Optimal Convergence Rates of Moving Finite Element Methods for Space-Time Fractional Differential Equations

Abstract

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 ₀𝐷^𝛼_𝑡𝑢−𝑝_𝑎𝐷^𝛽_𝑥𝑢−𝑞_𝑥𝐷^𝛽_𝑏()(],𝑢=𝑓,𝑥∈Ω∶=𝑎,𝑏,𝑡∈𝐼∶=0,𝑇𝑢(𝑎,𝑡)=0,𝑢(𝑏,𝑡)=0,𝑡∈𝐼,𝑢(𝑥,0)=𝜑(𝑥),𝑥∈Ω, (1)(2)(3) where 0<𝛼<1 and 1<𝛽<2, 𝑓 and 𝜑 are given functions, ₀𝐷^𝛼_𝑡 and _𝑎𝐷^𝛽_𝑥 represent left Caputo fractional differential operators for time and space, respectively, and _𝑥𝐷^𝛽_𝑏 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 time-fractional 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. Preliminaries

Define left Riemann-Liouville fractional integral as _𝑎𝐷_𝑥^−𝜎𝑢(𝑥)=1Γ(𝜎)^𝑥_𝑎(𝑥−𝜉)^𝜎−1𝑢(𝜉)𝑑𝜉,𝑥>𝑎,𝜎>0, (4) where 𝑎∈ℝ or 𝑎=−∞, and right Riemann-Liouville fractional integral as _𝑥𝐷_𝑏^−𝜎𝑢(𝑥)=1Γ(𝜎)^𝑏_𝑥(𝑥−𝜉)^𝜎−1𝑢(𝜉)𝑑𝜉,𝑥<𝑏,𝜎>0, (5) where 𝑏∈ℝ or 𝑏=+∞. The Caputo left and right fractional derivatives are defined by, respectively, _𝑎𝐷^𝜇_𝑥𝑢(𝑥)=_𝑎𝐷_𝑥^−𝜎𝐷^𝑛𝑢(𝑥),𝜎=𝑛−𝜇,𝑛−1≤𝜇<𝑛,_𝑥𝐷^𝜇_𝑏𝑢(𝑥)=_𝑥𝐷_𝑏^−𝜎𝐷^𝑛𝑢(𝑥),𝜎=𝑛−𝜇,𝑛−1≤𝜇<𝑛. (6) Define a functional space 𝐻^𝜇₀(Ω), 𝜇>0 as the closure of 𝐶^∞₀(Ω) under the norm ‖𝑢‖_𝐻^𝜇_(Ω)∶=‖𝑢‖²_𝐿2(Ω)+|𝜔|^𝜇ℱ(𝑢)²_𝐿2(ℝ)^1/2, (7) where ℱ(𝑢) denotes the Fourier transform of 𝑢, and 𝑢 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 𝑢∈𝐻^𝛾₀(Ω) such that ₀𝐷^𝛼_𝑡𝑢,𝑣+𝐵(𝑢,𝑣)=𝐹(𝑣),∀𝑣∈𝐻^𝛾₀(Ω),(𝑢(𝑥,0),𝑣)=𝜑(𝑥),𝑣,∀𝑣∈𝐻^𝛾₀(Ω), (8)(9) where 𝐵(𝑢,𝑣)∶=𝑝_𝑎𝐷^𝛾_𝑥𝑢,_𝑥𝐷^𝛾_𝑏𝑣+𝑞_𝑥𝐷^𝛾_𝑏𝑢,_𝑎𝐷^𝛾_𝑥𝑣,𝐹(𝑣)∶=𝑓,𝑣, (10) where (𝑢,𝑣) denotes 𝐿₂ inner product, ⟨⋅,⋅⟩ denotes the duality pairing of 𝐻^−𝜇(Ω), and 𝐻^𝜇₀(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 coercive and continuous properties over space 𝐻^𝛾₀(Ω): 𝐵(𝑢,𝑢)≳‖𝑢‖²_𝐻^𝛾_(Ω),∀𝑢∈𝐻^𝛾₀(Ω),|𝐵(𝑢,𝑣)|≲‖𝑢‖_𝐻^𝛾_(Ω)‖𝑣‖_𝐻^𝛾_(Ω),∀𝑢,𝑣∈𝐻^𝛾₀(Ω). (11)(12)

