Comp. Appl. Math. https://doi.org/10.1007/s40314-018-0609-3

Galerkin finite element method for time-fractional stochastic diffusion equations Guang-an Zou1

Received: 24 January 2018 / Revised: 7 March 2018 / Accepted: 12 March 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Abstract In this paper, Galerkin finite element method for solving the time-fractional stochastic diffusion equations with multiplicative noise is proposed and investigated. The pathwise regularity properties of solutions to the semidiscrete Galerkin approximations are demonstrated and the convergence of optimal rates are derived. And also we construct the fully discrete scheme which is based on the approximations of the Mittag–Leffler function and analyze the error estimates of convergence in L 2 -norm space. Finally, numerical results are conducted to confirm our theoretical findings. Keywords Time-fractional derivative · Stochastic diffusion equations · Galerkin finite element method · Error estimates Mathematics Subject Classification 65C30 · 65N30

1 Introduction In the past decades, fractional calculus and its applications has attracted considerable interests in various fields of science and engineering (Kilbas et al. 2006; Zhou 2014; Hilfer 2000; Podlubny 1999), many important phenomena can be successfully modeled using mathematical techniques inspired of fractional derivatives (Bhrawy et al. 2016; Machado et al. 2011). In this way, fractional differential equations have developed as one of the most rapidly growing area of research in recent years, there has been a significant development in theoretical analysis and numerical approximations of fractional differential equations, see Deng and Deng (2012), Jiang and Ma (2011), Mainardi (2010), Povstenko (2015), Zeng et al. (2013), Zhai et al. (2014), Bhrawy et al. (2015), Zaky (2017a, b), Bhrawy and Zaky (2017a), and Bhrawy

Communicated by José Tenreiro Machado.

B 1

Guang-an Zou [email protected]; [email protected] School of Mathematics and Statistics, Henan University, Kaifeng 475004, China

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G. Zou

and Zaky (2017b) and the references therein. On the other hand, stochastic perturbations are coming from many natural sources in the practically physical system, they can not be ignored and the presence of noises might give rise to some statistical features and important phenomena, then the stochastic differential equations are produced, which enable one to represent real-world physical phenomena with more accuracy (Oksendal 2013; Prévôt and Röckner 2007). Recently, some related works about the theoretical analysis of fractional differential equations driven by stochastic perturbations have been intensively investigated in Zou et al. (2018), Liu and Fu (2018), Chen et al. (2015), Mijena and Nane (2015), Zou et al. (2018), Zou and Wang (2017), and Huang and Shen (2016). As we know, it is usually very difficult to obtain analytical solutions for such problems. Therefore, numerical methods are employed for finding the approximate solutions to the fractional stochastic differential equations. However, it seems that there are less literatures related to numerical approximations of stochastic partial differential equations with fractional derivatives. In this work, we consider the approximate solutions of time-fractional stochastic diffusion equations on a bounded domain D ⊂ Rd (d = 1, 2, 3) as following: ⎧ α ⎪ ⎨∂t u + Au = f (u) + g(u)W˙ (t), x ∈ D, t ∈ (0, T ], (1.1) u|∂ D = 0, t ∈ (0, T ), ⎪ ⎩ u(0) = u 0 , x ∈ D, where g(u)W˙ (t) = g(u) ∂ W∂t(t) describes a state dependent random noise, and W (t) is a Ft -adapted Wiener process defined on a filtered probability space (, F , P, {Ft }t≥0 ). The operator A := − stands for the negative Laplacian operator with the domain D (A) := {v ∈ H 2 (D) : v|∂ D = 0}. We shall assume that the functions f and g are globally Lipschitz continuous. The initial value u 0 is F0 -measurable random variables. Here, ∂tα denotes the Caputo-type derivative of order α with respect to t defined by   t ∂u(x,s) ds 1 α (1−α) 0 ∂s (t−s)α , 0 < α < 1, ∂t u(x, t) = Γ∂u(x,t) (1.2) , α = 1, ∂t where Γ (·) stands for the gamma function. Denoting the Laplace transform of u by  u (z) =

L {u(t)}, then the Laplace transform of ∂tα is given by (see Kilbas et al. 2006): L {∂tα u(t)} = z α u (z) − z α−1 u(0).

(1.3)

Note that the considered problem (1.1) can be reduced to a class of stochastic reactiondiffusion equations for α = 1. In recent years, the stochastic diffusion equations have become a popular tool for modeling biological systems (Agbanusi and Isaacson 2014) and play a critical role in the chemical reaction process and biochemical networks (Chevalier and ElSamad 2012; Erban et al. 2014). The existence and uniqueness of solutions for these types of equations were investigated by several authors (Cerrai 2003; Kunze and Neerven 2012; Misiats et al. 2016). The study of random attractors and dynamical behavior for such equations can be found in Bates et al. (2009), Cao et al. (2015), Li and Guo (2008), and Wang and Zhou (2011). From a computational view point, the stochastic reaction-diffusion equations are usually handled by using finite difference methods (Kloeden et al. 2011; Wang and Gan 2013), finite element methods (Feng et al. 2017), Monte Carlo methods (Kerr et al. 2008) and other methods (Engblom et al. 2009; Ferm et al. 2010; Gyöngy 1999; Hellander and Löstedt 2011; Ramaswamy and Sbalzarini 2011; Kim et al. 2017). It should be pointed out that the extension of stochastic diffusion equations with fractional derivative might be used to model the random effects on transport of particles in anomalous diffusion process with memory

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Galerkin finite element method for time-fractional...

effects. The existence and uniqueness of mild solutions to the time-fractional stochastic diffusion equations have been studied in Zou et al. (2018). Zou et al. (2018) developed a semidiscrete finite element method for fractional stochastic diffusion-wave equations (the case f (u) = 0 and 1 < α < 2), which was mainly based on Mainardi’s Wright-type function. However, to the best of our knowledge, numerical methods for the given problem (1.1) are yet to be investigated. The contribution of this paper is to develop the fully discrete Galerkin finite element method for solving the time-fractional stochastic diffusion equations, which is based on the approximations of the Mittag–Leffler function, the key and difficulty of this paper is how to give the error estimates for our presented methods. Numerical example is also presented to test the efficiency of our algorithm. The rest of the paper is organized as follows. In Sect. 2, we give some notations and construct the semidiscrete Galerkin approximations and the fully discrete scheme for the equations. The pathwise regularity results of mild solution have been proved in Sect. 3. In Sect. 4, we prove the convergence error estimates for both semi-discrete and fully discrete finite element approximations for our given problem. Numerical example is discussed in Sect. 5. Conclusions are given in the final section.

