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

Authors: Y. J. Choi; S. K. Chung
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider finite element Galerkin solutions for the space fractional diffusion equation with a nonlinear source term. Existence, stability, and order of convergence of approximate solutions for the backward Euler fully discrete scheme have been discussed as well as for the semidiscrete scheme. The analytical convergent orders are obtained asO(k+hγ˜), where γ˜ is a constant depending on the order of fractional derivative. Numerical computations are presented, which confirm the theoretical results when the equation has a linear source term. When the equation has a nonlinear source term, numerical results show that the diffusivity depends on the order of fractional derivative as we expect.


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## Body

## 1. 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(1.1)〈(∆r)2〉=2dκμtμ,
where d is the spatial dimension and κμ is the diffusion constant. The classical normal diffusion case arises when the exponent μ=1. When μ≠1, anomalous diffusions arise. 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(1.2)∂u(x,t)∂t=κμ∇μu(x,t)+f(x,t,u)
with an initial condition
(1.3)u(x,0)=u0,x∈Ω⊂R
and boundary conditions
(1.4)u(x,t)=0,x∈∂Ω,0≤t≤T,
where κμ denotes the anomalous diffusion coefficient and ∂Ω is the boundary of the domain Ω. And the differential operator ∇μ is
(1.5)∇μ=12Dxμa+12xDbμ,
where aDxμ and xDbμ are called the left and the right Riemann-Liouville space fractional derivatives of order μ, respectively, defined by
(1.6)Dμu:=aDxμu(x)=DnaDxμ-nu(x)=1Γ(n-μ)dndxn∫ax‍(x-ξ)n-μ-1u(ξ)dξ,Dμ*u:=xDbμu(x)=(-D)nxDbμ-nu(x)=(-1)nΓ(n-μ)dndxn∫xb‍(ξ-x)n-μ-1u(ξ)dξ.
Here n is the smallest integer such that n-1≤μ<n.Throughout this paper, we will assume that the nonlinear source termf(x,t,u) is locally Lipschitz continuous with constants Cl and Cf such that
(1.7)‖f(u)-f(v)‖L2(Ω)≤Cl‖u-v‖L2(Ω),(1.8)‖f(u)‖L2(Ω)≤Cf‖u‖L2(Ω)
for u,v∈{w∈H0μ/2(Ω)∣∥w∥L2(Ω)≤l}.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. We will 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 Section2, 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+hγ~), where γ~=μ if μ≠3/2 and γ~=μ-ϵ,0<ϵ<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 section we will 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].For any given positive numberμ>0, define the seminorm
(2.1)|u|JLμ(R)=‖Dμu‖L2(R)
and the norm
(2.2)‖u‖JLμ(R)=(‖u‖L2(R)2+|u|JLμ(R)2)1/2,
where the left fractional derivative space JLμ(R) denotes the closure of C0∞(R) with respect to the norm ∥·∥JLμ(R).Similarly, we may define the right fractional derivative spaceJRμ(R) as the closure of C0∞(R) with respect to the norm ∥·∥JRμ(R), where
(2.3)‖u‖JRμ(R)=(‖u‖L2(R)2+|u|JRμ(R)2)1/2
and the seminorm
(2.4)|u|JRμ(R)=‖Dμ*u‖L2(R).Furthermore, with the help of Fourier transform we define a seminorm(2.5)|u|Hμ(R)=‖|ω|μu^‖L2(R)
and the norm
(2.6)‖u‖Hμ(R)=(‖u‖L2(R)2+|u|Hμ(R)2)1/2.
Here Hμ(R) denotes the closure of C0∞(R) with respect to ∥·∥Hμ(R). It is known in [24] that the spaces JLμ(R),JRμ(R), and Hμ(R) are all equal with equivalent seminorms and norms. Analogously, when the domain Ω is a bounded interval, the spaces JL,0μ(Ω),JR,0μ(Ω), and H0μ(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 givenμ>0 and a real valued function u(2.7)(Dμu,Dμ*u)=cos⁡(πμ)‖Dμu‖L2(R)2.Remark 2.2.