3. Convergence Analysis of Moving Finite Element Method

Define a temporal mesh 0≡𝑡₀<𝑡₁<⋯<𝑡_𝑀≡𝑇,Δ𝑡_𝑛∶=𝑡_𝑛−𝑡_𝑛−1,𝑛=1,…,𝑀,Δ𝑡=max_{1≤𝑛≤𝑀}Δ𝑡_𝑛. (13) Define spatial mesh (moving mesh) at time 𝑡_𝑛, 𝑎≡𝑥^𝑛₀<𝑥^𝑛₁<⋯<𝑥^𝑛_𝑁ℎ≡𝑏,𝑛=0,1,…,𝑀,_𝑛𝑘∶=𝑥^𝑛_𝑘−𝑥^𝑛_𝑘−1ℎ,𝑘=1,…,𝑁,_𝑛∶=max_{1≤𝑘≤𝑁}ℎ^𝑛_𝑘,ℎ_𝑛∶=max_{0≤ℓ≤𝑛}ℎ^ℓ. (14) Define a finite element space 𝒱^𝑛⊂𝐻^𝛾₀(Ω) on the above moving mesh as 𝒱^𝑛∶=𝑣∈𝐻^𝛾₀(Ω)∩𝐶⁰(Ω)∶𝑣∣_[𝑥^𝑛_{𝑘−1,𝑥}^𝑛_𝑘]∈𝑃_𝑚−1, (15) where 𝑃_𝑚−1 denotes the space of polynomials of degree less than or equal to 𝑚−1.

Then, the moving finite element method for the proposed problems is defined as follows: Find 𝑈^𝑛∈𝒱^𝑛⊂𝐻^𝛾₀(Ω), for 𝑛=1,…,𝑀, such that 1Γ(1−𝛼)^𝑛_𝑘=1𝑏^𝑛_𝑘𝑈^𝑘(𝑥)−𝑈^𝑘−1(𝑥),𝑣+𝐵𝑈^𝑛(𝑥),𝑣=𝐹^𝑛(𝑣),∀𝑣∈𝒱^𝑛,𝑈⁰(𝑥),𝑣=𝜑(𝑥),𝑣,∀𝑣∈𝒱⁰, (16)(17) where 𝐹^𝑛(𝑣)∶=⟨𝑓(⋅,𝑡_𝑛),𝑣⟩ and 𝑏^𝑛_𝑘=1Δ𝑡_𝑘^𝑡𝑘_𝑡𝑘−1𝑑𝑠𝑡_𝑛−𝑠^𝛼,1≤𝑘≤𝑛. (18) In the scheme (16), 𝐵(𝑈^𝑛,𝑣) is the discretization of 𝐵(𝑢,𝑣), and 1Γ(1−𝛼)^𝑛_𝑘=1𝑏^𝑛_𝑘𝑈^𝑘(𝑥)−𝑈^𝑘−1(𝑥),𝑣 (19) is the discretization of the time-fractional derivative (₀𝐷^𝛼_𝑡𝑢,𝑣) in (8). To do the convergence analysis, we introduce a fractional Ritz projection operator (an analog of the standard one in [16]), 𝑅_𝑛∶𝐻^𝛾₀(Ω)→𝒱^𝑛 defined via, for 𝑢∈𝐻^𝛾₀(Ω), 𝐵𝑢−𝑅_𝑛𝑢,𝑣=0,∀𝑣∈𝒱^𝑛. (20) For the fractional Ritz projection operator we have the following estimation—Lemma 2.