2 Notations and preliminaries Let H = L 2 (D) be a real separable Hilbert space with inner product (·, ·) and norm  · . Denote by L 2 (, H ) a Hilbert space of H -valued random variables equipped with the inner product E(·, ·) and norm E · , it is defined by 

v(ω)2 dP(ω) < ∞, ω ∈  , L 2 (, H ) = v : Ev2 = 

where E denotes the expectation. Assume that W = {W (t), t ≥ 0} is a U -valued Wiener process with covariance operator Q. We introduce the subspace U0 = Q 1/2 (U ) endowed with the inner product (Oksendal 2013): (u, v)U0 = (Q 1/2 u, Q 1/2 v), u, v ∈ U0 , and induced norm  · U0 , where Q −1/2 denotes the pseudo-inverse of Q 1/2 . Denote by L 02 = L 2 (U0 , H ) the space of Hilbert–Schmidt operators T : U0 → H endowed with the norm ϕ2L 0 = T r [(ϕ Q 1/2 )(ϕ Q 1/2 )∗ ] < ∞, 2

for any ϕ ∈ Itô isometry is useful for the stochastic integrals (Prévôt and Röckner 2007), that is t 2 t E v(s)dW (s) Ev(s)2L 2 ds, t ∈ [0, T ]. =

L 02 . Let {v(t)}t∈[0,T ] be an L 02 -valued predictable stochastic process, the following

0

(2.1)

0

0

r For any r > 0, let H˙ r be the domain of the fractional power A 2 , which can be defined by r

A2χ =

∞

k=1

˙r

r

λk2 (χ, ϕk )ϕk ,

r 2

 r 2

H = D (A ) = χ ∈ H : A χ = 2

∞

 λrk (χ, ϕk )2

<∞ ,

k=1

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G. Zou r

with the induced norms χr2 = A 2 χ2 =

∞  k=1

λrk (χ, ϕk )2 .

Applying the Laplace transform to both sides of (1.1) and using (1.3) we deduce that  u (z) − z α−1 u(0) + A u (z) =  f (z) + G(z), z α  denotes the Laplace transform of the term g(u)W˙ , and so where G(z)  u (z) =

1 z α−1  u0 + α [ f (z) + G(z)]. α z +A z +A

(2.2)

Here, we introduce the generalized Mittag–Leffler function E α,β (t) defined as follows: E α,β (t) =

∞

k=0

tk , Γ (αk + β)

(2.3)

and the Laplace transform of Mittag–Leffler function is shown that (see Haubold et al 2011)

∞ z α−β L {t β−1 E α,β (−λt α )} = e−zt t β−1 E α,β (−λt α )dt = α . (2.4) z +λ 0 Thus, we define the operators E1 (t) and E2 (t) as E1 (t)v = E α,1 (−At α ), E2 (t)v = t α−1 E α,α (−At α ),

so that the solution of (1.1) can be expressed by the Duhamel formula:

t

t u(t) = E1 (t)u 0 + E2 (t − s) f (u(s))ds + E2 (t − s)g(u(s))dW (s), 0

(2.5)

0

which is called a mild solution to the given problem (1.1). In order to ensure the existence and uniqueness of solution to (1.1), we shall assume that u 0 ∈ L 2 (; H˙ ν ) with ν ≥ 0 is F0 −measurable and bounded random variable, and the functions f and g satisfy the following global Lipschitz and growth conditions:  f (u) ≤ Cu,  f (u) − f (v) ≤ Cu − v,

(2.6)

g(u) L 2 ≤ Cu, g(u) − g(v) L 2 ≤ Cu − v,

(2.7)

and 0

0

for any u, v ∈ H and C > 0 is constant. Next, we shall construct the semidiscrete Galerkin approximations and fully discrete scheme of the equations. Let Vh denote standard piecewise linear finite element spaces defined on a family of quasiuniform triangulations of D and vanishing on ∂ D. We assume that the projection operator Ph is the standard L 2 -projection operator onto Vh and the Ritz projection Rh : H01 (D) → Vh , which defined by (Ph ψ, χ) = (ψ, χ), ∀χ ∈ Vh , (∇ Rh ψ, ∇χ) = (∇ψ, ∇χ), ∀χ ∈ Vh . where ψ ∈ L 2 (D). Therefore, the following uniform inequalities can be easily derived (see Kruse 2014; Thomée 1984) Ph χ ≤ Cχ, ∀χ ∈ H, Ph ψ − ψ + h∇(Ph ψ − ψ) ≤ Ch ψ H˙ q , ∀ψ ∈ H˙ q , q = 1, 2, Rh ψ − ψ + h∇(Rh ψ − ψ) ≤ Ch q ψ ˙ q , ∀ψ ∈ H˙ q , q = 1, 2. q

H

123

(2.8) (2.9) (2.10)

Galerkin finite element method for time-fractional...

