Optimal Convergence Rates of Moving Finite Element Methods for ...

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 792912, 6 pages http://dx.doi.org/10.1155/2013/792912

Research Article Optimal Convergence Rates of Moving Finite Element Methods for Space-Time Fractional Differential Equations Xuemei Gao1 and Xu Han2,3 1

School of Economic Mathematics and School of Finance, Southwestern University of Finance and Economics, Chengdu 611130, China School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China 3 Postal Savings Bank of China Shanghai Branch, Shanghai 200002, China 2

Correspondence should be addressed to Xuemei Gao; [email protected] Received 9 June 2013; Revised 16 August 2013; Accepted 22 August 2013 Academic Editor: Malgorzata Peszynska Copyright © 2013 X. Gao and X. Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper studies the moving finite element methods for the space-time fractional differential equations. An optimal convergence rate of the moving finite element method is proved for the space-time fractional differential equations.

1. Introduction Consider a time-dependent space fractional differential equation of the following form 𝛼 0 𝐷𝑡 𝑢

𝛽

− 𝑝 𝑎 𝐷𝑥𝛽 𝑢 − 𝑞 𝑥 𝐷𝑏 𝑢 = 𝑓,

𝑥 ∈ Ω := (𝑎, 𝑏) , 𝑢 (𝑎, 𝑡) = 0,

𝑡 ∈ 𝐼 := (0, 𝑇] ,

𝑢 (𝑏, 𝑡) = 0,

𝑢 (𝑥, 0) = 𝜑 (𝑥) ,

𝑡 ∈ 𝐼,

𝑥 ∈ Ω,

(1) (2) (3)

where 0 < 𝛼 < 1 and 1 < 𝛽 < 2, 𝑓 and 𝜑 are given functions, 𝛼 𝛽 0 𝐷𝑡 and 𝑎 𝐷𝑥 represent left Caputo fractional differential 𝛽 operators for time and space, respectively, and 𝑥 𝐷𝑏 denotes right Caputo fractional differential operator, 𝑝 and 𝑞 are two nonnegative constants satisfying 𝑝 + 𝑞 = 1. For some nonlinear reaction terms 𝑓 = 𝑓(𝑥, 𝑡, 𝑢), the above equation has finite-time blowup solution which means that the solution tends to infinity as time approaching to a finite time (see e.g., [1]). Moving mesh methods have great advantages in solving blowup problems (see, e.g., [2–9]). Therefore it is important to develop moving mesh methods for solving the fractional differential equations. Although there are many references for developing and analyzing numerical methods on fixed mesh for solving

fractional differential equations, the development of moving mesh methods for fractional differential equations is still in the early stage. Ma and Jiang [6] develop moving mesh collocation methods to solve nonlinear time fractional partial differential equations with blowup solutions. Jiang and Ma [10] analyze moving mesh finite element methods for time fractional partial differential equations and simulate the blow-up solutions. More recently, Ma et al. [11] provide a convergence analysis of moving finite element methods for space fractional differential equations with integer derivatives in time. The convergence rates of moving finite element methods for integer partial differential equations are established by Bank et al. [12–14]. However, fractional derivatives in time will raise much challenge in the convergence analysis of moving finite element methods. The technique using interpolation in the paper [10] is not possible to derive the optimal convergence rates. In this paper, by introducing a fractional Ritz-projection operator, we obtain the optimal convergence rate which is consistent with the numerical predictions in the paper [10]. Moreover, we study the space-time fractional differential equations which are more complex than the timefractional differential equations. Throughout the paper, we use notation 𝐴 ≲ 𝐵 and 𝐴 ≳ 𝐵 to denote 𝐴 ≤ 𝑐𝐵 and 𝐴 ≥ 𝑐𝐵, respectively, where 𝐶 is a generic positive constant independent of any functions and numerical discretization parameters.

2

Journal of Applied Mathematics Define spatial mesh (moving mesh) at time 𝑡𝑛 ,

2. Preliminaries

𝑛 ≡ 𝑏, 𝑎 ≡ 𝑥0𝑛 < 𝑥1𝑛 < ⋅ ⋅ ⋅ < 𝑥𝑁

Define left Riemann-Liouville fractional integral as −𝜎 𝑎 𝐷𝑥 𝑢 (𝑥)

𝑥

1 ∫ (𝑥 − 𝜉)𝜎−1 𝑢 (𝜉) 𝑑𝜉, Γ (𝜎) 𝑎

=

𝑥 > 𝑎, 𝜎 > 0, (4)

where 𝑎 ∈ R or 𝑎 = −∞, and right Riemann-Liouville fractional integral as −𝜎 𝑥 𝐷𝑏 𝑢 (𝑥)

𝑏 1 ∫ (𝑥 − 𝜉)𝜎−1 𝑢 (𝜉) 𝑑𝜉, Γ (𝜎) 𝑥

=

𝑥 < 𝑏, 𝜎 > 0, (5)

where 𝑏 ∈ R or 𝑏 = +∞. The Caputo left and right fractional derivatives are defined by, respectively, 𝜇 −𝜎 𝑛 𝑎 𝐷𝑥 𝑢 (𝑥) =𝑎 𝐷𝑥 𝐷 𝑢 (𝑥) ,

𝜎 = 𝑛 − 𝜇, 𝑛 − 1 ≤ 𝜇 < 𝑛,

𝜇 −𝜎 𝑛 𝑥 𝐷𝑏 𝑢 (𝑥) =𝑥 𝐷𝑏 𝐷 𝑢 (𝑥) ,

𝜎 = 𝑛 − 𝜇, 𝑛 − 1 ≤ 𝜇 < 𝑛.

