A finite element method for time fractional partial differential equations

Transcription

1 CHESTER RESEARCH ONLINE Research in Mathematics and its Applications ISSN Series Editors: CTH Baker, NJ Ford A finite element method for time fractional partial differential equations Neville J. Ford, Jingyu Xiao and Yubin Yan 212:RMA:7 211 NJ Ford, J Xiao & Y Yan

2 Chester Research Online Research in Mathematics and its Applications ISSN Please reference this article as follows: Ford, N. J., Xiao, J., & Yan, Y. (211). A finite element method for time fractional partial differential equations. Fractional Calculus and Applied Analysis, 14(3), doi: /s :RMA:7

3 Chester Research Online Research in Mathematics and its Applications ISSN Author Biographies Prof. Neville J. Ford Neville Ford is Dean of Research at the University of Chester, where he has been employed since He holds degrees from Universities of Oxford (MA), Manchester (MSc) and Liverpool (PhD). He founded the Applied Mathematics Research Group in 1991 and became Professor of Computational Applied Mathematics in 2. He has research interests in theory, numerical analysis and modelling using functional differential equations, including delay, mixed and fractional differential equations. Jingyu Xiao Jingyu Xiao is a PhD student at the Harbin Institute of Technology in China, and held a 21/11 Chinese Scholarship Council Fellowship to study at the University of Chester. Dr Yubin Yan Yubin Yan graduated from China in 1985 with a BSc in Mathematics and gained his MSc in Mathematics in 1991 from the Harbin institute of Technology in China. In 1997, he visited the Chalmers University of Technology in Sweden for one year. Yubin completed his PhD in Mathematics at Chalmers in 23. Before he joined Chester in 27 Yubin worked as a research associate, first in in the Department of Mathematics at the University of Manchester and then in the Department of Automatic Control and System Engineering at the University of Sheffield. 212:RMA:7

4 Chester Research Online Research in Mathematics and its Applications ISSN A FINITE ELEMENT METHOD FOR TIME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS Neville J Ford 1, Jingyu Xiao 2, Yubin Yan 3 Abstract In this paper, we consider the finite element method for time fractional partial differential equations. The existence and uniqueness of the solutions are proved by using the Lax-Milgram Lemma. A time stepping method is introduced based on a quadrature formula approach. The fully discrete scheme is considered by using a finite element method and optimal convergence error estimates are obtained. The numerical examples at the end of the paper show that the experimental results are consistent with our theoretical results. MSC 21 : Primary 65M12: Secondary 65M6; 65M6; 65M7;35S1 Key Words and Phrases: Fractional partial differential equations, finite element method, error estimates, numerical examples 1. Introduction In this paper, we will consider the finite element method for the time fractional partial differential equation R D α t u(t, x) u(t, x) = f(t, x), t [, T ], x Ω, (1.1) u(, x) =, x Ω, (1.2) u(t, x) =, t [, T ], x Ω, (1.3) where < α < 1 and Ω is the bounded open domain in R d, d = 1, 2, 3 and Ω is the boundary of Ω. Here = denotes the Laplacian operator with respect to the x variable, R Dα t u(t, x) denotes the x 2 1 x 2 2 x 2 3 left c Year Diogenes Co., Sofia pp. xxx xxx 212:RMA:7

5 Chester Research Online Research in Mathematics and its Applications ISSN N.J. Ford, J. Xiao, Y. Yan Riemann-Liouville fractional derivative with respect to the time variable t defined by R D α t u(t, x) = 1 Γ(1 α) t t u(τ, x) dτ, < α < 1, (1.4) (t τ) α where Γ denotes the Gamma function. Time fractional partial differential equations have many applications in areas such as diffusion processes, electromagnetics, electrochemistry, material science, turbulent flow, chaotic dynamics, etc. [3], [4], [14], [15], [24], [25], [28], [29]. Analytical solutions of time fractional partial differential equations have been studied using Green s functions or Fourier-Laplace transforms [26], [23], [3], [31]. Numerical methods for fractional ordinary differential equations were studied in, for example, [6], [7], [8], [9] [12], [13]. Numerical methods for fractional partial differential equations were also studied by some authors. Liu et al. [22] employed the finite difference method in both space and time and analyzed the stability condition. Sun and Wu [32] proposed a finite difference method for the fractional diffusion-wave equation. Langlands and Henry [18] considered an implicit numerical scheme for fractional diffusion equation. Lin and Xu [2] proposed a finite difference method in time and Legendre spectral method in space. Li and Xu [19] proposed a time-space spectral method for time-space fractional partial differential equation based on a weak formulation and a detailed error analysis was carried out. Recently, Ervin and Roop [1], [11] used finite element methods to find the variational solution of the fractional advection dispersion equation, in which the fractional derivative depends on the space, related to the nonlocal operator, but the time derivative term is of first order, related to the local operator. Adolfsson et al. [1], [2] considered an efficient numerical method to integrate the constitutive response of fractional order viscoelasticity based on the finite element method. Li et al. [16] considered a time fractional partial differential equation by using the finite element method and obtained error estimates in both semidiscrete and fully discrete cases. Jiang et al. [17] considered a high-order finite element method for the time fractional partial differential equaions and proved the optimal order error estimates. In this paper, we will use the framework in Li and Xu [19] in which the authors introduced suitable spaces and norms in which the time fractional differential problem can be formulated into an elliptic problem. Using these spaces, we introduce a finite element method for time fractional partial differential equation and obtain the optimal order error estimates both in semidiscrete and fully discrete cases. 212:RMA:7