The semidiscrete Galerkin finite element approximation of (1.1) is described by (∂tα u h , χ) + (∇u h , ∇χ) = ( f (u h ), χ) + (g(u h )W˙ (t), χ), with u h (0) = Ph u 0 for all χ ∈ Vh , with the discrete version of Laplacian operator Ah = −h : Vh → Vh defined by (h ψ, χ) = −(∇ψ, ∇χ), ∀ ψ, χ ∈ Vh . N be the eigenpairs of the discrete Laplacian A , we define the space by Let {λkh , ϕkh }k=1 h   ∞ r 2 r h r h 2 2 (λ ) (χ, ϕ ) < ∞ , H˙ h = χ ∈ Vh : A χ = k

h

k

k=1 r

with the norms χ2h,r = Ah2 χ2 =

∞  k=1

(λkh )r (χ, ϕkh )2 . Finally, the finite element approxi-

mation of (1.1) can be written as ∂tα u h (t) + Ah u h (t) = Ph f (u h (t)) + Ph g(u h (t))W˙ (t),

(2.11)

with u h (0) = Ph u 0 = u h0 . Similarly, the mild solution u h (t) of the discrete problem (2.11) admits

t

t Eh2 (t − s)Ph f (u h (s))ds + Eh2 (t − s)Ph g(u h (s))dW (s), u h (t) = Eh1 (t)Ph u 0 + 0

0

(2.12)

where the discrete analogues Eh1 (t) and Eh2 (t) are defined as following Eh1 (t)vh = E α,1 (−Ah t α ), Eh2 (t)vh = t α−1 E α,α (−Ah t α ).

Let tn = nk (n = 0, 1, . . . , N ) with a fixed time step size k > 0, we define a piecewiseconstant approximation U n ≈ u h (tn ), then the explicit time discretization of the finite element solutions (2.12) by using the Taylor series: U n = Eh1 (tn )Ph u 0 +

n−1

Eh2 (tn − t j )Ph f (U j )k +

j=0

n−1

Eh2 (tn − t j )Ph g(U j )W j , (2.13)

j=0

where W = W (t j+1 )− W (t j ) denotes the Wiener increments, for the discrete analogue of Mittag–Leffler function in the operators Eh1 (t j ) and Eh2 (t j ), the maximum number of terms N taken into account is chosen such that   N  

t kj   R N (t j ) =  E α,β (t j ) − <ε  Γ (αk + β)  j

k=0

is smaller than a given accuracy ε, which can ensure the Taylor series are sufficient to approximate the generalized Mittag–Leffler function (Seybold and Hilfer 2008).

3 Pathwise regularity results In this section, we will focus on the proof of the pathwise spatial-temporal (Sobolev-Hölder) regularity properties of mild solution. First of all, we shall introduce and prove the following lemmas, which will be needed below.

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G. Zou

Lemma 3.1 (see Haubold et al 2011; Seybold and Hilfer 2008) Let 0 < α < 2 and β ∈ R be arbitrary and πα/2 < μ < min(π, πα). Then there exists a constant C > 0 such that |E α,β (z)| ≤

C , μ ≤ |arg(z)| ≤ π. 1 + |z|

Moreover, the Mittag–Leffler function E α,β (z) satisfies the following differentiation formula d m β−1 [t E α,β (t α )] = t β−1−m E α,β−m (t α ), dt m where the condition β − m > 0 should be satisfied. Lemma 3.2 For any 0 < α < 1 and 0 ≤ σ ≤ ν ≤ 2, there exists certain constant C > 0 such that m d (ν−σ )α+2m ≤ Ct − 2 E (t)χ χh,σ , h1 dt m h,ν

and

m d dt m Eh2 (t)χ

≤ Ct

[2−(ν−σ )]α−2(m+1) 2

χh,σ .

h,ν

Proof For 0 < α < 1 and 0 ≤ σ ≤ ν ≤ 2, by means of Lemma 3.1, we get 2 m   2 ν2 d m d α A E (t)χ = E (−A t ) χ h h dt m α,1 dt m h1 h,ν |Ah t α |(ν−σ ) σ2 2 ≤ Ct (−(σ −ν)−2m) sup χ A h α 2 t∈(0,∞) (1 + |A h t |) ≤ Ct −(ρ−ν)α−2m χ2h,σ , and

2 m d E (t)χ dt m h2

h,ν

ν2 d m  α−1  2 α A t = E (−A t ) χ α,α h h dt m 2 ν = t α−1−m Ah2 E α,α−m (−Ah t α )χ |Ah t α |(ν−σ ) σ2 2 χ A h α 2 t∈(0,∞) (1 + |A h t |)

≤ Ct [2−(ν−σ )]α−2(m+1) sup

≤ Ct [2−(ν−σ )]α−2(m+1) χ2h,σ , 

where we have used |Ah t α |(ν−σ ) /(1 + |Ah t α |)2 ≤ C for 0 ≤ σ ≤ ν ≤ 2. Lemma 3.3 For 0 ≤ t1 < t2 ≤ T and 0 ≤ σ ≤ ν ≤ 2, there exists some constant C > 0 such that (ν−σ )α [Eh1 (t2 ) − Eh1 (t1 )]χh,ν ≤ C(t2 − t1 ) 2 χh,σ , and [Eh2 (t2 ) − Eh2 (t1 )]χh,ν ≤ C(t2 − t1 )

123

2−[2−(ν−σ )]α 2

χh,σ .

Galerkin finite element method for time-fractional...

Proof For 0 < T0 ≤ t1 < t2 ≤ T and 0 ≤ σ ≤ ν ≤ 2, using Lemma 3.2 (m = 1), we have [Eh1 (t2 ) − Eh1 (t1 )]χh,ν

t 2 d = Eh1 (t)χdt dt t1 h,ν

t2 (ν−σ )α+2 ≤ Ct − 2 χh,σ dt t1  (ν−σ )α  )α 2C − 2 − (ν−σ 2 = − t2 t χh,σ (ν − σ )α 1 (ν−σ )α 2C ≤ (t − t1 ) 2 χh,σ , (ν−σ )α 2 (ν − σ )αT0

and t 2 d χdt [Eh2 (t2 ) − Eh2 (t1 )]χh,ν = E (t) h2 t1 dt h,ν

t2 [2−(ν−σ )]α−4 2 ≤ Ct χh,σ dt t1

= ≤

2C 2 − [2 − (ν − σ )]α 2C {2 − [2 − (ν

  [2−(ν−σ )]α−2 [2−(ν−σ )]α−2 2 2 − t2 χh,σ t1

2−[2−(ν−σ )]α − σ )]α}T0

(t2 − t1 )

2−[2−(ν−σ )]α 2

where we have used t2a − t1a ≤ C(t2 − t1 )a for 0 ≤ t1 < t2 ≤ T and 0 ≤ a ≤ 1.