It follows from (2.7) that we may use the following norm:
(2.8)‖u‖H0μ/2(R)2=‖u‖L2(R)2+κμ|cos⁡(π⋅μ2)||u|H0μ/2(R)2
instead of the norm ∥u∥Hμ(R).For the seminorm onH0μ(Ω) with Ω=(a,b), the following fractional Poincaré-Friedrich’s inequality holds. For the proof, we refer to [9, 24].Lemma 2.3.
Foru∈H0μ(Ω), there is a positive constant C such that
(2.9)‖u‖L2(Ω)≤C|u|H0μ(Ω)
and for 0<s<μ,s≠n-1/2,n-1≤μ<n,n∈ℕ,
(2.10)|u|H0s(Ω)≤C|u|H0μ(Ω).Hereafter, a positive numberC 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, if u∈Lp(Ω), p≥1, then
(2.11)Dax-μDax-νu(x)=Dax-μ-νu(x),∀x∈Ω,xDb-μDxb-νu(x)=Dxb-μ-νu(x),∀x∈Ω,
and specially
(2.12)(Dax-μu,v)L2(Ω)=(u,Dxb-μv)L2(Ω),∀u,v∈L2(Ω).In the rest of this section, we will consider a weak problem for (1.2)–(1.4) with 1<μ<2: find a function u∈H0μ/2(Ω) such that
(2.13)(ut,v)=(κμ∇μu,v)+(f(u),v),∀v∈H0μ/2(Ω).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 allv∈H0μ/2(Ω), the following inequality holds:
(2.14)-(κμ∇μv,v)≥κμ|cos⁡(π⋅μ2)||v|H0μ/2(Ω)2.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)-(κμ∇μv,v)=-κμ2{(Daxμv,v)+(Dxbμv,v)}=-κμ2{∫ab‍(D2Dax-(2-μ)v)vdx+∫ab‍((-D)2Dxb-(2-μ)v)vdx}=κμ2{∫ab‍(DDax-(2-μ)v)Dvdx+∫ab‍(DDxb-(2-μ)v)Dvdx}=κμ2{∫ab‍(Dax-(2-μ)Dv)Dvdx+∫ab‍(Dxb-(2-μ)Dv)Dvdx}=κμ2{∫ab‍(Dax-(2-μ)/2Dax-(2-μ)/2Dv)Dvdx+∫ab‍(Dxb-((2-μ)/2)Dxb-(2-μ)/2Dv)Dvdx}=κμ2{∫ab‍(Dax-(2-μ)/2Dv)(Dxb-(2-μ)/2Dv)dx+∫ab‍(xDb-(2-μ)/2Dv)(Dax-(2-μ)/2Dv)dx}=-κμ(Dμ/2v,D(μ/2)*v)=-κμcos⁡(π⋅μ2)‖Dμ/2v‖L2(Ω)2≥κμ|cos⁡(π⋅μ2)||v|H0μ/2(Ω)2.
This completes the proof.We consider the stability of a weak solutionu for (2.13).Theorem 2.5.
Letu be a solution of (2.13). Then there is a constant C such that
(2.16)‖u(t)‖L2(Ω)≤C‖u(0)‖L2(Ω).Proof.
Takingv=u(t) in (2.13), we obtain
(2.17)(ut,u)-(κμ∇μu,u)=(f(u),u).
Since the second term on the left hand side is nonnegative from Lemma 2.4, we have
(2.18)12ddt‖u‖L2(Ω)2≤12ddt‖u‖L2(Ω)2+κμ|cos⁡(π⋅μ2)||u|H0μ/2(Ω)2≤‖f(u)‖L2(Ω)‖u‖L2(Ω)≤Cf‖u‖L2(Ω)2.
Integrating both sides with respect to t, we obtain
(2.19)‖u(t)‖L2(Ω)2≤‖u(0)‖L2(Ω)2+C∫0t‍‖u(s)‖L2(Ω)2ds.
An application of Gronwall's inequality gives that there is a constant C such that
(2.20)‖u(t)‖L2(Ω)2≤C‖u(0)‖L2(Ω)2.
This completes the proof.