6 Chester Research Online Research in Mathematics and its Applications ISSN A FEM FOR TIME FRACTIONAL PDES... 3 The paper is organized as follows. In Section 2, we consider the existence and uniqueness of the solution of the time fractional partial differential equation. In Section 3, we introduce a time discretization scheme and prove the error estimate. In Section 4, we consider the finite element method and obtain the optimal order error estimates in space discretization. Finally in Section 5, we give two numerical examples and show that the numerical results are consistent with the theoretical results. 2. Existence and uniqueness Let C (, T ) denote the space of infinitely differentiable functions on (, T ) and C (, T ) denote the space of infinitely differentiable functions on (, T ) with compact support in (, T ). Let C (, T ) denote the space of infinitely differentiable functions on (, T ) with compact support in (, T ]. Then we introduce the following Sobolev space H α (, T ), < α < 1 which is the closure of C (, T ) with respect to the norm H α (,T ), where H α (,T ) denotes the norm in the usual fractional Sobolev space H α (, T ) [21]. Further, let L 2 (Ω), H 1 (Ω), H 2 (Ω) denote the usual Sobolev spaces with corresponding norms L2 (Ω), H 1 (Ω) and H 2 (Ω), respectively. Denote H 1(Ω) = {v H1 (Ω), v Ω = } with norm H 1 (Ω). Define the space, with < α < 1, B α( (, T ) Ω ) = H α( (, T ), L 2 (Ω) ) L 2 ( (, T ), H 1 (Ω) ). Here B α( (, T ) Ω ) is a Banach space with respect to the norm v ((,T ) Ω) = ( v H α ((,T ),L 2 (Ω)) + v L2 ((,T ),H 1 (Ω)) ) 1/2. We have the following existence and uniqueness theorem. Theorem 2.1. Assume that < α < 1 and f L 2 ((, T ) Ω). Then the system (1.1) - (1.3) has a unique solution in B α( (, T ) Ω ). Further the following stability result holds: u B α/2 ((,T ) Ω C f L 2 ((,T ) Ω). (2.1) The proof of Theorem 2.1 can be found in [19]. For completeness, and because we use the approach later, we will give the ideas of the proof of this theorem here. Recall that the right Riemann-Liouville fractional integral is defined as R t D α T v(t) = 1 Γ(1 α) t t v(τ) dτ, < α < 1. (2.2) (t τ) α 212:RMA:7

7 Chester Research Online Research in Mathematics and its Applications ISSN N.J. Ford, J. Xiao, Y. Yan Definition 2.1. Define Hr α (, T ) as the closure of C (, T ) with respect to the norm H α r (,T ), that is, { } Hr α (, T ) = v L 2 (, T ) v n C (, T ), such that v n v H α r (,T ). Here the norm H α r (,T ) is defined by ( 1/2, v H α r (,T ) = v 2 L 2 (,T ) + v 2 Hr α (,T )) where the seminorm H α r (,T ) is defined by v H α r (,T ) = R t D α T v L2 (,T ). Definition 2.2. Define Hc α (, T ) as the closure of C (, T ) with respect to the norm H α c (,T ), that is, { } Hc α (, T ) = v L 2 (, T ) v n C (, T ), such that v n v H α c (,T ). Here the norm H α c (,T ) is defined by ( ) 1/2, v H α c (,T ) = v 2 L 2 (,T ) + v 2 Hc α(,t ) where the seminorm H α c (,T ) is defined by v H α c (,T ) = ( Dα t v, R t DT α v ) 1/2 L 2, (,T ) where (, ) L2 (,T ) denotes the inner product in L 2 (, T ). Lemma 2.1. Let < α < 1. We have H α r (, T ) = H α c (, T ) = H α (, T ), and the norms H α r (,T ), H α c (,T ) and H α (,T ) are equivalent. P r o o f. We first prove Hr α (, T ) = Hc α (, T ). In fact, v Hr α (, T ), there exists a sequence v n C (, T ) such that v n v H α r (,T ), n, which implies that v n v m H α r (,T ), n, m, that is, {v n } is a Cauchy sequence in Hr α (, T ). We will show that the norm H α r (,T ) is equivalent to the norm H α c (,T ) in C (, T ). Assuming this for the moment, we see that {v n } is also a Cauchy sequence in H α c (,T ). Thus there exists v Hc α (, T ) such that v n v H α c (,T ), n. 212:RMA:7