χh,σ ,



Remark 3.1 For 0 ≤ t1 < t2 ≤ T and 0 ≤ σ ≤ ν ≤ 2, the operators E1 (t) and E2 (t) satisfy [E1 (t2 ) − E1 (t1 )]χν ≤ C(t2 − t1 ) and [E2 (t2 ) − Eh2 (t1 )]χν ≤ C(t2 − t1 )

(ν−σ )α 2

χσ ,

2−[2−(ν−σ )]α 2

χσ .

Now we will prove the pathwise regularity results of mild solution to semidiscrete Galerkin approximations (2.11). Theorem 3.1 For any 0 < α < 1 and 0 ≤ σ ≤ ν ≤ 2, let u h (t) be the unique mild solution to (2.11), then there exists a constant C such that sup Eu h (t)h,ν ≤ CEu h0 h,σ .

t∈[0,T ]

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G. Zou

Proof For 0 < T0 ≤ t ≤ T , from (2.12), by virtue of Hölder’s inequality, Lemma 3.2 with m = 0, Itô isometry (2.1) and the assumption on f and g, we have

t Eu h (t) 2h,ν ≤ 3EEh1 (t) Ph u 0 2h,ν + 3E Eh2 (t − s) Ph f (u h (s)) ds2h,ν 0

t + 3E Eh2 (t − s) Ph g (u h (s)) dW (s) 2h,ν 0

≤ Ct −(ν−σ )α Eu h0 2h,σ   t   t 2 +3 dt sup EEh2 (t − s) Ph f (u h (s)) h,ν ds 0 s∈[0,T ]

0



+3

0

≤ Ct

t



EEh2 (t − s) Ph g (u h (s)) 2L 2 U , H˙ ν ds 0( 0 h) 

−(ν−σ )α

Eu h0 2h,σ

+C

t

‡

(t − s)

2α−2

0

sup

s∈[0,T ]

 Eu h (s) 2h,ν ds

,

and the extended Gronwall’s lemma (Kruse 2014) yields −(ν−σ )α

sup Eu h (t)2h,ν ≤ Ct −(ν−σ )α Eu h0 2h,σ ≤ C T0

t∈[0,T ]

Eu h0 2h,σ < ∞. 

The proof is complete.

Remark 3.2 For 0 ≤ σ ≤ ν ≤ 2, let u(t) be a unique mild solution of (1.1) with P(u(t) ∈ H˙ ν ) = 1 for any t ∈ [0, T ], then there exists a constant C such that sup Eu(t)ν ≤ CEu 0 2σ .

t∈[0,T ]

(3.1)

The proof of this result is similar to that of Theorem 3.1 and hence is omitted here. Theorem 3.2 For 0 < α < 1, 0 ≤ t1 < t2 ≤ T and 0 ≤ σ ≤ ν ≤ 2. Then the unique mild solution u h (t) to (2.11) is Hölder continuous and it satisfies Eu h (t2 ) − u h (t1 )h,ν ≤ C(t2 − t1 )γ , )α [2−(ν−σ )]α 2−[2−(ν−σ )]α where γ = min{ (ν−σ , , } > 0. 2 2 2

Proof For 0 ≤ t1 < t2 ≤ T , from the mild solution (2.12), we get

t2 u h (t2 ) − u h (t1 ) = [Eh1 (t2 ) − Eh1 (t1 )]u h0 + Eh2 (t2 − s)Ph f (u h (s))ds 0

t2

t1 Eh2 (t1 − s)Ph f (u h (s))ds + Eh2 (t2 − s)Ph g(u h (s))dW (s) − 0 0

t1 Eh2 (t1 − s)Ph g(u h (s))dW (s) − 0

=: I1 + I2 + I3 , (3.2) where I1 = [Eh1 (t2 ) − Eh1 (t1 )]u h0 ,

123

(3.3)

Galerkin finite element method for time-fractional...

and

t2

I2 =



0

Eh2 (t1 − s)Ph f (u h (s))ds

0 t1

=

t1

Eh2 (t2 − s)Ph f (u h (s))ds −



t2

[Eh2 (t2 − s) − Eh2 (t1 − s)]Ph f (u h (s))ds +

0

Eh2 (t2 − s)Ph f (u h (s))ds

t1

=: I21 + I22 , (3.4) and

I3 =

t2



0

=

t1

Eh2 (t2 − s)Ph g(u h (s))dW (s) −

Eh2 (t1 − s)Ph g(u h (s))dW (s)

0 t1



[Eh2 (t2 − s) − Eh2 (t1 − s)]Ph g(u h (s))dW (s) +

0

t2

Eh2 (t2 − s)Ph g(u h (s))dW (s)

t1

= : I31 + I32 .

(3.5) 

Based on the application of Lemma 3.3, we have EI1 h,ν = EEh1 (t2 ) − Eh1 (t1 )]u h0 h,ν ≤ C(t2 − t1 )

(ν−σ )α 2

Eu h0 h,σ .

(3.6)

Making use of Hölder inequality, Lemma 3.3, and the assumption on f , it follows that

EI21 2h,ν = E

t1

0



t1

≤

[Eh2 (t2 − s) − Eh2 (t1 − s)]Ph f (u h (s)) ds2h,ν  

t1

ds 0

0



t1

≤ Ct1



sup E[Eh2 (t2 − s) − Eh2 (t1 − s)]Ph f (u h (s)) 2h,ν ds

s∈[0,T ]

(t2 − t1 )2−[2−(ν−σ )]α sup Eu h (s) 2h,σ ds s∈[0,T ]

0

≤ C T Eu h0 2h,σ (t2 − t1 )2−[2−(ν−σ )]α .