## 3. 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).LetSh be a partition of Ω with a grid parameter h such that Ω¯={∪K∣K∈Sh} and h=maxK∈ShhK, where hK is the width of the subinterval K. Associated with the partition Sh, we may define a finite-dimensional subspace Vh⊂H0μ/2(Ω) with a basis {φi}i=1N of piecewise polynomials. Then the semidiscrete variational problem is to find uh∈Vh such that
(3.1)(uh,t,v)=(κμ∇μuh,v)+(f(uh),v),∀v∈Vh,(3.2)uh(x,0)=u0,(3.3)uh(a,t)=uh(b,t)=0.
Since uh can be represented as
(3.4)uh(x,t)=∑i=1N‍αi(t)φi(x),
we may rewrite (3.1) in a matrix form:
(3.5)Au˙(t)+Bu=F(u),
where N×N matrices A and B and vectors u and F are
(3.6)A=(aij),aij=(φi,φj),B=(bij),bij=-κμ2[(Dμ/2φi,D(μ/2)*φj)+(Dμ/2φj,D(μ/2)*φi)],F(u)=(Fj),Fj=(f(∑l=1N‍αlφl),φj),u=(α1(t),α2(t),…,αN(t))T.It follows from∑i,j=1N‍αiαj(φi,φj)=(∑i=1N‍αiφi,∑j=1N‍αjφj)≥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)‖uh‖L2(Ω)≤C‖u0‖L2(Ω).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 Vh⊂H0μ/2(Ω) is chosen so that the interpolation Ihu of u satisfies an approximation property [9, 28]: for u∈Hγ(Ω), 0<γ≤n, and 0≤s≤γ, there exists a constant C depending only on Ω such that
(3.8)‖u-Ihu‖Hs(Ω)≤Chγ-s‖u‖Hγ(Ω).
Since the norm ∥·∥Hs(Ω) is equivalent to the seminorm |·|Hs(Ω), we may replace (3.8) by the relation
(3.9)‖u-Ihu‖Hs(Ω)≤Chγ-s|u|Hγ(Ω).Further we need an adjoint problem to findw∈Hμ(Ω)∩H0μ/2(Ω) satisfying
(3.10)-κμ∇μw=g,inΩ,w=0,on⁡∂Ω.
Bai and Lü [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
(3.11)‖w‖Hμ(Ω)≤C‖g‖L2(Ω),μ≠32,(3.12)‖w‖Hμ-ϵ(Ω)≤C‖g‖L2(Ω),μ=32,0<ϵ<12.Letu~h=Phu be the elliptic projection Ph:H0μ/2(Ω)→Vh of the exact solution u, which is defined by
(3.13)-κμ(∇μ(u-u~h),v)=0,∀v∈Vh.
Let θ=uh-u~h and ρ=u~h-u. Then the error is expressed as
(3.14)eh=uh-u=(uh-u~h)+(u~h-u)=θ+ρ.First, we consider the following estimates onρ.Lemma 3.1.
Letu~h be a solution of (3.13) and let u∈Hμ(Ω)∩H0μ/2(Ω) be the solution of (2.13). Let ρ(t)=u~h(t)-u(t). Then there is a constant C such that
(3.15)‖ρ(t)‖L2(Ω)≤Chγ~‖u(t)‖Hγ(Ω),‖ρt(t)‖L2(Ω)≤Chγ~‖u(t)‖Hγ(Ω),
where γ~=μ if μ≠3/2 and γ~=μ-ϵ, 0<ϵ<1/2 if μ=3/2.Proof.
It follows from the fractional Poincaré-Friedrich’s inequality and the adjoint property (2.12) that for ψ,χ∈Vh⊂H0μ/2(Ω)(3.16)(Dμψ,χ)=∫ab‍(Dμ/2ψ)D(μ/2)*χdx≤|ψ|JL,0μ/2(Ω)|χ|JR,0μ/2(Ω)≤C‖ψ‖H0μ/2(Ω)‖χ‖H0μ/2(Ω).
Similarly we obtain
(3.17)(Dμ*ψ,χ)=∫ab‍(D(μ/2)*ψ)Dμ/2χdx≤C‖ψ‖H0μ/2(Ω)‖χ‖H0μ/2(Ω).
It follows from Lemma2.4 that for v∈Vh(3.18)κμ|cos⁡(π⋅μ2)||u-u~h|H0μ/2(Ω)2≤-κμ(∇μ(u-u~h),u-u~h)≤-κμ(∇μ(u-u~h),u-v)-κμ(∇μ(u-u~h),v-u~h)≤C‖u-u~h‖H0μ/2(Ω)‖u-v‖H0μ/2(Ω).