(6)

𝜇

Define a functional space 𝐻0 (Ω), 𝜇 > 0 as the closure of 𝐶0∞ (Ω) under the norm 󵄩2 󵄩 𝑢)󵄩󵄩󵄩𝐿2 (R) ) ‖𝑢‖𝐻𝜇 (Ω) := (‖𝑢‖2𝐿2 (Ω) + 󵄩󵄩󵄩|𝜔|𝜇 F (̃

1/2

,

(7)

where F(̃ 𝑢) denotes the Fourier transform of 𝑢̃, and 𝑢̃ is the extension of 𝑢 by zero outside of Ω. Let 𝛾 := 𝛽/2. Then, the variational form of problem (1) with boundary conditions (2) and initial condition (3) is given by the following (see [15] for the derivation). Find 𝛾 𝑢 ∈ 𝐻0 (Ω) such that ( 0 𝐷𝛼𝑡 𝑢, V) + 𝐵 (𝑢, V) = 𝐹 (V) , (𝑢 (𝑥, 0) , V) = (𝜑 (𝑥) , V) ,

∀V ∈ ∀V ∈

𝛾 𝐻0

𝛾 𝐻0

(Ω) ,

(Ω) ,

(8) (9)

where 𝐵 (𝑢, V) :=

𝑝 ⟨𝑎 𝐷𝛾𝑥 𝑢,𝑥

𝛾 𝐷𝑏 V⟩

+

𝛾 𝑞 ⟨ 𝑥 𝐷𝑏 𝑢,𝑎

𝐷𝑥𝛾 V⟩ ,

𝐹 (V) := ⟨𝑓, V⟩ ,

(10)

where (𝑢, V) denotes 𝐿 2 inner product, ⟨⋅, ⋅⟩ denotes the 𝜇 duality pairing of 𝐻−𝜇 (Ω), and 𝐻0 (Ω), 𝜇 ≥ 0. The properties of the bilinear form 𝐵(⋅, ⋅) are given by the following Lemma 1 whose proof can be found in [15]. Lemma 1. The bilinear form 𝐵(⋅, ⋅) satisfies the following 𝛾 coercive and continuous properties over space 𝐻0 (Ω): 𝐵 (𝑢, 𝑢) ≳ ‖𝑢‖2𝐻𝛾 (Ω) , |𝐵 (𝑢, V)| ≲ ‖𝑢‖𝐻𝛾 (Ω) ‖V‖𝐻𝛾 (Ω) ,

𝛾

∀𝑢 ∈ 𝐻0 (Ω) , 𝛾

∀𝑢, V ∈ 𝐻0 (Ω) .

(11) (12)

3. Convergence Analysis of Moving Finite Element Method

ℎ𝑛 := max ℎ𝑘𝑛 , 1≤𝑘≤𝑁

𝑘 = 1, . . . , 𝑁,

(14)

ℎ𝑛 := max ℎℓ . 0≤ℓ≤𝑛

𝛾

Define a finite element space V𝑛 ⊂ 𝐻0 (Ω) on the above moving mesh as 𝛾

𝑛 V𝑛 := {V ∈ 𝐻0 (Ω) ∩ 𝐶0 (Ω) : V|[𝑥𝑘−1 ,𝑥𝑘𝑛 ] ∈ 𝑃𝑚−1 } ,

(15)

where 𝑃𝑚−1 denotes the space of polynomials of degree less than or equal to 𝑚 − 1. Then, the moving finite element method for the proposed 𝛾 problems is defined as follows: Find 𝑈𝑛 ∈ V𝑛 ⊂ 𝐻0 (Ω), for 𝑛 = 1, . . . , 𝑀, such that 𝑛 1 ∑ 𝑏𝑘𝑛 (𝑈𝑘 (𝑥) − 𝑈𝑘−1 (𝑥) , V) + 𝐵 (𝑈𝑛 (𝑥) , V) Γ (1 − 𝛼) 𝑘=1 (16)

= 𝐹𝑛 (V) ,

∀V ∈ V𝑛 ,

(𝑈0 (𝑥) , V) = (𝜑 (𝑥) , V) ,

∀V ∈ V0 ,

(17)

1 ≤ 𝑘 ≤ 𝑛.