(3.7)

Using the Hölder inequality, Lemma 3.2 with m = 0, Theorem 3.1 and (2.6) we get

EI22 2h,ν = E

t2 t1



Eh2 (t2 − s) Ph f (u h (s)) ds2h,ν

t2

≤C

[2−(ν−σ )]α−2

(t2 − s)

 ds

t



 1 × ≤

t2

sup E (t2 − s)

t1 s∈[0,T ]

2−[2−(ν−σ )]α 2

Eh2 (t2 − s) Ph f

(u h (s)) 2h,ν ds

C Eu h0 2h,σ (t2 − t1 )[2−(ν−σ )]α . [2 − (ν − σ )]α − 1

(3.8)

123

G. Zou

Applying the Lemma 3.3, Lemma 3.2 with m = 0, Itô isometry and the assumption on g, we obtain

t1 EI31 2h,ν = E [Eh2 (t2 − s) − Eh2 (t1 − s)]Ph g(u h (s))dW (s)2h,ν 0  t1  2 = E[E2 (t2 − s) − E2 (t1 − s)]Ph g(u h (s)) L 2 (U , H˙ ν ) ds 0

0



t1

≤C

0

h

(t2 − t1 )2−[2−(ν−σ )]α sup Eu h (s)2h,σ ds s∈[0,T ]

0

≤ C T Eu h0 2h,σ (t2 − t1 )2−[2−(ν−σ )]α , and

EI32 2h,ν = E

=

t1

t2

t1



≤C

t2

E2 (t2 − s)Ph g(u h (s))dW (s)2h,ν

EE2 (t2 − s)Ph g(u h (s))2L 2 (U 0

t2

t1

≤

(3.9)

˙ ν) 0,H h

ds

(t2 − s)[2−(ν−σ )]α−2 sup Eu(s)2ν,σ ds) s∈[0,T ]

C T0−1 [2 − (ν − σ )]α

Eu h0 2h,σ (t2 − t1 )[2−(ν−σ )]α .

(3.10)

Taking expectation on both side of (3.2), combining (3.3)–(3.10), we conclude that Eu h (t2 ) − u h (t1 )h,ν ≤ C(t2 − t1 )γ , )α [2−(ν−σ )]α 2−[2−(ν−σ )]α , , } > 0. where we take γ = min{ (ν−σ 2 2 2

Remark 3.3 For 0 ≤ σ ≤ ν ≤ 2 and 0 ≤ t1 < t2 ≤ T , the unique mild solution u(t) to (1.1) is Hölder continuous and satisfies Eu(t2 ) − u(t1 )ν ≤ C(t2 − t1 )min{

(ν−σ )α [2−(ν−σ )]α 2−[2−(ν−σ )]α , , } 2 2 2

.

(3.11)

The proof of (3.11) is similar to that of the above results and we omit it here.

4 Error estimates In this section, we will prove the error estimates for both the semidiscrete and fully discrete approximation schemes. First of all, we need the error estimates for the corresponding deterministic equations. Lemma 4.1 For any 0 < α < 1, there exists a certain constant C > 0 such that [Eh1 (t)Ph − E1 (t)]χ ≤ Ch 2 χ2 , and [Eh2 (t)Ph − E2 (t)]χ ≤ Ch 2 χ2 . Proof (1) Let u(t) = E1 (t)χ and u h (t) = Eh1 (t)Ph χ be the mild solutions to the deterministic homogeneous equations as follows ∂tα u(t) + Au(t) = 0, t > 0 with u(0) = χ,

123

Galerkin finite element method for time-fractional...

and

∂tα u h (t) + Ah u h (t) = 0, t > 0 with u h (0) = Ph χ.

We can divide the error u h (t) − u(t) into e(t) = u h (t) − u(t) = (u h (t) − Ph u(t)) + (Ph u(t) − u(t)) := ρ(t) + θ (t).

(4.1) 

For the estimation of θ (t), by (2.9) and (3.1) we get θ (t) = Ph u(t) − u(t) ≤ Ch 2 u(t)2 ≤ Ch 2 χ2 .

(4.2)

Then, using Ph ∂tα θ (t) = ∂tα Ph (Ph u(t) − u(t)) = 0 and the identity Ah Rh = Ph A, we get the following equation for ρ(t): ∂tα ρ(t) + Ah ρ(t) = Ah (Rh − Ph )u(t), t > 0, ρ(0) = 0. In view of (2.12), ρ(t) can be represented by

t ρ(t) = Eh2 (t − s)Ah (Rh − Ph )u(s)ds. 0

By means of Lemma 3.2 (m = 0) and the estimates (2.8)–(2.10), we obtain

t Eh2 (t − s)Ah (Rh − Ph )u(s)ds ρ(t) ≤ 0

t (t − s)α−1 Ah (Rh − Ph )u(s)ds ≤C 0 α 2

≤ C T h χ2 .

(4.3)

Therefore, combining (4.1)–(4.3), by virtue of triangle inequality, we have e(t) ≤ θ (t) + ρ(t) ≤ Ch 2 χ2 . (2) Taking the Laplace transform to H (t) = t α−1 E α,α (−Ah t α )Ph − t α−1 E α,α (−At α ) and combining (2.4) yields (z) = L {H (t)} = (z α I + Ah )−1 Ph − (z α I + A)−1 . H Based on (z α I + Ah )−1  ≤ C z −α and (z α I + A)−1  ≤ C z −α wherein z ∈ φ = {z ∈ C : |argz| ≤ φ} and (2.9), we have (z)χ ≤ C z −α h 2 χ2 , z ∈ φ , H then the Laplace inversion formula yields



1 (z)χdz ≤ 1 (z)χdz ≤ Ch 2 χ2 , H (t)χ =  e zt H e zt H 2πi Γθ 2π Γθ where Γθ is a line parallel to the imaginary axis in the right half-plane (McLean and Thomée 2004). So we can deduce that [Eh2 (t)Ph − E2 (t)]χ ≤ Ch 2 χ2 . This finishes the proof of the lemma.