Using the equivalence of seminorms and norms, we obtain
(3.19)‖u-u~h‖H0μ/2(Ω)≤Cinf⁡v∈Vh‖u-v‖H0μ/2(Ω)≤C‖u-Ihu‖H0μ/2(Ω).
In case ofμ≠3/2 and v∈Vh, 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)(ρ,ρ)=-κμ(∇μw,ρ)=-κμ(∇μ(w-v),ρ)-κμ(∇μρ,v)=-κμ(∇μ(w-v),ρ)≤C‖w-v‖H0μ/2(Ω)‖ρ‖H0μ/2(Ω).
Taking v=Ihw in the previously mentioned inequalities, we have
(3.21)‖ρ‖L2(Ω)2≤C‖w-Ihw‖H0μ/2(Ω)‖ρ‖H0μ/2(Ω)≤Chμ/2‖w‖Hμ(Ω)‖u-Ihu‖H0μ/2(Ω)≤Chμ/2‖ρ‖L2(Ω)hμ/2‖u‖Hμ(Ω).
Thus we obtain
(3.22)‖ρ‖L2(Ω)≤Chμ‖u‖Hμ(Ω).
We now differentiate (3.13). Then we obtain -κμ(∇μρt,v)=0 for all v∈Vh. Using the previous duality arguments again, we have
(3.23)‖ρt‖L2(Ω)≤Chμ‖u‖Hμ(Ω).
In case ofμ=3/2, we can similarly prove (3.15) by applying the assumption (3.12). This completes the proof.We now consider the estimates onθ.Lemma 3.2.
Letuh and u~h be the solutions of (3.1)–(3.3) and (3.13), respectively. Let θ(t)=uh(t)-u~h(t). Then there is a constant C such that
(3.24)‖θ(t)‖L2(Ω)≤Chγ~,
where γ~=μ if μ≠3/2 and γ~=μ-ϵ, 0<ϵ<1/2 if μ=3/2.Proof.
It follows from (3.1) and (3.13) that for v∈Vh,
(3.25)(θt,v)-κμ(∇μθ,v)=(f(uh)-f(u),v)-(ρt,v).
Replacing v=θ in (3.25), we obtain
(3.26)12ddt‖θ‖L2(Ω)2≤Cl‖uh-u‖L2(Ω)‖θ‖L2(Ω)+‖ρt‖L2(Ω)‖θ‖L2(Ω).
Using Young's inequality
(3.27)ddt‖θ‖L2(Ω)2≤C(‖uh-u~h‖L2(Ω)+‖u~h-u‖L2(Ω))‖θ‖L2(Ω)+‖ρt‖L2(Ω)‖θ‖L2(Ω)≤C(‖θ‖L2(Ω)+‖ρ‖L2(Ω)+‖ρt‖L2(Ω))‖θ‖L2(Ω)≤C1‖θ‖L2(Ω)2+C2‖ρ‖L2(Ω)2+C3‖ρt‖L2(Ω)2.
Integration on time t gives
(3.28)‖θ(t)‖L2(Ω)2≤‖θ(0)‖L2(Ω)2+C∫0t‍‖θ‖L2(Ω)2ds+C∫0t‍(‖ρ‖L2(Ω)2+‖ρt‖L2(Ω)2)ds.
Applying Gronwall's inequality, we obtain
(3.29)‖θ(t)‖L2(Ω)2≤C1‖θ(0)‖L2(Ω)2+C2∫0t‍(‖ρ‖L2(Ω)2+‖ρt‖L2(Ω)2)ds.
Since
(3.30)‖θ(0)‖L2(Ω)≤‖uh(0)-u(0)‖L2(Ω)+‖u~h(0)-u(0)‖L2(Ω)≤Chγ~‖u0‖Hγ(Ω),
we obtain the desired inequality
(3.31)‖θ(t)‖L2(Ω)≤Chγ~,
where γ~=μ if μ≠3/2 and γ~=μ-ϵ, 0<ϵ<1/2, if μ=3/2.Combining Lemmas3.1 and 3.2, we obtain the following error estimates.Theorem 3.3.