(18)

where 𝐹𝑛 (V) := ⟨𝑓(⋅, 𝑡𝑛 ), V⟩ and 𝑏𝑘𝑛 =

𝑑𝑠 1 𝑡𝑘 , ∫ Δ𝑡𝑘 𝑡𝑘−1 (𝑡𝑛 − 𝑠)𝛼

In the scheme (16), 𝐵(𝑈𝑛 , V) is the discretization of 𝐵(𝑢, V), and 𝑛 1 ∑ 𝑏𝑘𝑛 (𝑈𝑘 (𝑥) − 𝑈𝑘−1 (𝑥) , V) Γ (1 − 𝛼) 𝑘=1

(19)

is the discretization of the time-fractional derivative (0 𝐷𝑡𝛼 𝑢, V) in (8). To do the convergence analysis, we introduce a fractional Ritz projection operator (an analog of the standard 𝛾 𝛾 one in [16]), 𝑅𝑛 : 𝐻0 (Ω) → V𝑛 defined via, for 𝑢 ∈ 𝐻0 (Ω), 𝐵 (𝑢 − 𝑅𝑛 𝑢, V) = 0,

∀V ∈ V𝑛 .

(20)

For the fractional Ritz projection operator we have the following estimation—Lemma 2. Lemma 2. For the fractional Ritz projection operator defined 𝛾 by (20) and 𝑢 ∈ 𝐻0 (Ω) ∩ 𝐻𝑟 (Ω) (𝛾 ≤ 𝑟 ≤ 𝑚), one has the following estimation: 󵄩 󵄩󵄩 𝑛 𝑟 󵄩󵄩𝑢 − 𝑅𝑛 𝑢󵄩󵄩󵄩𝐿2 (Ω) ≲ (ℎ ) ‖𝑢‖𝐻𝑟 (Ω) .

(21)

Proof. The proof of this lemma can be obtained by simply modifying the proof for Theorem 4.4 in [15].

Define a temporal mesh 0 ≡ 𝑡0 < 𝑡1 < ⋅ ⋅ ⋅ < 𝑡𝑀 ≡ 𝑇, Δ𝑡𝑛 := 𝑡𝑛 − 𝑡𝑛−1 ,

𝑛 ℎ𝑘𝑛 := 𝑥𝑘𝑛 − 𝑥𝑘−1 ,

𝑛 = 0, 1, . . . , 𝑀,

𝑛 = 1, . . . , 𝑀, Δ𝑡 = max Δ𝑡𝑛 . 1≤𝑛≤𝑀

(13)

We will also need the following lemma (see [10]) for proving our main results.

Journal of Applied Mathematics

3

Lemma 3. Suppose that positive numbers 𝜀𝑛 , 𝑛 = 0, 1, . . . , 𝑀, satisfy 𝑛

𝑛 ) 𝜀𝑘−1 + 𝑏1𝑛 𝜇 + 𝜅, 𝑏𝑛𝑛 𝜀𝑛 ≤ ∑ (𝑏𝑘𝑛 − 𝑏𝑘−1

𝑛 = 1, . . . , 𝑀, (22)

𝑘=2

where 𝑏𝑘𝑛 , 𝑘 = 1, . . . , 𝑛, are given by (18), and 𝜅, 𝜇 are positive numbers. Then we have 𝜅 𝜀𝑛 ≤ 𝜇 + 𝑛 , 𝑛 = 1, . . . , 𝑀. (23) 𝑏1

Theorem 4. Assume that the solution of (1)–(3) satisfies 𝑢 ∈ 𝛾 𝐻0 (Ω) ∩ 𝐻𝑟 (Ω) (𝛾 ≤ 𝑟 ≤ 𝑚). Then, the convergence estimation for the moving finite element method (16)-(17) is given by, for 𝑛 = 1, . . . , 𝑀, 󵄩󵄩 𝑛󵄩 𝑟 󵄩󵄩𝑢 (⋅, 𝑡𝑛 ) − 𝑈 󵄩󵄩󵄩𝐿2 (Ω) ≲ ℎ𝑛 [max ‖𝑢 (𝑥, 𝑡)‖𝐻𝑟 (Ω) 0≤𝑡≤𝑇 (24)

𝑛

1 ∑ 𝑏𝑛 (𝑢 (𝑥, 𝑡𝑘 ) − 𝑢 (𝑥, 𝑡𝑘−1 )) − 0 𝐷𝛼𝑡 𝑢, Γ (1 − 𝛼) 𝑘=1 𝑘 (25)

where 𝑢(𝑥, 𝑡) is the exact solution. From [10], we can derive that 2−𝛼 󵄩 󵄩󵄩 𝑛 󵄩󵄩 󵄩 󵄩󵄩T 󵄩󵄩𝐿2 (Ω) ≲ (Δ𝑡) max 󵄩󵄩󵄩𝑢𝑡𝑡 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (Ω) . (26) 0≤𝑡≤𝑇 From (8) we have the identity 𝑛 1 ∑ 𝑏𝑘𝑛 (𝑢 (⋅, 𝑡𝑘 ) − 𝑢 (⋅, 𝑡𝑘−1 ) , V) + 𝐵 (𝑢 (⋅, 𝑡𝑛 ) , V) Γ (1 − 𝛼) 𝑘=1

∀V ∈ V𝑛 . (27)

Let 𝑒𝑛 = 𝑢(⋅, 𝑡𝑛 ) − 𝑈𝑛 . Then subtracting (16) by (27) gives the error equation

(28)

𝜎 := 𝑅𝑛 𝑢 (⋅, 𝑡𝑛 ) − 𝑈 ,

(29)

where 𝑅𝑛 is the fractional Ritz projection operator defined by (20). Then, 𝑒𝑛 = 𝜎𝑛 − 𝜂𝑛 .