8 Chester Research Online Research in Mathematics and its Applications ISSN Hence, we have A FEM FOR TIME FRACTIONAL PDES... 5 v v L2 (,T ) v v n L2 (,T ) + v n v L2 (,T ) v v n H α r (,T ) + v n v H α c (,T ), n, (2.3) which implies that v = v and therefore H α r (, T ) H α c (, T ). Similarly, we can prove H α c (, T ) H α r (, T ). Hence we get H α c (, T ) = H α r (, T ). Denote the norm equivalence by H α r (,T ) = H α c (,T ). We now need to prove that the norm H α r (,T ) is equivalent to the norm H α c (,T ) in C (, T ). In fact, v C (, T ), let ṽ be the extension of v by zero outside (, T ). Then we have where we use the fact that which follows from v H α r (,T ) = ṽ H α r (R) = ṽ H α c (R) = v H α c (,T ), v H α c (,T ) = ṽ H α c (R), ṽ H α c (R) = ( Dα t ṽ, t D α + ṽ ) 1/2 = ( Dα t ṽ, t D α T ṽ ) 1/2 = v H α c (,T ). Here we also use the fact that H α r (R) = H α c (R) and the norm H α r (R) = H α c (R) which can be found in [11]. Next we prove that the norm H α r (,T ) is equivalent to the norm H α c (,T ) in space H α r (, T ) = H α c (, T ). In fact, following the ideas of the proof above, v H α r (, T ), there exists a sequence v n C (, T ) such that v n v H α r (,T ), n, and v n v H α c (,T ), n. Thus v H α r (,T ) v v n H α r (,T ) + v n H α r (,T ) v v n H α r (,T ) + C v n H α c (,T ) v v n H α r (,T ) + C v n v H α c (,T ) + C v H α c (,T ). Let n, we get v H α r (,T ) C v H α c (,T ) for any v Hc α (, T ). Similarly we can prove v H α c (,T ) C v H α r (,T ) for any v Hr α (, T ). Thus the norm H α r (,T ) is equivalent to the norm H α c (,T ) in Hr α (, T ) = Hc α (, T ) Finally we turn to the proof of Hc α (, T ) = H α (, T ) and the norm H α c (,T ) is equivalent to the norm H α (,T ) in Hc α (, T ) = H α (, T ). The 212:RMA:7

9 Chester Research Online Research in Mathematics and its Applications ISSN N.J. Ford, J. Xiao, Y. Yan arguments of the proof are the same as the proof for Hr α (, T ) = Hc α (, T ) above. Therefore it suffices to prove that the norm H α c (,T ) is equivalent to the norm H α (,T ) in C (, T ) which follows from v H α c (,T ) = ṽ H α c (R) = ṽ H α c (R) = F ( t Dα T ṽ ) L2 (R) = (iw) α F(ṽ) L2 (R) = ṽ H α (R) = v H α (,T ). Here we used the Plancherel Theorem [26] and remark that F(ṽ) denotes the Fourier transform of ṽ. Lemma 2.2. [26] We have (1) If < p < 1, < q < 1, v() =, t >, then D p+q t v(t) = ( )( ) ( )( ) Dp t v(t) = Dp t v(t), w H p+q (, T ) Dq t (2) Let < α < 1. Then we have ( Dα t w, v ) L 2 (,T ) = ( w, t D α T v ) L 2 (,T ), w Hα (, T ), v C (, T ). Dq t Lemma 2.3. Let < α < 1. Then for any w H α (, T ), v H α/2 (, T ), we have ( Dα t w, v ) L 2 (,T ) = ( Dα/2 t w, t D α/2 T v) L 2 (,T ). P r o o f. Since < α < 1, we have [21] H α/2 (, T ) = H α/2 (, T ). Thus, v H α/2 (, T ), there exists a sequence v n C (, T ) such that v v n H α/2 (,T ), n. By Lemma 2.2, we have, for any w H α (, T ) with w() =, ) ( ( ) ) ( Dα t w, v n L 2 = (,T ) Dα/2 t )( Dα/2 t w, vn L 2 (,T ) = ( Dα/2 t w, t D α/2 T v ) n L 2 (,T ). We now prove that ) ( Dα t w, v n L 2 (,T ) ( Dα t w, v ) L 2, n, (2.4) (,T ) ( Dα/2 t w, t D α/2 T v n )L 2 (,T ) ( Dα/2 t w, t D α/2 T v) L 2, n, (2.5) (,T ) 212:RMA:7