123

G. Zou

Theorem 4.1 For 0 < α < 1 and t ∈ [0, T ], let u h (t) and u(t) be the mild solutions of equations (2.11) and (1.1), respectively. Then there exists a constant C > 0, which is independent of h, such that sup Eu h (t) − u(t) ≤ Ch 2 .

t∈[0,T ]

Proof From (2.12) and (2.5), we have u h (t) − u(t) = [Eh1 (t)Ph − E1 (t)]u 0

t

t + Eh2 (t − s)Ph f (u h (s))ds − E2 (t − s) f (u(s))ds 0 0

t

t Eh2 (t − s)Ph g(u h (s))dW (s) − E2 (t − s)g(u(s))dW (s) + 0

0

:= J1 + J2 + J3 ,

(4.4)

where J1 = [Eh1 (t)Ph − E1 (t)]u 0 , and J2 = =

t 0

t 0

Eh2 (t − s)Ph f (u h (s))ds −

t 0

E2 (t − s) f (u(s))ds

Eh2 (t − s)Ph [ f (u h (s)) − f (u(s))]ds +

t 0

[Eh2 (t − s)Ph − E2 (t − s)] f (u(s))ds

:= J21 + J22 ,

and

J3 =

t

(4.6)

Eh2 (t − s)Ph g(u h (s))dW (s) −

0

=

0

(4.5)

t

E2 (t − s)g(u(s))dW (s)

0 t

Eh2 (t − s)Ph [g(u h (s)) − g(u(s))]dW (s)



t

+

[Eh2 (t − s)Ph − E2 (t − s)]g(u(s))dW (s)

0

:= J31 + J32 .

(4.7) 

Making use of Lemma 4.1, we obtain EJ1  = E[Eh1 (t)Ph − E1 (t)]u 0  ≤ Ch 2 Eu 0 2 .

(4.8)

Applying the Hölder inequality, Lemma 3.2 (m = 0), Lemma 4.1, Remark 3.2 and the assumption on f , we deduce that

t EJ21 2 = E Eh2 (t − s)Ph [ f (u h (s)) − f (u(s))]ds2 0  

 

t

≤

t

dt 0



≤ CT 0

123

t

sup EEh2 (t − s)Ph [ f (u h (s)) − f (u(s))]2 ds

0 s∈[0,T ]

(t − s)(2α−2) sup Eu h (s) − u(s)2 ds, s∈[0,T ]

(4.9)

Galerkin finite element method for time-fractional...

and



t

EJ22 2 = E 0 t



≤

[Eh2 (t − s) Ph − E2 (t − s)] f (u (s)) ds2  

t

dt 0

≤ CT h



4



sup E[Eh2 (t − s) Ph − E2 (t − s)] f (u (s)) 2 ds

0 s∈[0,T ] t



Eu (s) 22 ds

sup

0 s∈[0,T ]

≤ C † T 2 Eu 0 2 h 4 .

(4.10)

Using Lemma 3.2 (m = 0), Lemma 4.1, Remark 3.2, Itô isometry (2.1) and the assumption on g, we obtain

t Eh2 (t − s)Ph [g(u h (s)) − g(u(s))]dW (s)2 EJ31 2 = E 0

t = EEh2 (t − s)Ph [g(u h (s)) − g(u(s))]2L 0 ds 2 0

t (t − s)(2α−2) sup Eu h (s) − u(s)2 ds, (4.11) ≤C s∈[0,T ]

0

and

t EJ32 2 = E [Eh2 (t − s)Ph − E2 (t − s)]g(u(s))dW (s)2 0

t = E[Eh2 (t − s)Ph − E2 (t − s)]g(u(s))2L 0 ds 2 0  

≤ Ch 4

t

sup Eu(s)22 ds

0 s∈[0,T ]

≤ C ‡ T Eu 0 2 h 4 .

(4.12)

Taking expectation on (4.4) and together with (4.5)–(4.12) and with the application of integral version of Gronwall’s lemma gives sup Eu h (t) − u(t)2 ≤ Ch 4 .

t∈[0,T ]

This completes the proof of Theorem 4.1. Theorem 4.2 For any 0 < α < 1, let U n and u(tn ) satisfy (2.13) and (1.1), respectively. Then there exists a constant C > 0 such that EU n − u(tn )2 ≤ C(k 2α + k 4−2α + k 2 h 4 + h 4 ).

123

G. Zou

Proof From (2.13) and (2.5), we have U n − u(tn ) = [Eh1 (tn )Ph − E1 (tn )]u 0 +

− +

n−1

Eh2 (tn − t j )Ph f (U j )k

j=0 tn

0 n−1

E2 (tn − s) f (u(s))ds

(4.13)



tn

Eh2 (tn − t j )Ph g(U j )W j −

E2 (tn − s)g(u(s))dW (s)

0

j=0

:= L 1 + L 2 + L 3 , where we define L 1 = [Eh1 (tn )Ph − E1 (tn )]u 0 ,

(4.14)

and L2 =

n−1



Eh2 (tn − t j )Ph f (U )k − j

=

E2 (tn − s) f (u(s))ds

0

j=0 n−1

tn

t j+1

[Eh2 (tn − t j )Ph f (U j ) − E2 (tn − s) f (u(s))]ds

j=0 t j

=

n−1

t j+1

Eh2 (tn − t j )Ph [ f (U j ) − f (u(t j ))]ds

j=0 t j

+

n−1

t j+1

Eh2 (tn − t j )Ph [ f (u(t j )) − f (u(s))]ds

j=0 t j

+

n−1

t j+1

[Eh2 (tn − t j )Ph − E2 (tn − t j )] f (u(s))ds

j=0 t j

+

n−1

t j+1

[E2 (tn − t j ) − E2 (tn − s)] f (u(s))ds

j=0 t j

:= L 21 + L 22 + L 23 + L 24 ,

123

(4.15)