(𝜂 − 𝜂

𝑘−1

, V) ∀V ∈ V𝑛 .

Choosing V = 𝜎𝑛 in (31) and using Cauchy-Schwartz inequality with noting that 𝐵(𝜎𝑛 , 𝜎𝑛 ) ≥ 0 (see (11)), we get 𝑛

󵄩 󵄩 𝑛 𝑏𝑛𝑛 󵄩󵄩󵄩𝜎𝑛 󵄩󵄩󵄩𝐿2 (Ω) ≤ ∑ (𝑏𝑘𝑛 − 𝑏𝑘−1 ) 𝑘=2

𝑛

(32)

󵄩 󵄩 + ∑ 𝑏𝑘𝑛 󵄩󵄩󵄩󵄩𝜂𝑘 − 𝜂𝑘−1 󵄩󵄩󵄩󵄩𝐿2 (Ω)

Using Lemma 3, we obtain that 󵄩 0󵄩 󵄩󵄩 𝑛 󵄩󵄩 󵄩󵄩𝜎 󵄩󵄩𝐿2 (Ω) ≤ 󵄩󵄩󵄩󵄩𝜎 󵄩󵄩󵄩󵄩𝐿2 (Ω) 𝑛

󵄩 󵄩 + 𝑇 [ ∑ 𝑏𝑘𝑛 󵄩󵄩󵄩󵄩𝜂𝑘 − 𝜂𝑘−1 󵄩󵄩󵄩󵄩𝐿2 (Ω) 𝑘=1

(33)

󵄩 󵄩 + Γ (1 − 𝛼) 󵄩󵄩󵄩T𝑛 󵄩󵄩󵄩𝐿2 (Ω) ] . Now we estimate the error 󵄩 󵄩󵄩 𝑘 󵄩󵄩𝜂 − 𝜂𝑘−1 󵄩󵄩󵄩 2 󵄩𝐿 (Ω) 󵄩 󵄩 = 󵄩󵄩󵄩(𝑅𝑘 𝑢 (⋅, 𝑡𝑘 ) − 𝑢 (⋅, 𝑡𝑘 )) − (𝑅𝑘−1 𝑢 (⋅, 𝑡𝑘−1 ) 󵄩 − 𝑢 (⋅, 𝑡𝑘−1 ))󵄩󵄩󵄩𝐿2 (Ω) 󵄩 = 󵄩󵄩󵄩(𝑅𝑘 (𝑢 (⋅, 𝑡𝑘 ) − 𝑢 (⋅, 𝑡𝑘−1 ))) − (𝑢 (⋅, 𝑡𝑘 ) − 𝑢 (⋅, 𝑡𝑘−1 ))

(34)

󵄩 ≤ 󵄩󵄩󵄩(𝑅𝑘 (𝑢 (⋅, 𝑡𝑘 ) − 𝑢 (⋅, 𝑡𝑘−1 )))

󵄩 󵄩 + 󵄩󵄩󵄩(𝑅𝑘 𝑢 (⋅, 𝑡𝑘−1 ) − 𝑅𝑘−1 𝑢 (⋅, 𝑡𝑘−1 ))󵄩󵄩󵄩𝐿2 (Ω) .

𝑛

𝜂 := 𝑅𝑛 𝑢 (⋅, 𝑡𝑛 ) − 𝑢 (⋅, 𝑡𝑛 ) ,

𝑘

󵄩 − (𝑢 (⋅, 𝑡𝑘 ) − 𝑢 (⋅, 𝑡𝑘−1 ))󵄩󵄩󵄩𝐿2 (Ω)

Define 𝑛

(31)

∑ 𝑏𝑘𝑛 𝑘=1

󵄩 + (𝑅𝑘 𝑢 (⋅, 𝑡𝑘−1 ) − 𝑅𝑘−1 𝑢 (⋅, 𝑡𝑘−1 ))󵄩󵄩󵄩𝐿2 (Ω)

∀V ∈ V𝑛 . 𝑛

+

𝑛

󵄩 󵄩 + Γ (1 − 𝛼) 󵄩󵄩󵄩T𝑛 󵄩󵄩󵄩𝐿2 (Ω) .

Proof. Define the local truncation error as

𝑛 1 ∑ 𝑏𝑘𝑛 (𝑒𝑘 − 𝑒𝑘−1 , V) + 𝐵 (𝑒𝑛 , V) = (T𝑛 , V) , Γ (1 − 𝛼) 𝑘=1

𝑘=2

𝑘=1

0≤𝑡≤𝑇

= (T𝑛 , V) + 𝐹𝑛 (V) ,

𝑛

𝑛 ) (𝜎𝑘−1 , V) + 𝑏1𝑛 (𝜎0 , V) = ∑ (𝑏𝑘𝑛 − 𝑏𝑘−1

󵄩 󵄩 󵄩 󵄩 × 󵄩󵄩󵄩󵄩𝜎𝑘−1 󵄩󵄩󵄩󵄩𝐿2 (Ω) + 𝑏1𝑛 󵄩󵄩󵄩󵄩𝜎0 󵄩󵄩󵄩󵄩𝐿2 (Ω)

󵄩 󵄩 + (Δ𝑡)2−𝛼 max 󵄩󵄩󵄩𝑢𝑡𝑡 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (Ω) .