10 Chester Research Online Research in Mathematics and its Applications ISSN A FEM FOR TIME FRACTIONAL PDES... 7 It is easy to prove (2.4). For (2.5), we have ( Dα/2 t w, t D α/2 T v n )L 2 (,T ) ( Dα/2 t w, t D α/2 T v) L 2 (,T ) D α/2 t w L2 (,T ) t D α/2 T v n t D α/2 T v L2 (,T ) D α/2 t w L2 (,T ) v n v α/2 H r (,T ) C D α/2 t w L2 (,T ) v n v H α/2 (,T ), n, where we used Lemma 2.1 in the last inequality. Together these estimates complete the proof of the Lemma 2.3. Proof of Theorem 2.1. The weak formulation of (1.1)-(1.3) is to find u B α/2( (, T ) Ω ) such that A(u, v) = F(v), v B α/2( (, T ) Ω ), (2.6) where the bilinear forms A(, ) and F(v) are defined by using Lemma 2.3, and A(u, v) = ( Dα t u, v ) L 2 ((,T ) Ω) + ( u, v) L 2 ((,T ) Ω) = ( Dα/2 t u, t D α/2 v) L 2 ((,T ) Ω) + ( u, v) L 2 ((,T ) Ω), T F(v) = (f, v) L2 ((,T ) Ω). It is easy to prove the continuities of the bilinear form A(, ) and the right hand functional F(v), that is, there exists a constant C >, such that A(u, v) C u B α/2 ((,T ) Ω) v B α/2 ((,T ) Ω), and F(v) f L2 ((,T ) Ω) v B α/2 ((,T ) Ω). (2.7) We next prove the coercivity of the bilinear operator A(, ) on B α/2( (, T ) Ω ). Note that [2] ( Dα/2 t ϕ, t D α/2 T ϕ) L 2 = ( (,T ) Dα/2 t ϕ, t D α/2 ϕ ) L 2 (R) = cos ( α 2 π) D α/2 t ϕ L2 (R), ϕ C (, T ), where ϕ is the extension of ϕ by zero outside of (, T ). Thus we find that ( Dα/2 t v, t D α/2 T v) L 2 (,T ) is nonnegative for v Hα/2 (, T ), < α < 1 since cos ( α 2 π) is nonnegative for < α < :RMA:7

11 Chester Research Online Research in Mathematics and its Applications ISSN N.J. Ford, J. Xiao, Y. Yan Combining this with Lemma 2.1, we have A(v, v) = ( Dα/2 t v, t D α/2 v) L 2 ((,T ) Ω) + ( v, v) L 2 ((,T ) Ω) T C ( Dα/2 t v, D α/2 t v ) L 2 ((,T ) Ω) + ( v, v) L 2 ((,T ) Ω) C v B α/2 ((,T ) Ω), (2.8) where we applied the Poincaré inequalities in the last inequality, that is and ϕ L2 (Ω) = ϕ H 1 (Ω), ϕ H 1 (Ω), D α/2 t ψ L2 (,T ) = ψ H α/2 (,T ), ψ H α/2 (, T ). By using the well-known Lax-Milgram Lemma, there exists a unique solution u B α/2 ((, T ) Ω) such that (2.6) holds. To prove the stability estimate (2.1), we take v = u in (2.6) to get, by using (2.8) and (2.7), C u B α/2 ((,T ) Ω) A(u, u) = F(u) C f L 2 ((,T ) Ω) u B α/2 ((,T ) Ω), which implies that The proof is complete. u B α/2 ((,T ) Ω) C f L 2 ((,T ) Ω). 3. Time discretization In this section, we will consider the time discretization of (1.1)- (1.3). Define A =, D(A) = H 1 (Ω) H2 (Ω). Then the system (1.1)-(1.3) can be written in the abstract form or, equivalently, Note that R Dt α u(t) + Au(t) = f(t), < t < T, < α < 1, (3.1) u() = u, (3.2) R D α t [u u ](t) + Au(t) = f(t), < t < T, < α < 1. (3.3) R Dt α u(t) = 1 Γ( α) d dt t u(τ) (t τ) α dτ = 1 t u(τ) dτ, Γ( α) (t τ) α+1 where the integral must be interpreted as a Hadamand finite-part integral [6]. Let = t < t 1 < < t n = T be a partition of [, T ]. Then, for fixed t j, j = 1, 2,... n, we have 212:RMA:7