Galerkin finite element method for time-fractional...

and L3 =

t

n−1 

Eh2 (tn − t j )Ph g(U j )W j − 0n E2 (tn − s)g(u(s))dW (s)

j=0

=

n−1  j=0

=

t j+1 t j [Eh2 (tn

n−1 

t j+1 tj

j=0

+

n−1 

t j+1 tj

j=0

+

n−1  j=0

+

− t j )Ph g(U j ) − E2 (tn − s)g(u(s))]dW (s)

Eh2 (tn − t j )Ph [g(U j ) − g(u(t j ))]dW (s) Eh2 (tn − t j )Ph [g(u(t j )) − g(u(s))]dW (s)

t j+1 t j [Eh2 (tn

n−1  j=0

− t j )Ph − E2 (tn − t j )]g(u(s))dW (s)

t j+1 t j [E2 (tn

− t j ) − E2 (tn − s)]g(u(s))dW (s)

:= L 31 + L 32 + L 33 + L 34 .

(4.16)

For any 0 < α < 1, using Lemma 4.1, it follows that EL 1 2 = E[Eh1 (tn )Ph − E1 (tn )]u 0 2 ≤ Ch 4 Eu 0 22 .

(4.17)

Applying the Hölder inequality, Lemma 3.2 (m = 0), Lemma 4.1, Remarks 3.1–3.3, and the assumption on f , we get the following estimates by 2 n−1 t j+1 2 j EL 21  = E E (t − t )P [ f (U ) − f (u(t ))]ds h2 n j h j j=0 t j  

 

n−1 t j+1 t j+1

j 2 ≤C ds EEh2 (tn − t j )Ph [ f (U ) − f (u(t j ))] ds tj

j=0

≤ Ck

n−1 t j+1

tj

(tn − t j )2α−2 EU j − u(t j )2 ds

j=0 t j

≤ Ck 2α

n−1

(n − j)2α−2 EU j − u(t j )2 ,

(4.18)

j=0

123

G. Zou

and

2 n−1 t

j+1      EL 22 2 = E E − t [ f u t t P − f (u (s))]ds h2 n j h j j=0 t j    

t j+1 n−1 t j+1

     2 ≤C ds EEh2 tn − t j Ph [ f u t j − f (u (s))] ds tj

j=0

≤ Ck

n−1

tj

t j+1



tn − t j

2α−2

  Eu t j − u (s) 2 ds

j=0 t j

≤ C † k 2α , (4.19) and

2 n−1 t

j+1     2 EL 23  = E [Eh2 tn − t j Ph − E2 tn − t j ] f (u (s)) ds j=0 t j    



n−1 t j+1 t j+1

    2 ≤C ds EEh2 tn − t j Ph − E2 tn − t j ] f (u (s))  ds tj

j=0

≤ Ckh 4

n−1

tj t j+1

j=0 t j

sup Eu (s) 22 ds

s∈[0,T ]

≤ Ck 2 h 4 Eu 0 22 ,

(4.20)

and

2 n−1 t j+1   t − [ E − t E − s)] f ds EL 24 2 = E (t (u (s)) 2 n j 2 n j=0 t j    

t j+1 n−1 t j+1

  2 ≤C ds E[E2 tn − t j − E2 (tn − s)] f (u (s))  ds j=0

≤ Ck

n−1

tj t j+1

tj



s − tj

j=0 t j

2−2α

sup Eu (s) 2 ds

s∈[0,T ]

≤ Ck 4−2α Eu 0 2 . (4.21)

123

Galerkin finite element method for time-fractional...

By means of Lemma 3.2 (m = 0), Lemma 4.1, Remarks 3.1–3.3, Itô isometry (2.1) and the global Lipschitz and growth conditions (2.7), we can obtain 2 n−1 t   j+1      2 j EL 31  = E − g u t E − t [g U t P ]dW (s) h2 n j h j j=0 t j  

n−1   t j+1

   2   j ≤C EEh2 tn − t j Ph [g U − g u t j ] L 0 ds ≤C

2

tj

j=0 n−1



t j+1

tn − t j

2α−2

  EU j − u t j 2 ds

j=0 t j n−1

≤ Ct1−1 k 2α

  (n − j)2α−2 EU j − u t j 2 ,

(4.22)

j=0

and

2 n−1 t

j+1      2 EL 32  = E Eh2 tn − t j Ph [g u t j − g (u (s))]dW (s) j=0 t j   n−1 t j+1

     2 ≤C EEh2 tn − t j Ph [g u t j − g (u (s))] L 0 ds n−1

≤C

2

tj

j=0

t j+1



tn − t j

2α−2

  Eu t j − u (s))2 ds

j=0 t j

≤ Ct1−1 k 2α ,

(4.23)

and EL 33 2 = E

n−1

t j+1

    [Eh2 tn − t j Ph − E2 tn − t j ]g (u (s)) dW (s) 2

j=0 t j

≤C

 n−1

j=0

≤ Ch 4

t j+1



EEh2 tn − t j

tj

n−1

t j+1

j=0 t j









Ph − E2 tn − t j ]g (u (s)) 2L 0 ds 2

sup Eu (s) 22 ds

s∈[0,T ]

≤ Ct1−1 k 2 h 4 Eu 0 2 ,

(4.24)

123

G. Zou

and

2 n−1 t

j+1 [ E (t − t ) − E (t − s)]g(u(s))dW (s) EL 34 2 = E 2 n j 2 n j=0 t j   n−1 t j+1

2 ≤C E[E2 (tn − t j ) − E2 (tn − s)]g(u(s)) L 0 ds ≤C

2

tj

j=0 n−1

t j+1

(s − t j )2−2α sup Eu(s)2 ds s∈[0,T ]

j=0 t j

≤ Ct1−1 k 4−2α Eu 0 2 .