T𝑛 (𝑥) :=

𝑏𝑛𝑛 (𝜎𝑛 , V) + Γ (1 − 𝛼) 𝐵 (𝜎𝑛 , V)

+ Γ (1 − 𝛼) (T𝑛 , V) ,

Proof. The proof can be found in [10, Lemma 2.4].

󵄩 󵄩 +max 󵄩󵄩󵄩𝑢𝑡 (𝑥, 𝑡)󵄩󵄩󵄩𝐻𝑟 (Ω) ] 0≤𝑡≤𝑇

Using (20), which tells us that 𝐵(𝜂𝑛 , V) = 0 for all V ∈ V𝑛 , we rewrite the error equation (28) as

(30)

Using Taylor theorem and Lemma 2, we have 󵄩 󵄩󵄩 󵄩󵄩(𝑅𝑘 (𝑢 (⋅, 𝑡𝑘 ) − 𝑢 (⋅, 𝑡𝑘−1 )) − (𝑢 (⋅, 𝑡𝑘 ) − 𝑢 (⋅, 𝑡𝑘−1 )))󵄩󵄩󵄩𝐿2 (Ω) 𝑟󵄩 󵄩 ≲ Δ𝑡𝑘 (ℎ𝑘 ) 󵄩󵄩󵄩𝑢𝑡 (⋅, 𝑡𝑘 )󵄩󵄩󵄩𝐻𝑟 (Ω) .

(35)

4


If the fixed spatial mesh is taken, then 𝑅𝑘 ≡ 𝑅𝑘−1 and thereby 󵄩 󵄩󵄩 󵄩󵄩(𝑅𝑘 𝑢 (⋅, 𝑡𝑘−1 ) − 𝑅𝑘−1 𝑢 (⋅, 𝑡𝑘−1 ))󵄩󵄩󵄩𝐿2 (Ω) = 0. (36) Therefore, for moving spatial mesh, it is reasonable to assume that 󵄩 󵄩󵄩 󵄩󵄩(𝑅𝑘 𝑢 (⋅, 𝑡𝑘−1 ) − 𝑅𝑘−1 𝑢 (⋅, 𝑡𝑘−1 ))󵄩󵄩󵄩𝐿2 (Ω) (37) 𝑟󵄩 󵄩 ≲ Δ𝑡𝑘 (ℎ𝑘 ) 󵄩󵄩󵄩𝑢 (⋅, 𝑡𝑘−1 )󵄩󵄩󵄩𝐻𝑟 (Ω) , which is verified numerically by examples in the next section. In addition subtracting (17) by (9) gives that (𝑒0 , V) = 0,

∀V ∈ V0 .

Taking V = 𝜎0 into (38) we derive that 󵄩󵄩 0 󵄩󵄩 󵄩󵄩 0 󵄩󵄩 0 0 󵄩󵄩󵄩𝜎0 󵄩󵄩󵄩2 󵄩󵄩 󵄩󵄩𝐿2 (Ω) = (𝜂 , 𝜎 ) ≤ 󵄩󵄩󵄩𝜂 󵄩󵄩󵄩𝐿2 (Ω) 󵄩󵄩󵄩𝜎 󵄩󵄩󵄩𝐿2 (Ω) . Consequently, it follows from Lemma 2 that 0 0 𝑟 󵄩󵄩󵄩𝜎0 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩𝐿2 (Ω) ≤‖ 𝜂 ‖𝐿2 (Ω) ≲ (ℎ ) ‖𝑢 (𝑥, 0)‖𝐻𝑟 (Ω) .

(38)

(39)

(40)

Combining (26), (34), (35), (37), and (40) into (33) gives that 󵄩 󵄩󵄩 𝑛 󵄩󵄩 󵄩 𝑟 󵄩󵄩𝜎 󵄩󵄩𝐿2 (Ω) ≲ ℎ𝑛 [max ‖𝑢 (𝑥, 𝑡)‖𝐻𝑟 (Ω) + max 󵄩󵄩󵄩𝑢𝑡 (𝑥, 𝑡)󵄩󵄩󵄩𝐻𝑟 (Ω) ] 0≤𝑡≤𝑇 0≤𝑡≤𝑇

4. Numerical Studies of Fractional Ritz Projections In this section, we verify assumption (37) via numerical examples. To this end, we calculate the fractional Ritz projection (defined by (20)) for a given function 𝑔(𝑥). Example 6. Let 𝑔(𝑥) = 𝑥𝜇1 − 𝑥𝜇2 , 𝜇1 , 𝜇2 > 0, the moving meshes {𝑥𝑘𝑛 }𝑁 𝑘=0 be generated by de Boor’ algorithm [19] based on equidistribution principle and satisfy 󵄨 󵄨󵄨 𝑛+1 󵄨󵄨ℎ𝑘 − ℎ𝑘𝑛 󵄨󵄨󵄨 ≲ Δ𝑡𝑛+1 min (ℎ𝑘𝑛+1 , ℎ𝑘𝑛 ) , (44) 󵄨 󵄨 which is often used in the analysis (see, e.g., [17, 18]). The bilinear form is given by 1 1 𝛾 𝛾 (45) ⟨ 𝐷𝛾 𝑢, 𝐷 V⟩ + ⟨𝑥 𝐷1 𝑢, 0 𝐷𝛾𝑥 V⟩ . 2 0 𝑥 𝑥 1 2 We calculate the fractional Ritz projection 𝑅𝑛 𝑔 on the 1storder FEM spaces V𝑛 and verify the error estimation (37). 𝐵 (𝑢, V) :=