Letuh 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)‖u(t)-uh(t)‖L2(Ω)≤C(u)hμ,μ≠32,‖u(t)-uh(t)‖L2(Ω)≤C(u)hμ-ϵ,μ=32,0<ϵ<12.


## 4.  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 letk=T/M for a positive integer M and tm=mk. Let um be the solution of the backward Euler method defined by
(4.1)um+1-umk=κμ∇μum+1+f(um+1)
with an initial condition
(4.2)u0(x)=u0,x∈Ω=(a,b)
and boundary conditions
(4.3)um+1(a)=um+1(b)=0,m=0,1,…,M-1.
Then we get the fully discrete variational formulation of (1.2) to find um+1∈H0μ/2(Ω) such that for all v∈H0μ/2(Ω)(4.4)(um+1,v)-k(κμ∇μum+1,v)=(kf(um+1),v)+(um,v).
Thus a finite Galerkin solution uhm+1∈Vh⊂H0μ/2(Ω) is a solution of the equation
(4.5)(uhm+1,vh)-kκμ(∇μuhm+1,vh)=k(f(uhm+1),vh)+(uhm,vh),∀vh∈Vh
with an initial condition
(4.6)uh0=u0
and boundary conditions
(4.7)uhm+1(a)=uhm+1(b)=0,m=0,1,…,M-1.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 solutionuhm+1∈Vh⊂H0μ/2(Ω) of (4.5)–(4.7).Proof.
Let(4.8)G(uhm+1)=uhm+1-kκμ∇μuhm+1-kf(uhm+1)-uhm.
Then G(v) is obviously a continuous function from Vh to Vh. In order to show the existence of solution for G(v)=0, we adopt the mathematical induction. Assume that uh0,uh1,…,uhm exist for m<M. It follows from (1.8), Lemma 2.4, and Young's inequality that
(4.9)(G(v),v)=(v,v)-(uhm,v)-k(κμ∇μv,v)-k(f(v),v)≥‖v‖L2(Ω)2-‖uhm‖L2(Ω)‖v‖L2(Ω)+kκμ|cos⁡(π⋅μ2)||v|H0μ/2(Ω)2-Cfk‖v‖L2(Ω)2≥‖v‖L2(Ω)2-‖uhm‖L2(Ω)‖v‖L2(Ω)-Cfk‖v‖L2(Ω)2≥‖v‖L2(Ω)2-12(‖uhm‖L2(Ω)2+‖v‖L2(Ω)2)-Cfk‖v‖L2(Ω)2=(12-Cfk)‖v‖L2(Ω)2-12‖uhm‖L2(Ω)2.
If we take sufficiently small k so that k<1/2Cf and ∥v∥L2(Ω)>∥uhm∥L2(Ω)/(1-2Cfk), then the Brouwer's fixed-point theorem implies the existence of a solution.
For the proof of the uniqueness of solutions, we assume thatu and v are two solutions of (4.5). Then we obtain
(4.10)(u-v,ψ)=kκμ(∇μ(u-v),ψ)+k(f(u)-f(v),ψ),∀ψ∈Vh⊂H0μ/2(Ω).
Replacing ψ=u-v in the above equation and applying Lemma 2.4, we obtain
(4.11)‖u-v‖L2(Ω)2≤-kκμ|cos⁡(π⋅μ2)||u-v|H0μ/2(Ω)+k‖f(u)-f(v)‖L2(Ω)‖u-v‖L2(Ω)≤k‖f(u)-f(v)‖L2(Ω)‖u-v‖L2(Ω)≤kCl‖u-v‖L2(Ω)2.
This implies u-v=0 since u(0)=v(0).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(4.12)‖um+1‖L2(Ω)≤C‖u0‖L2(Ω).Proof.
It follows from (1.8), Lemma 2.4, and Young's inequality that by taking v=um+1 in (4.4), we obtain
(4.13)0=(um+1,um+1)-k(κμ∇μum+1,um+1)-k(f(um+1),um+1)-(um,um+1)≥‖um+1‖L2(Ω)2+kκμ|cos⁡(π⋅μ2)||um+1|H0μ/2(Ω)2-Cfk‖um+1‖L2(Ω)2-‖um‖L2(Ω)‖um+1‖L2(Ω)≥12‖um+1‖L2(Ω)2+kκμ|cos⁡(π⋅μ2)||um+1|H0μ/2(Ω)2-Cfk‖um+1‖L2(Ω)2-12‖um‖L2(Ω)2.