12 Chester Research Online Research in Mathematics and its Applications ISSN A FEM FOR TIME FRACTIONAL PDES... 9 R Dt α [u u ](t j ) = 1 tj u(τ) u() dτ (t τ) 1+α Γ( α) = t α 1 j u(t j t j w) u() Γ( α) w 1+α dw = t α j Γ( α) 1 g(w)w 1 α dw, where g(w) = u(t j t j w) u(). Now, for every j, we replace the integral by a first-degree compound quadrature formula with the equispaced nodes, 1 j, 2 j,..., j j and obtain 1 g(w)w 1 α dw = j α kj g(k/j) + R j (g), k= where the weights α kj satisfy that [6] 1, for k =, α(1 α)j α α kj = 2k 1 α (k 1) 1 α (k + 1) 1 α, for k = 1, 2,..., j 1, (α 1)k α (k 1) 1 α + k 1 α, for k = j, and the remainder term R j (g) satisfies Thus we have R j (g) Cj α 2 k= sup t T g (t). R Dt α [u u ](t j ) = t α ( j j ( α kj u(tj t k ) u() ) ) + R j (g) Γ( α) where Γ(2 α)w kj = t α j = t α ( w kj u(tj t k ) u() ) + t α j Γ( α) R j(g), k= 1, for k =, 2k 1 α + (k 1) 1 α + (k + 1) 1 α, for k = 1, 2,..., j 1, (α 1)k α + (k 1) 1 α k 1 α, for k = j. Let t = t j. We can write (3.3) as j ( w kj u(tj t k ) u() ) +Au(t j ) = f(t j ) t α j Γ( α) R j(g), j = 1, 2, 3,.... k= (3.4) 212:RMA:7

13 Chester Research Online Research in Mathematics and its Applications ISSN N.J. Ford, J. Xiao, Y. Yan Denote U j u(t j ) as the approximation of u(t j ). We can define the following time stepping method j t α ( U j k U ) + AU j = f j, f j = f(t j ). (3.5) k= Let e j = U j u(t j ) denote the error. Then we have the following error estimate: Theorem 3.1. Let U n and u(t n ) be the solutions of (3.5) and (3.1), respectively. Then we have U n u(t n ) C t 2 α. In order to prove this Theorem, we need the following Lemma. Lemma 3.1. [6] For < α < 1, let the sequence {d j } j = 1, 2,... be given by d 1 = 1 and j 1 d j = 1 + α(1 α)j α α kj d j k, j = 2, 3,, where α kj is as in (3.4). Then, 1 d j k=1 sin πα πα(1 α) jα, j = 1, 2,.... Proof of Theorem 3.1. Subtracting (3.5) from (3.4), we get the error equation j t α ( w kj e j k e ) + Ae j = t α j Γ( α) R j(g), or Thus, k= e j = ( t α w j + A ) 1 ( t α j w kj e j k t α ) j Γ( α) R j(g) k=1 = (α j + Γ( α)t α j A) 1( j ) α kj e j k R j (g). k=1 e j (α j + Γ( α)t α j A) 1 (( j k=1 ) α kj e j k + R j (g). 212:RMA:7

14 Chester Research Online Research in Mathematics and its Applications ISSN A FEM FOR TIME FRACTIONAL PDES Note that A is a positive definite elliptic operator, we have, since α j < and Γ( α) <, (αj + Γ( α)t α j A) 1 = sup (α j + Γ( α)t α j λ) 1 < α 1 Hence e j α 1 j λ> ( j α kj e j k + Cj α 2 n 2 k=1 sup t T j = α(1 α)j α α kj e j k + α(1 α)cn 2 k=1 Note that e = u() U =. Denote d 1 = 1, ) u (t) sup t T j 1 d j = 1 + α(1 α)j α α kj d j k, j = 2, 3,..., n. Then we have by induction, k=1 e j Cα(1 α)n 2 sup t T u (t) d j. By Lemma 3.1, we get e j sin πα C sup u (t) j α n 2 C t 2 α. π t T The proof is complete. 4. Space discretization j. u (t). In this section, we will consider the space discretization of (1.1) - (1.3). The variational form of (1.1) - (1.3) is to find u(t) H 1 (Ω), such that, ( Dα t u(t), v) L2 (Ω) + ( u(t), v) L2 (Ω) = (f(t), v) L2 (Ω), v H 1 (Ω). (4.1) Let T denote a partition of Ω into disjoint triangles such that no vertex of any triangle lies on the interior of a side of another triangle and such that the union of the triangles determines a polygonal domain Ω h Ω with boundary vertices on Ω. Let h denote the maximal length of the sides of the triangulation T h. We assume that the triangulations are quasiuniform in the sense that the triangles of T h are of essentially the same size. Let r be any nonnegative integer. We denote by H r (Ω) the norm in H r (Ω). Let S h H 1 be a family of finite element space with the accuracy 212:RMA:7