On meshes {𝑥𝑘𝑛 }𝑁 𝑘=0 , we construct piecewise linear finite element spaces V𝑛 = span {𝜙𝑘𝑛 (𝑥) , 𝑘 = 1, . . . , 𝑁 − 1} ,

(46)

𝜙𝑘𝑛 (𝑥),

󵄩 󵄩 + (Δ𝑡)2−𝛼 max 󵄩󵄩󵄩𝑢𝑡𝑡 (𝑥, 𝑡)󵄩󵄩󵄩𝐿2 (Ω) . 0≤𝑡≤𝑇

(41) Finally, by applying Lemma 2 and (41) to 󵄩󵄩 𝑛 󵄩󵄩 󵄩 𝑛󵄩 󵄩 𝑛󵄩 󵄩󵄩𝑒 󵄩󵄩𝐿2 (Ω) ≤ 󵄩󵄩󵄩𝜂 󵄩󵄩󵄩𝐿2 (Ω) + 󵄩󵄩󵄩𝜎 󵄩󵄩󵄩𝐿2 (Ω) , (42) which is led by the triangle inequality, we complete the proof of this theorem. Remark 5. In the above proof, we use assumption (37). Now we give comments on the assumption. For fixed meshed, the finite element spaces for time level 𝑡𝑘−1 and 𝑡𝑘 are equal. Therefore, the Ritz-projections of 𝑢 on the finite element spaces remain unchanged and, thus, the left-hand side of (37) is zero, that is, 󵄩 󵄩󵄩 󵄩󵄩(𝑅𝑘 𝑢 (⋅, 𝑡𝑘−1 ) − 𝑅𝑘−1 𝑢 (⋅, 𝑡𝑘−1 ))󵄩󵄩󵄩𝐿2 (Ω) = 0. (43) For moving spatial mesh, the finite element spaces for time level 𝑡𝑘−1 and 𝑡𝑘 are not the same and the different structure highly depends on the mesh movement. However, the difference between the adjacent finite element spaces will not be significant unless the mesh movement is too fast. Therefore, it is reasonable to assume the inequality (37) holds. For integer partial differential equations, assumptions on the mesh movement are generally required for proving the optimal convergence rates for moving finite element methods (see, e.g., [12–14]) and moving finite difference methods (see, e.g., [17]). Not surprisingly, conditions on the mesh movement are needed to prove the optimal convergence rates for moving mesh methods for the fractional differential equations. Numerical examples in the next section show that if the mesh satisfies the condition (44), which is normally used in the papers addressing the convergence analysis of moving mesh methods (see, e.g., [17, 18]), then (37) is verified.

𝑘 = 1, . . . , 𝑁 − 1 are hat functions. So the where fractional Ritz projection of function 𝑔 on the finite element spaces V𝑛 can be written as 𝑁−1

𝑅𝑛 𝑔 (𝑥) = ∑ 𝑔𝑘𝑛 𝜙𝑘𝑛 (𝑥) .

(47)

𝑘=1

Inserting (47) into (20) gives that 𝑁−1

𝐵 (𝑔 − ∑ 𝑔𝑘𝑛 𝜙𝑘𝑛 (𝑥) , V) = 0,

∀V ∈ V𝑛 .

(48)

𝑘=1

Thus, we may obtain a system of algebraic equations by taking V = 𝜙𝑖𝑛 (𝑥), 𝑖 = 1, . . . , 𝑁 − 1: Ag𝑛 = b,

(49)

for unknown vector 𝑇

𝑛 ) , g𝑛 := (𝑔1𝑛 , . . . , 𝑔𝑁−1

(50)

with matrix A and b given by 𝐵 (𝜙1𝑛 , 𝜙1𝑛 ) 𝑛 𝑛 A = ( 𝐵 (𝜙1 , 𝜙2 ) .. . 𝑛 𝐵 (𝜙1𝑛 , 𝜙𝑁−1 )

⋅⋅⋅

𝑛 𝐵 (𝜙𝑁−1 , 𝜙1𝑛 )

⋅⋅⋅ .. .

𝑛 𝐵 (𝜙𝑁−1 , 𝜙2𝑛 ) ) , .. .