Lemma 2. For the fractional Ritz projection operator defined by (20) and 𝑢 ∈ 𝐻^𝛾₀(Ω) ∩ 𝐻^𝑟(Ω)  (𝛾≤𝑟≤𝑚), one has the following estimation: 𝑢−𝑅_𝑛𝑢_𝐿2(Ω)≲ℎ^𝑛𝑟‖𝑢‖_𝐻^𝑟_(Ω). (21)

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

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

Lemma 3. Suppose that positive numbers 𝜀_𝑛, 𝑛=0,1,…,𝑀, satisfy 𝑏^𝑛_𝑛𝜀_𝑛≤^𝑛_𝑘=2𝑏^𝑛_𝑘−𝑏^𝑛_𝑘−1𝜀_𝑘−1+𝑏^𝑛₁𝜇+𝜅,𝑛=1,…,𝑀, (22) where 𝑏^𝑛_𝑘, 𝑘=1,…,𝑛, are given by (18), and 𝜅, 𝜇 are positive numbers. Then we have 𝜀_𝑛𝜅≤𝜇+𝑏^𝑛₁,𝑛=1,…,𝑀. (23)

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

Theorem 4. Assume that the solution of (1)–(3) satisfies 𝑢∈𝐻^𝛾₀(Ω)∩𝐻^𝑟(Ω)(𝛾≤𝑟≤𝑚). Then, the convergence estimation for the moving finite element method (16)-(17) is given by, for 𝑛=1,…,𝑀, 𝑢⋅,𝑡_𝑛−𝑈^𝑛_𝐿2(Ω)≲ℎ^𝑟_𝑛max_{0≤𝑡≤𝑇}‖𝑢(𝑥,𝑡)‖_𝐻^𝑟_(Ω)+max_{0≤𝑡≤𝑇}𝑢_𝑡(𝑥,𝑡)_𝐻𝑟(Ω)+(Δ𝑡)^2−𝛼max_{0≤𝑡≤𝑇}𝑢_𝑡𝑡(𝑥,𝑡)_𝐿2(Ω). (24)