(4.25)

Taking expectation on (4.13) and combining (4.14)–(4.25) and by the discrete version of Gronwall’s lemma, we have EU n − u(tn )2 ≤ C(k 2α + k 4−2α + k 2 h 4 + h 4 ). 

This ends the proof of Theorem 4.2.

5 Numerical example In this section, the numerical results are performed to illustrate visually the previously claimed strong convergence rates. To this end, we consider the semi-linear time-fractional stochastic diffusion equations in one space dimensional as follows: ⎧ ∂ 2 u(x,t) ∂ W (x,t) α ⎪ ⎨∂t u(x, t) = ∂ x 2 + sin[u(x, t)] + u(x, t) ∂t , (x, t) ∈ (0, 2) × (0, 2], u(0, t) = u(2, t) = 0, t ∈ (0, 2), ⎪ ⎩ u(x, 0) = x, x ∈ (0, 2). The “exact” solution is approximated by using the fully discretization with a very small time step size kexact = 2−12 and taking h exact = 1/200 for spatial discretization. The error en := U n − u(tn ) can be measured by the normalized error Een 2 in discrete L 2 -norm, where the expected values E·2 are approximated by computing averages over 200 samples. Table 1 shows the numerical errors and convergence rates in spatial direction by taking different spatial mesh sizes, where the fixed and sufficiently small time step sizes are taken. We can see that the expected spatial rate of convergence are in agreement with the theoretical results O (h 4 ). Table 2 exhibits the numerical errors in temporal direction with different α. We Table 1 Numerical errors and convergence rates α = 0.3

α = 0.6

Een 2

Rate

Een 2

Rate

Een 2

1/5

2.5576e−02

–

2.4908e−02

–

2.3772e−02

–

1/10

1.6437e−03

3.96

1.5898e−03

3.97

1.4963e−03

3.99

1/20

1.0415e−04

3.97

1.0004e−04

3.98

9.4157e−05

3.99

1/40

6.6473e−06

3.97

6.2099e−06

3.99

5.6852e−06

4.01

h

123

α = 0.9 Rate

Galerkin finite element method for time-fractional... Table 2 Numerical errors and convergence rates α = 0.3

α = 0.6

Een 2

Rate

Een 2

Rate

Een 2

Rate

2−4

3.4305e−02

–

3.4028e−02

–

3.3288e−02

–

2−5

2.2167e−02

0.63

1.4508e−02

1.23

9.4290e−03

1.82

2−6

1.4127e−02

0.64

6.1847e−03

1.23

2.5261e−03

1.83

2−7

9.0656e−03

0.64

2.5823e−03

1.24

7.4074e−04

1.83

k

(a)

α = 0.9

(b) 4

u(x,t)

u(x,t)

4 2 0 2 1

x

0 0

1

2 0 2

2

1 0 0

x

t

(c)

2

1

t

(d) 4

u(x,t)

E[u(x,t)]

4 2

3 2

0 2 1

x

0 0

1

t

2

1

0

0.5

1

t

1.5

2

Fig. 1 The numerical results with α = 0.5. Plots of a and b show the numerical simulations of u(x, t) from two independent pathwise of the noises, the time series of results a and b are shown in d with red and blue line, respectively. c The expected values E[u(x, t)] by the mean of 200 independent realizations, the time series of results are shown in d with black line

know that the expected temporal rate of convergence is closer to the theoretical convergence order O (k 2α ), that is, the numerical results are in consistent with the theoretical ones. Figures 1 and 2 show the numerical results obtained by fully discrete scheme with k = 2−6 and h = 1/20 for α = 0.5, 0.9, respectively. For details, Figs. 1 and 2a, b display the numerical solutions of u(x, t) from two independent realizations of the noise. The expected values E[u(x, t)] by the mean of 200 independent samples are reported in Figs. 1 and 2c. Figs. 1 and 2d exhibit the time series of the corresponding results in Figs. 1 and 2a–c. The results demonstrate that the numerical solutions are different when imposed by different paths of the noise, but the expectation of the solutions are relatively stable.

6 Conclusions In this paper, we present a Galerkin finite element method for solving the time-fractional stochastic diffusion equations driven by multiplicative noise. The pathwise regularity prop-

123

G. Zou

(a)

(b) 4

u(x,t)

u(x,t)

4 2 0 2 1

x

0 0

1

2 0 2

2

1

x

t

(c)

0 0

2

1

t

(d) 4

u(x,t)

E[u(x,t)]

4 2

3 2

0 2 1

x

0 0

1

t

2

1

0

0.5

1

t

1.5

2

Fig. 2 The numerical results with α = 0.9. Plots of a and b show the numerical simulations of u(x, t) from two independent pathwise of the noises, the time series of results a and b are shown in d with red and blue line, respectively. c The expected values E[u(x, t)] by the mean of 200 independent realizations, the time series of results are shown in d with black line

erties of mild solution to the semidiscrete Galerkin approximations are proved. We establish the strong convergence error estimates for both semidiscrete and fully discrete schemes in a semigroup framework. Numerical results obtained from the introduced example have a good agreement with the theoretical ones. The presented methods and analytical techniques in this study can also be extended to other linear (or nonlinear) time-fractional stochastic partial differential equations. However, it should be noted that the calculation accuracy of the algorithm is relatively low due to the influence of random terms, the high-order schemes (e.g., using the discontinuous Galerkin methods or the Runge–Kutta method in time) provide an interesting direction for our future research. Acknowledgements We would like to thank the reviewers for giving us constructive comments and suggestions which would help us to improve the quality of the paper. This work is supported by National Nature Science Foundation of China (Grant No. 11626085).

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