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