(51)

𝑛 𝑛 ⋅ ⋅ ⋅ 𝐵 (𝜙𝑁−1 , 𝜙𝑁−1 )

𝑇

𝑛 )) . b = (𝐵 (𝑔, 𝜙1𝑛 ) , 𝐵 (𝑔, 𝜙2𝑛 ) , . . . , 𝐵 (𝑔, 𝜙𝑁−1

We check the rate (37) in the following way: For fixed time meshsize Δ𝑡𝑛 ≡ 1/𝑀 (𝑀 fixed), calculate the space rate for varying 𝑁, 󵄨󵄨󵄨 log (Error (𝑁) /Error (𝑁 − 1)) 󵄨󵄨󵄨 󵄨󵄨 , (52) Rate for space := 󵄨󵄨󵄨 󵄨󵄨 log ((𝑁 − 1) /𝑁) 󵄨 󵄨󵄨


5

Table 1: Rate for space for 𝑔(𝑥) = 𝑥𝜇1 − 𝑥𝜇2 with 𝜇1 = 7/2, 𝜇2 = 7/3 for Example 6 using 1st-order FEM. 𝑁 8 16 32 64 128 256

𝛾 = 0.4 Error 2.2257𝑒 − 3 4.9491𝑒 − 4 1.3344𝑒 − 4 3.0457𝑒 − 5 7.4518𝑒 − 6 1.8840𝑒 − 6

Rate — 2.1 1.9 2.1 2.0 2.0

𝛾 = 0.7 Error 2.3272𝑒 − 3 5.4391𝑒 − 4 1.4496𝑒 − 4 3.3110𝑒 − 5 7.9799𝑒 − 6 1.9807𝑒 − 6

Rate — 2.1 1.9 2.1 2.1 2.0

𝑛 (𝑥), 𝑘 = 1, . . . , 3𝑁 − 1 are basis functions defined where 𝜙𝑘/3 by

𝜙𝑘𝑛

𝑛 , 𝑥𝑘𝑛 ] ; ℓ𝑛 (𝑥) , 𝑥 ∈ [𝑥𝑘−1 { { 𝑘−1,3 𝑛 𝑛 𝑛 (𝑥) = {ℓ𝑘,0 (𝑥) , 𝑥 ∈ [𝑥𝑘 , 𝑥𝑘+1 ] ; { otherwise, {0,

𝑘 = 1, . . . , 𝑁 − 1, 𝑛 ℓ𝑘,𝑗

𝑛 𝜙𝑘+𝑗/3 ={ 0,

(𝑥) ,

(56)

𝑛 ]; [𝑥𝑘𝑛 , 𝑥𝑘+1

𝑥∈ otherwise,

𝑗 = 1, 2, 𝑘 = 0, 1, . . . , 𝑁 − 1, Table 2: Rate for time for 𝑔(𝑥) = 𝑥𝜇1 − 𝑥𝜇2 with 𝜇1 = 7/2, 𝜇2 = 7/3 for Example 6 using 1st-order FEM. 𝛾 = 0.4

𝑀 32 64 128 256 512 1024

Error 7.8436𝑒 − 5 4.2750𝑒 − 5 2.2223𝑒 − 5 1.1253𝑒 − 5 5.6105𝑒 − 6 2.8238𝑒 − 6

Rate — 0.9 0.9 1.0 1.0 1.0

𝛾 = 0.7 Error 7.8611𝑒 − 5 4.2814𝑒 − 5 2.2225𝑒 − 5 1.1265𝑒 − 5 5.6164𝑒 − 6 2.8266𝑒 − 6

Rate — 0.9 0.9 1.0 1.0 1.0

𝑛 where ℓ𝑘,𝑗 (𝑥), 𝑗 = 0, 1, 2, 3, are the cubic Lagrange basis 𝑛 𝑛 , 𝑥𝑘+2/3 , functions with respect to local mesh points 𝑥𝑘𝑛 , 𝑥𝑘+1/3 𝑛 𝑥𝑘+1 , where 𝑛 𝑥𝑘+1/3 =: 𝑥𝑘𝑛 + 𝑛 𝑥𝑘+2/3

=:

𝑥𝑘𝑛

𝑛 (𝑥𝑘+1 − 𝑥𝑘𝑛 ) , 3

(57)

𝑛 2 (𝑥𝑘+1 − 𝑥𝑘𝑛 ) + . 3

So, the fractional Ritz projection of function 𝑔 on the finite element spaces V𝑛 can be written as 3𝑁−1

𝑛 𝑛 𝑅𝑛 𝑔 (𝑥) = ∑ 𝑔𝑘/3 𝜙𝑘/3 (𝑥) .

for fixed 𝑁, calculate the time rate for varying time meshsize Δ𝑡𝑛 = 1/𝑀,

Inserting (58) into (20) gives that 3𝑁−1

󵄨󵄨 log (Error (𝑀) /Error (𝑀 − 1)) 󵄨 Rate for time := 󵄨󵄨󵄨 log ((𝑀 − 1) /𝑀) 󵄨󵄨

󵄨󵄨 󵄨󵄨 󵄨󵄨 . 󵄨󵄨

𝑛 𝑛 𝐵 (𝑔 − ∑ 𝑔𝑘/3 𝜙𝑘/3 (𝑥) , V) = 0,

(53)

Example 7. We calculate the fractional Ritz projection for function 𝑔(𝑥) = 𝑥𝜇1 − 𝑥𝜇2 , 𝜇1 , 𝜇2 > 0 on the following constructed 3rd-order FEM spaces V𝑛 . Also, like Example 6, we restrict the moving meshes {𝑥𝑘𝑛 }𝑁 𝑘=0 to satisfy (44) and we use the bilinear form 1 1 𝛾 𝛾 ⟨ 𝐷𝛾 𝑢, 𝐷 V⟩ + ⟨𝑥 𝐷1 𝑢,0 𝐷𝑥𝛾 V⟩ . 2 0 𝑥 𝑥 1 2