Then
(4.14)12‖um+1‖L2(Ω)2≤12‖um+1‖L2(Ω)2+kκμ|cos⁡(π⋅μ2)||um+1|H0μ/2(Ω)2≤Cfk‖um+1‖L2(Ω)2+12‖um‖L2(Ω)2.
Adding the above inequality fromm=0 to m, we obtain
(4.15)(1-2Cfk)‖um+1‖L2(Ω)2≤‖u0‖L2(Ω)2+2Cfk∑j=1m‍‖uj‖L2(Ω)2.
Applying the discrete Gronwall's inequality with sufficiently small k such that k<1/2Cf, we obtain the desired result.The following theorem is an error estimate for the fully discrete problem (4.4).Theorem 4.3.
Letu be the exact solution of (1.2) and let um be the solution of (4.4). Then there is a constant C such that
(4.16)‖u(tm)-um‖L2(Ω)≤Ck.Proof.
Letem=u(tm)-um be the error at tm. It follows from (1.2) and (4.4) that for any v∈H0μ/2(Ω)(4.17)(em+1,v)-k(κμ∇μem+1,v)=k(f(u(tm+1))-f(um+1),v)+(em,v)+(krm+1,v),
where r=O(k). Taking v=em+1,
(4.18)‖em+1‖L2(Ω)2≤‖em+1‖L2(Ω)2+kκμ|cos⁡(π⋅μ2)||em+1|H0μ/2(Ω)2≤k‖f(u(tm+1))-f(um+1)‖L2(Ω)‖em+1‖L2(Ω)+‖em‖L2(Ω)‖em+1‖L2(Ω)+‖krm+1‖L2(Ω)‖em+1‖L2(Ω).
Applying the locally Lipschitz continuity of f and Young's inequality, we obtain
(4.19)‖em+1‖L2(Ω)2≤kCl‖em+1‖L2(Ω)2+‖em‖L2(Ω)‖em+1‖L2(Ω)+‖krm+1‖L2(Ω)‖em+1‖L2(Ω)≤kCl‖em+1‖L2(Ω)2+ɛ1‖em‖L2(Ω)2+14ɛ1‖em+1‖L2(Ω)2+ɛ2‖krm+1‖L2(Ω)2+14ɛ2‖em+1‖L2(Ω)2.
That is,
(4.20)(1-14ɛ1-14ɛ2)‖em+1‖L2(Ω)2≤kCl‖em+1‖L2(Ω)2+ɛ1‖em‖L2(Ω)2+ɛ2‖krm+1‖L2(Ω)2.
Denoting ɛ0=1-1/4ɛ1-1/4ɛ2 and adding the above equation from m=0 to m, we obtain
(4.21)(ɛ0-kCl)‖em+1‖L2(Ω)2≤ɛ1‖e0‖L2(Ω)2+(kCl+ɛ1-ɛ0)∑i=1m‍‖ei‖L2(Ω)2+ɛ2∑i=1m+1‍‖kri‖L2(Ω)2.
Applying the discrete Gronwall's inequality with sufficiently small k such that (ɛ0-ɛ1)/Cl<k<ɛ0/Cl, we obtain the desired result since ∑i=1m+1‍∥kri∥L2(Ω)≤Ck and ∥e0∥L2(Ω)=∥u(0)-u0∥L2(Ω)=0.As in the previous section, denoteθm+1=uhm+1-u~hm+1 and ρm+1=u~hm+1-u(tm+1). Here u~hm+1 is the elliptic projection of u(tm+1) defined in (3.13). Then
(4.22)ehm+1=θm+1+ρm+1.Theorem 4.4.
Letu be the exact solution of (1.2)–(1.4) and let {uhm}m=0M be the solution of (4.5)–(4.7). Then when μ≠3/2(4.23)‖u(tm+1)-uhm+1‖L2(Ω)≤Ck+Chμ‖u(tm+1)‖Hμ(Ω)
and when μ=3/2,0<ϵ<1/2,
(4.24)‖u(tm+1)-uhm+1‖L2(Ω)≤Ck+Chμ-ϵ‖u(tm+1)‖Hμ-ϵ(Ω).Proof.
