right Riemann-Liouville space fractional derivatives of order μ, respectively, defined by D μ u : n−μ−1 u ξ dξ.
and H μ R are all equal with equivalent seminorms and norms.Analogously, when the domain Ω is a bounded interval, the spaces J μ L,0 Ω , J μ R,0 Ω , and H μ 0 Ω are equal with equivalent seminorms and norms24, 27 .

h − u m 1 h and ρ m 1 u m 1 h− u t m 1 . Here u m 1 h 22 Theorem 4 . 4 .
is the elliptic projection of u t m 1 defined in 3.13 .Thene m 1 h θ m 1 ρ m 1 .4.Let u be the exact solution of 1.2 -1.4 and let {u m h } M m 0 be the solution of 4.5 -4.7 .Then when μ / 3/2

Figure 2 :Figure 3 :
Figure 2: Log-log plots of the error for the rate of convergence.
Let e m u t m − u m be the error at t m .It follows from 1.2 and 4.4 that for any v ∈ H

Table 1 :
L 2 -error and order of convergence in x when μ 1.6.

Table 2 :
L 2 -error and order of convergence in x when μ 1.8.

Table 3 :
L 2 -error and order of convergence in t when μ 1.8.

Table 4 :
L 2 -error and order of convergence in x when μ 1.5.