15 Chester Research Online Research in Mathematics and its Applications ISSN N.J. Ford, J. Xiao, Y. Yan of order r 2, i.e., S h consists of continuous functions on the closure Ω of Ω which are polynomials of degree at most r 1 in each triangle of T h and which vanish outside Ω h, such that, for small h, with v H s (Ω) H 1 (Ω) [33] inf χ Sh v χ L2 (Ω) + h (v χ) L2 (Ω) Ch s v L2 (Ω), for 1 s r. (4.2) The semidiscrete problem of (4.1) is to find the approximate solution u h (t) = u h (, t) S h for each t such that ( Dα t u h (t), χ) L2 (Ω) +( u h (t), χ) L2 (Ω) = (f(t), χ) L2 (Ω), χ S h. (4.3) Let R h : H 1 (Ω) S h be the elliptic projection, or Ritz projection, defined by ( (R h u), χ) = ( u, χ), χ S h. (4.4) We then have, Lemma 4.1. [33] Assume that (4.2) holds. Then, with R h defined by (4.4) we have, with v H s (Ω) H 1(Ω), R h v v L2 (Ω) + h (R h v v) L2 (Ω) Ch s v H s (Ω), for 1 s r. Lemma 4.2. We have Let < α < 1 and assume that w H α ((, T ), L 2 (Ω)). ( Dα t w(t), w(t) ) L 2 (Ω) dt = = ( Dα/2 ( Dα/2 P r o o f. We have, by Lemmas 2.3, 2.1, t w(t), t D α/2 T t w(t), D α/2 t w(t)) L 2 (Ω) dt w(t) ) L 2 (Ω) dt ( Dα t w(t), w(t) ) L 2 dt (4.5) (Ω) ( = Dα t w(x, t) ) w(x, t) dxdt Ω ( = Dα t w(x, ), w(x, ) ) L 2 dx (4.6) (,T ) Ω = ( Dα/2 t w(x, ), t D α/2 T w(x, )) L 2 (,T ) dx = Ω ( Dα/2 t w(t), t D α/2 T w(t)) L 2 (Ω) dt. 212:RMA:7

16 Chester Research Online Research in Mathematics and its Applications ISSN Similarly, we can prove A FEM FOR TIME FRACTIONAL PDES ( Dα t w(t), w(t) ) L 2 (Ω) dt = ( Dα/2 t w(t), D α/2 t w(t) ) L 2 (Ω) dt. The proof is complete. We now come to the main theorem in this section. Theorem 4.1. Let u h and u be the solutions of (4.1) and (4.3). Then D α/2 t (u h (t) u(t)) 2 L 2 (Ω) dt Ch2r P r o o f. We write D α/2 t u(t) 2 H r (Ω) dt. u h u = θ + ρ, where θ = u h R h u, ρ = R h u u. The second term is easily bounded by Lemma 3.1 and has the obvious estimates D α/2 t ρ(t) 2 L 2 (Ω) dt Ch2r D α/2 t u(t) 2 H r (Ω) dt. (4.7) In order to estimate θ, we note that by our definitions, ( D α t θ(t), χ) L2 (Ω) + ( θ(t), χ) L2 (Ω) = ( D α t (u h (t) R h u(t)), χ) L2 (Ω) + ( (u h (t) R h u(t)), χ) L2 (Ω) = ( D α t u h (t), χ) L2 (Ω) + ( u h (t), χ) L2 (Ω) ( D α t R h u(t), χ) L2 (Ω) + ( R h u(t), χ) L2 (Ω) = (f(t), χ) L2 (Ω) ( D α t R h u(t), χ) L2 (Ω) + ( R h u(t), χ) L2 (Ω) = ( D α t u(t), χ) L2 (Ω) ( D α t R h u(t), χ) L2 (Ω) = ((I R h ) D α t R h u(t), χ) L2 (Ω) = ( D α t ρ(t), χ) L2 (Ω), χ S h. Choose χ = θ(t) and integrating on both sides with respect to t on [, T ], we obtain ( Dα t θ(t), θ(t) ) L 2 (Ω) dt+ ( T θ(t), χ )L 2 (Ω) dt = ( Dα t ρ(t), θ(t) ) L 2 (Ω) dt. 212:RMA:7

17 Chester Research Online Research in Mathematics and its Applications ISSN N.J. Ford, J. Xiao, Y. Yan By Lemma 4.2, we have, for any small ɛ >, = D α/2 t θ(t) 2 L 2 (Ω) dt + ( D α t ρ(t), θ(t)) L2 (Ω) dt = ( C ɛ D α/2 θ(t) 2 L 2 (Ω) dt ( D α/2 t ρ(t), D α/2 t θ(t)) L2 (Ω) dt t ρ(t) 2 L 2 (Ω) dt + ɛ D α/2 t θ(t) 2 L 2 (Ω) For sufficiently small ɛ >, we get, by (4.7) C D α/2 t θ(t) 2 L 2 (Ω) dt + θ(t) 2 L 2 (Ω) dt ) dt. D α/2 t ρ(t) 2 L 2 (Ω) dt Ch2r D α/2 t u(t) 2 H r (Ω) dt. (4.8) Combining (4.7) with (4.8), we complete the proof of the theorem. Corollary 4.1. Let u h and u be the solutions of (4.1) and (4.3). Then u h (t) u(t) 2 L 2 (Ω) dt Ch2r D α/2 t u(t) 2 H r (Ω) dt. (4.9) = P r o o f. Note that, by Theorem 4.1, We have u h (t) u(t) 2 L 2 (Ω) dt = u h (x, t) u(x, t) 2 dxdt Ω u h (x, t) u(x, t) 2 dtdx Ω Ω D α t (u h (t) u(t) 2 L 2 (Ω) dt Ch2r which is (4.9). The proof of the Lemma is complete. 5. Numerical simulations D α t (u h (x, t) u(x, t)) 2 dtdx D α/2 t u(t) 2 H r (Ω) dt, In this section, we present some numerical results by using the finite element method for solving the time-fractional partial differential equation (1.1) (1.3). The numerical results are consistent with our theoretical results. We can see that convergence rate of numerical solutions is of order 212:RMA:7