Since we know the estimates onρ from Lemma 3.1, we have only to show boundedness of θm+1. Using the property (3.13), we obtain for v∈Vh(4.25)(θm+1,v)-k(κμ∇μθm+1,v)=k(f(uhm+1)-f(u(tm+1)),v)+(uhm-u(tm),v)-(krm+1,v)-(ρm+1,v),
where r=O(k).
Takingv=θm+1 and applying Lemma 2.4, the locally Lipschitz continuity of f, Young's inequality, and the triangle inequality, we obtain
(4.26)‖θm+1‖L2(Ω)2≤‖θm+1‖L2(Ω)2+kκμ|cos⁡(π⋅μ2)||θm+1|H0μ/2(Ω)2≤k‖f(uhm+1)-f(u(tm+1))‖L2(Ω)‖θm+1‖L2(Ω)+‖ehm‖L2(Ω)‖θm+1‖L2(Ω)+‖krm+1‖L2(Ω)‖θm+1‖L2(Ω)+‖ρm+1‖L2(Ω)‖θm+1‖L2(Ω)≤kCl‖ehm+1‖L2(Ω)‖θm+1‖L2(Ω)+(‖θm‖L2(Ω)+‖ρm‖L2(Ω))‖θm+1‖L2(Ω)+‖krm+1‖L2(Ω)‖θm+1‖L2(Ω)+‖ρm+1‖L2(Ω)‖θm+1‖L2(Ω)≤kCl(1+14ɛ6)‖θm+1‖L2(Ω)2+(14ɛ3+14ɛ4+14ɛ5+14ɛ6)‖θm+1‖L2(Ω)2+ɛ3‖θm‖L2(Ω)2+ɛ4‖krm+1‖L2(Ω)2+ɛ5‖ρm‖L2(Ω)2+(1+kCl)ɛ6‖ρm+1‖L2(Ω)2.
This implies that
(4.27)(1-14ɛ3-14ɛ4-14ɛ5-14ɛ6)‖θm+1‖L2(Ω)2≤kCl(1+14ɛ6)‖θm+1‖L2(Ω)2+ɛ3‖θm‖L2(Ω)2+ɛ4‖krm+1‖L2(Ω)2+ɛ5‖ρm‖L2(Ω)2+(1+kCl)ɛ6‖ρm+1‖L2(Ω)2.
Denote ɛ7=1-1/4ɛ3-1/4ɛ4-1/4ɛ5-1/4ɛ6 and ɛ8=1+1/4ɛ6. Then adding the above inequality from m=0 to m, we obtain
(4.28)(ɛ7-kClɛ8)‖θm+1‖L2(Ω)2≤ɛ3‖θ0‖L2(Ω)2+(kClɛ8+ɛ3-ɛ7)∑i=1m‍‖θi‖L2(Ω)2+ɛ4∑i=1m+1‍‖kri‖L2(Ω)2+ɛ5∑i=0m‍‖ρi‖L2(Ω)2+(1+kCl)ɛ6∑i=1m+1‍‖ρi‖L2(Ω)2.
Applying the discrete Gronwall's inequality with sufficiently small k such that (ɛ7-ɛ3)/ɛ8Cl<k<ɛ7/Clɛ8,
(4.29)‖θm+1‖L2(Ω)2≤C1‖θ0‖L2(Ω)2+C2∑i=1m+1‍‖kri‖L2(Ω)2+C3∑i=0m+1‍‖ρi‖L2(Ω)2.
Also, using Lemma 3.1 and the initial conditions (1.3) and (4.6), we obtain
(4.30)‖θ0‖L2(Ω)≤‖uh0-u(0)‖L2(Ω)+‖u~h0-u(0)‖L2(Ω)≤Chγ~‖u0‖Hγ(Ω).
Since ∑i=1m+1‍∥kri∥L2(Ω)≤Ck, we get
(4.31)‖θm+1‖L2(Ω)≤Ck+C(u)hγ~,
where γ~=μ if μ≠3/2 and γ~=μ-ϵ, 0<ϵ<1/2, if μ=3/2. Thus we obtain the desired result.