(54)

(59)

𝑘=1

Ag𝑛 = b,

(60)

for unknown vector 𝑇

𝑛 𝑛 𝑛 , 𝑔2/3 , . . . , 𝑔(3𝑁−1)/3 ) , g𝑛 := (𝑔1/3

(61)

with matrix A and b given by 𝑛 𝑛 𝐵 (𝜙1/3 , 𝜙1/3 ) 𝑛 𝑛 A = ( 𝐵 (𝜙1/3 , 𝜙2/3 ) .. . 𝑛 𝑛 𝐵 (𝜙 , 𝜙 1/3 (3𝑁−1)/3 ) (

⋅⋅⋅

𝑛 𝑛 𝐵 (𝜙(3𝑁−1)/3 , 𝜙1/3 )

𝑛 𝑛 ⋅⋅⋅ 𝐵 (𝜙(3𝑁−1)/3 , 𝜙2/3 ) ), .. .. . . 𝑛 𝑛 ⋅ ⋅ ⋅ 𝐵 (𝜙(3𝑁−1)/3 , 𝜙(3𝑁−1)/3 )) 𝑇

𝑛 𝑛 𝑛 b = (𝐵 (𝑔, 𝜙1/3 ) , 𝐵 (𝑔, 𝜙2/3 ) , . . . , 𝐵 (𝑔, 𝜙(3𝑁−1)/3 )) .

On meshes {𝑥𝑘𝑛 }𝑁 𝑘=0 , we construct finite element spaces (3rdorder piecewise polynomials) 𝑛 V𝑛 = span {𝜙𝑘/3 (𝑥) , 𝑘 = 1, . . . , 3𝑁 − 1} ,

∀V ∈ V𝑛 .

Thus, we may obtain a system of algebraic equations by taking 𝑛 V = 𝜙𝑖/3 (𝑥), 𝑖 = 1, . . . , 3𝑁 − 1:

From Tables 1 and 2, we can see that the convergence order for space is 2 and the convergence order for time is 1, which are consistent with (37) where 𝑟 = 2 for the use of linear finite element methods.

𝐵 (𝑢, V) :=

(58)

𝑘=1

(55)

(62) We check the convergence rate (37) in the same way as Example 6. From the numerics in Tables 3 and 4, we can see that the convergence order for space is 4 and the convergence order for time is 1, which are consistent with (37) where 𝑟 = 4 for the use of 3rd-order finite element methods.

6


Table 3: Rate for space for 𝑔(𝑥) = 𝑥𝜇1 − 𝑥𝜇2 with 𝜇1 = 7/2, 𝜇2 = 7/3 for Example 7 using 3rd-order FEM. 𝑁 4 8 16 32 64 128 256

𝛾 = 0.4 Error 9.3105𝑒 − 3 7.2803𝑒 − 4 5.0060𝑒 − 5 3.2670𝑒 − 6 2.0783𝑒 − 7 1.3076𝑒 − 8 8.4731𝑒 − 10

Rate — 3.7 3.8 3.9 4.0 4.0 3.9

𝛾 = 0.7 Error 9.3105𝑒 − 3 7.2803𝑒 − 4 5.0059𝑒 − 5 3.2670𝑒 − 6 2.0783𝑒 − 7 1.3083𝑒 − 8 7.7842𝑒 − 10

[4] Rate — 3.8 3.9 3.9 4.0 4.0 4.0

Table 4: Rate for time for 𝑔(𝑥) = 𝑥𝜇1 − 𝑥𝜇2 with 𝜇1 = 7/2, 𝜇2 = 7/3 for Example 7 using 3rd-order FEM. 𝑀 16 32 64 128 256 512 1024 2048

𝛾 = 0.4 Error 5.9067𝑒 − 4 1.1010𝑒 − 4 3.2024𝑒 − 5 1.3049𝑒 − 5 6.4572𝑒 − 6 3.1807𝑒 − 6 1.5882𝑒 − 6 7.9375𝑒 − 7

Rate — 2.4 1.7 1.2 1.0 1.0 1.0 1.0

𝛾 = 0.7 Error 5.9068𝑒 − 4 1.1011𝑒 − 4 3.2025𝑒 − 5 1.3049𝑒 − 5 6.4572𝑒 − 6 3.1808𝑒 − 6 1.5882𝑒 − 6 7.9375𝑒 − 7

Rate — 2.4 1.7 1.2 1.0 1.0 1.0 1.0

5. Conclusions This paper studied the moving finite element methods for space-time fractional differential equations. The proof using interpolation (see [10]) was not possible to give the optimal convergence rates. However, using fractional Ritz projection operator proposed in this paper, the optimal convergence rates were obtained, although a natural assumption, which was numerically verified, was used. The proposed moving finite element methods can be readily implemented and applied to the nonlinear fractional differential equations with blowup solutions. These further studies on the applications will be carried out elsewhere.

[5]

[6]

[7]

[8] [9] [10]

[11]

[12]

[13] [14]

[15]

[16]

[17]

Acknowledgment The work was supported by the Scientific Research Fund of Southwestern University of Finance and Economics.

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