18 Chester Research Online Research in Mathematics and its Applications ISSN A FEM FOR TIME FRACTIONAL PDES α as the time stepsize tends to zero and of order 2 as the space step size tends to zero, on condition that the exact solution is smooth. Example 5.1. Consider R Dt α u(x, t) d2 u(x, t) = f(x, t), t [, T ], < x < 1, (5.1) dx2 u(x, ) =, < x < 1, (5.2) u(, t) = u(1, t) =, t [, T ], (5.3) where 2 f(x, t) = Γ(3 α) t2 α sin(2πx) + 4π 2 sin(2πx)t 2. The exact solution is u(x, t) = t 2 sin 2πx. The main purpose of these experiments is to check the convergence rate of the numerical solutions with respect to the fractional order α. We use the linear finite element method and therefore the convergence order is O( t 2 α + x 2 ). In the first test, we fix T = 1, α =.5 and x =.1 which is small enough such that the space discretization errors are negligible as compared with the time errors. We choose stepsize t = 1/2 i (i = 1,, 5), then we obtain Table 1 with the estimated convergence rate when α =.5, tending to a limit close to 1.5. In the same way, we can plot the errors in the logscale as functions of the log( t 1 ) for α =.2,.5,.9 in Figures 1, 2, 3 and obtain the convergence rates. Here we investigate both the L 2 -norm and the H 1 -norm in space. x t H 1 -norm 2-norm estimated cvgce. rates e Table 1. Example 1, Fix α =.5 x =.1 In Figures 1-3, one can observe that the error curves are all nearly straight lines. The convergence orders are the slopes of the lines respectively. By example 1, we can observe that the convergence order of the method with respect to the time step is 2 α, which have been shown in Table :RMA:7

19 Chester Research Online Research in Mathematics and its Applications ISSN N.J. Ford, J. Xiao, Y. Yan α est. cvgce. rate α est. cvgce. rate Table 2. Convergence rates in Example H1 norm 2 norm 7 errors in logscale t in logscale Figure 1. Example 1, H 1 norm and L 2 norm of errors with α =.2 x = H1 norm 2 norm 6 errors in logscale t in logscale Figure 2. Example 1, H 1 norm and L 2 norm of errors with α =.5 x =.1 212:RMA:7

20 Chester Research Online Research in Mathematics and its Applications ISSN A FEM FOR TIME FRACTIONAL PDES H1 norm 2 norm errors in logscale t in logscale Figure 3. Example 1, H 1 norm and L 2 norm of errors with α =.9 x =.1 On the other hand, if we fix t small enough, then the convergence rate of space discretization errors can be shown clearly (see Table 3 and figure 4). In this case α =.5, the limiting vale of the convergence rate is consistent with the value of 2 that is expected from the theory. x t H 1 -norm 2-norm est. cvge. rates Table 3. Example 1, Fix α =.5 t =.1 Example 5.2. Consider R Dt α u(x, t) d2 u(x, t) = f(x, t), t [, T ], < x < 1, (5.4) dx2 u(x, ) =, x Ω, (5.5) u(, t) = u(1, t) =, t [, T ]. (5.6) The exact solution is u(x, t) = sin πt sin πx. In this example, we fix T = 1, α =.5 and x =.2. We obtain the convergence rate in Table 4. In Figure 5, we plot the errors in logscale 212:RMA:7

21 Chester Research Online Research in Mathematics and its Applications ISSN N.J. Ford, J. Xiao, Y. Yan 1 H1 norm 2 norm 2 errors in logscale x in logscale Figure 4. Example 1, H 1 norm and L 2 norm of errors with α =.5 t =.1 as functions of time stepsize log( t 1 ) for α =.5. Here we can observe much better convergence than predicted by the theory, and this is worthy of further investigation. x t H 1 -norm 2-norm est. cvge. rates Table 4. Example 2, Fix α =.5 x = Acknowledgements The work of the second author was carried out during her stay at the University of Chester, which is supported financially by China Scholarship Council (CSC[21]36, No ), P. R. China. References [1] K. Adolfsson, M. Enelund, and S. Larsson, Adaptive discretization of integrodifferential equation with a weakly singular convolution kernel, Comput. Methods Appl. Mech. Engrg. 192(23) [2] K. Adolfsson, M. Enelund, and S. Larsson, Adaptive discretization of fractional order viscoelasticity using sparse time history, Comput. Methods Appl. Mech. Engrg. 193(24) :RMA:7