## 5. Numerical Experiments

In this section, we present numerical results for the Galerkin approximations which supports the theoretical analysis discussed in the previous section.LetSh denote a uniform partition of Ω and let Vh denote the space of continuous piecewise linear functions defined on Sh. 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 Vh 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:(5.1)∂u(x,t)∂t=∇μu(x,t)+2tt2+1u(x,t)-(t2+1)×({x2-μ+(1-x)2-μ}Γ(3-μ)-6{x3-μ+(1-x)3-μ}Γ(4-μ)+12{x4-μ+(1-x)4-μ}Γ(5-μ))
with an initial condition
(5.2)u(x,0)=x2(1-x)2,x∈[0,1]
and boundary conditions
(5.3)u(0,t)=u(1,t)=0.
In this case, the exact solution is
(5.4)u(x,t)=(t2+1)x2(1-x)2.Tables1, 2, and 4 show the order of convergence and L2-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 L2-errors and orders of convergence for the Galerkin approximate solution when μ=1.8 and the spatial step size h=0.0625.Table 1
L
2-error and order of convergence in x when μ=1.6.
h
∥
u
-
u
h
∥
L
2
(
Ω
)
Error
Order
1/4
8.37811
e
-
03
—
1/8
2.73537
e
-
03
1.615
1/16
8.75752
e
-
04
1.643
1/32
2.83167
e
-
04
1.629Table 2
L
2-error and order of convergence in x when μ=1.8.
h
∥
u
-
u
h
∥
L
2
(
Ω
)
Error
Order
1/4
8.03045
e
-
03
—
1/8
2.28959
e
-
03
1.810
1/16
6.32962
e
-
04
1.855
1/32
1.76406
e
-
04
1.843Table 3
L
2-error and order of convergence in t when μ=1.8.
k
∥
u
-
u
h
∥
L
2
(
Ω
)
Error
Ratio
1/20
4.20420
e
-
03
—
1/30
2.94873
e
-
03
0.951
1/40
2.31793
e
-
03
0.954
1/50
1.93046
e
-
03
0.961Table 4
L
2-error and order of convergence in x when μ=1.5.
h
∥
u
-
u
h
∥
L
2
(
Ω
)
Error
Order
1/4
5.47750
e
-
03
—
1/8
2.20129
e
-
03
1.315
1/16
8.86858
e
-
04
1.312
1/32
3.57629
e
-
04
1.310According to Tables1–3, we may find the order of convergence of O(k+hμ) 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+hμ-ϵ), 0<ϵ<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 usingh=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.Figure 1
Exact and numerical solutions withμ=1.6 and μ=1.8.Figure 2
Log-log plots of the error for the rate of convergence.Example 5.2.
We consider a space fractional diffusion equation with a nonlinear Fisher type source term which is described as(5.5)∂u(x,t)∂t=κμ∇μu(x,t)+λu(x,t)(1-βu(x,t))
with an initial condition
(5.6)u(x,0)=u0(x)
and boundary conditions
(5.7)u(-1,t)=u(1,t)=0.
In fact, we will consider the case of κμ=0.1, β=1 in (5.5) with an initial condition
(5.8)u0(x)={e-10x,x≥0,e10x,x<0.For numerical computations, we have to take care of the nonlinear termf(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 λun+1(1-βun+1) by λun+1(1-βun). 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.Figure 3
Numerical solutions for (5.5) with (5.8).Example 5.3.
We now consider (5.5) with κμ=0.1, β=1 and boundary conditions
(5.9)lim⁡|x|→∞u(x,t)=0.
We will consider an initial condition with a sharp peak in the middle as
(5.10)u0(x)=sech2(10x)
and an initial condition with a flat roof in the middle as
(5.11)u0(x)={e-10(x-1),x>1,1,-1<x≤1,e10(x+1),x≤-1.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 4
Numerical solutions att=1 for (5.5) and (5.10) with λ=0.25.Figure 5
Numerical solutions att=4 for (5.5) and (5.10) with λ=1.Figure6 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 6
Numerical solutions for (5.5) and (5.10) with λ=1.Figure7 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.Figure 7
Numerical solutions for (5.5) and (5.11) with λ=0.25.


## 6.   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 ofO(k+hγ~), 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.



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