22 Chester Research Online Research in Mathematics and its Applications ISSN A FEM FOR TIME FRACTIONAL PDES errors in logscale t in logscale Figure 5. Example 2, H 1 norm and L 2 norm of errors with α =.5 x =.2 [3] F. Amblard, A. C. Maggs, B. Yurke, A. N. Pargellis, and S. Leibler, Subdiffusion and anomalous local viscoelasticity in actin networks, Phys. Rev. Lett. 77(1996) 447. [4] E. Barkai, R. Metzler, and J. Klafter, Form continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E 61(2) [5] M. Caputo, The Green function of the diffusion of fluids in porous media with memory, Rend. Fis. Acc. Lincei (Ser. 9) 7(1996) [6] K. Diethelm, Generalized compound quadrature formulae for finite-part integrals, IMA J. Numer. Anal. 17(1997) [7] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electronic Transactions on Numerical Analysis 5(1997) 1-6. [8] K. Diethelm, N.J. Ford, A.D. Freed, Yu. Luchko: Algorithms for the fractional calculus: A selection of numerical methods. Computer Methods in Applied Mechanics and Engineering 194(25) [9] N. J. Ford and A. C. Simpson, The numerical solution of fractional differential equations: speed versus accuracy, Numer. Algorithms 26(21) [1] V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numerical. Meth. P. D. E. 22(26) [11] V. J. Ervin and J. P. Roop, Variational solution of fractional advection dispersion equations on bounded domain in R d, Numerical. Meth. P. D. E. 23(27) [12] N.J Ford and J.A. Connolly, Comparison of numerical methods for fractional differential equations. Comm. Pure Appl. Anal. 5(26) [13] N.J Ford and J.A. Connolly, Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations,j. Comput.Appl. Math. 229(29) [14] B. Henry and S. Wearne, Fractional reaction-diffusion, Physica A 276(2) [15] B. Henry and S. Wearne, Existence of turring instabilities in a two-species fractional reaction-diffusion system, SIAM J. Appl. Math. 62(22) :RMA:7

23 Chester Research Online Research in Mathematics and its Applications ISSN N.J. Ford, J. Xiao, Y. Yan [16] C. Li, Z. Zhao, and Y. Chen, Numerical approximation and error estimates of a time fractional order diffusion equation, Proceedings of the ASME 29 International Design Engineering Technical Conference and Computers and Information in Engineering Conference IDETC/CIE 29, San Diego, Californian, USA. [17] Y. Jiang and J. Ma, High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math. 235(211) [18] T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys. 25(25) [19] X. J. Li and C. J. Xu, Existence and uniqeness of the weak solution of the spacetime fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys. 8(21) [2] Y. M. Lin and C. J. Xu, Finite difference/spectral approximation for the time fractional diffusion equations, J. Comput. Phys. 2(27) [21] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Volume 1, Springer-Verlag, [22] F. Liu, S. Shen, V. Anh and I. Turner, Analysis of a discrete non-markovian random walk approximation for the time fractional diffusion equation, ANZIAMJ 46 (25) [23] F. Liu, V. Anh, I. Turner, and P. Zhuang, Time fractional advection dispersion equation, J. Comput. Appl. Math. 13(23) [24] H. P. Müller, R. Kimmich, and J. Weis, NMR flow velocity mapping in random percolation model objects: Evidence for a power-law dependence of the volume-averaged velocity on the probe-volume radius, Phys. Rev. E 54(1996) [25] R. R. Nigmatullin, Realization of the generalized transfer equation in a medium with fractal geometry, Physica B 133(1986) [26] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, [27] J. P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domain in R 2, J. Comp. Appl. Math. 193 (26) [28] H. Scher and M. Lax, Stochastic transport in a disordered solid, Phys. Rev. B 7(1973) [29] H. Scher and E. Montroll, Anomalous transit-time dispersion in amorphous solids, Phys. Rev. B 12(1975) [3] W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 3(1989) [31] W. Wyss, The fractional diffusion equation, J. Math. Phys. 27(1989) [32] Z. Z. Sun and X. N. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56(26) [33] V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer- Verlag, Berlin, Department of Mathematics University of Chester Parkgate Road Chester, CH1 4BJ, UK njford@chester.ac.uk Received: May 8, :RMA:7

24 Chester Research Online Research in Mathematics and its Applications ISSN A FEM FOR TIME FRACTIONAL PDES Department of Mathematics Harbin Institute of Technology Harbin, 151, P.R. China xjingyu23@gmail.com 3 Department of Mathematics University of Chester Parkgate Road Chester, CH1 4BJ, UK y.yan@chester.ac.uk 212:RMA:7