Proof. Define the local truncation error as 𝒯^𝑛(𝑥)1∶=Γ(1−𝛼)^𝑛_𝑘=1𝑏^𝑛_𝑘𝑢𝑥,𝑡_𝑘−𝑢𝑥,𝑡_𝑘−1−₀𝐷^𝛼_𝑡𝑢, (25) where 𝑢(𝑥,𝑡) is the exact solution. From [10], we can derive that 𝒯^𝑛_𝐿2(Ω)≲(Δ𝑡)^2−𝛼max_{0≤𝑡≤𝑇}𝑢_𝑡𝑡(𝑥,𝑡)_𝐿2(Ω). (26) From (8) we have the identity 1Γ(1−𝛼)^𝑛_𝑘=1𝑏^𝑛_𝑘𝑢⋅,𝑡_𝑘−𝑢⋅,𝑡_𝑘−1,𝑣+𝐵𝑢⋅,𝑡_𝑛=𝒯,𝑣_𝑛,𝑣+𝐹^𝑛(𝑣),∀𝑣∈𝒱^𝑛. (27) Let 𝑒^𝑛=𝑢(⋅,𝑡_𝑛)−𝑈^𝑛. Then subtracting (16) by (27) gives the error equation 1Γ(1−𝛼)^𝑛_𝑘=1𝑏^𝑛_𝑘𝑒^𝑘−𝑒^𝑘−1,𝑣+𝐵𝑒^𝑛,𝑣=𝒯^𝑛,𝑣,∀𝑣∈𝒱^𝑛. (28) Define 𝜎^𝑛∶=𝑅_𝑛𝑢⋅,𝑡_𝑛−𝑈^𝑛,𝜂^𝑛∶=𝑅_𝑛𝑢⋅,𝑡_𝑛−𝑢⋅,𝑡_𝑛, (29) where 𝑅_𝑛 is the fractional Ritz projection operator defined by (20). Then, 𝑒^𝑛=𝜎^𝑛−𝜂^𝑛. (30) Using (20), which tells us that 𝐵(𝜂^𝑛,𝑣)=0 for all 𝑣∈𝒱^𝑛, we rewrite the error equation (28) as 𝑏^𝑛_𝑛𝜎^𝑛,𝑣+Γ(1−𝛼)𝐵𝜎^𝑛=,𝑣_𝑛𝑘=2𝑏^𝑛_𝑘−𝑏^𝑛_𝑘−1𝜎^𝑘−1,𝑣+𝑏^𝑛₁𝜎⁰+,𝑣_𝑛𝑘=1𝑏^𝑛_𝑘𝜂^𝑘−𝜂^𝑘−1()𝒯,𝑣+Γ1−𝛼^𝑛,𝑣,∀𝑣∈𝒱^𝑛. (31) Choosing 𝑣=𝜎^𝑛 in (31) and using Cauchy-Schwartz inequality with noting that 𝐵(𝜎^𝑛,𝜎^𝑛)≥0 (see (11)), we get 𝑏^𝑛_𝑛𝜎^𝑛_𝐿2(Ω)≤^𝑛_𝑘=2𝑏^𝑛_𝑘−𝑏^𝑛_𝑘−1×𝜎^𝑘−1_𝐿2(Ω)+𝑏^𝑛₁𝜎⁰_𝐿2(Ω)+^𝑛_𝑘=1𝑏^𝑛_𝑘𝜂^𝑘−𝜂^𝑘−1_𝐿2(Ω)+Γ(1−𝛼)𝒯^𝑛_𝐿2(Ω). (32) Using Lemma 3, we obtain that 𝜎^𝑛_𝐿2(Ω)≤𝜎⁰_𝐿2(Ω)+𝑇^𝑛_𝑘=1𝑏^𝑛_𝑘𝜂^𝑘−𝜂^𝑘−1_𝐿2(Ω)+Γ(1−𝛼)𝒯^𝑛_𝐿2(Ω). (33) Now we estimate the error 𝜂^𝑘−𝜂^𝑘−1_𝐿2(Ω)=𝑅_𝑘𝑢⋅,𝑡_𝑘−𝑢⋅,𝑡_𝑘−𝑅_𝑘−1𝑢⋅,𝑡_𝑘−1−𝑢⋅,𝑡_{𝑘−1𝐿2(Ω)}=𝑅_𝑘𝑢⋅,𝑡_𝑘−𝑢⋅,𝑡_𝑘−1−𝑢⋅,𝑡_𝑘−𝑢⋅,𝑡_𝑘−1+𝑅_𝑘𝑢⋅,𝑡_𝑘−1−𝑅_𝑘−1𝑢⋅,𝑡_{𝑘−1𝐿2(Ω)}≤𝑅_𝑘𝑢⋅,𝑡_𝑘−𝑢⋅,𝑡_𝑘−1−𝑢⋅,𝑡_𝑘−𝑢⋅,𝑡_{𝑘−1𝐿2(Ω)}+𝑅_𝑘𝑢⋅,𝑡_𝑘−1−𝑅_𝑘−1𝑢⋅,𝑡_{𝑘−1𝐿2(Ω)}. (34) Using Taylor theorem and Lemma 2, we have 𝑅_𝑘𝑢⋅,𝑡_𝑘−𝑢⋅,𝑡_𝑘−1−𝑢⋅,𝑡_𝑘−𝑢⋅,𝑡_{𝑘−1𝐿2(Ω)}≲Δ𝑡_𝑘ℎ^𝑘𝑟𝑢_𝑡⋅,𝑡_{𝑘𝐻𝑟(Ω)}. (35) If the fixed spatial mesh is taken, then 𝑅_𝑘≡𝑅_𝑘−1 and thereby 𝑅_𝑘𝑢⋅,𝑡_𝑘−1−𝑅_𝑘−1𝑢⋅,𝑡_{𝑘−1𝐿2(Ω)}=0. (36) Therefore, for moving spatial mesh, it is reasonable to assume that 𝑅_𝑘𝑢⋅,𝑡_𝑘−1−𝑅_𝑘−1𝑢⋅,𝑡_{𝑘−1𝐿2(Ω)}≲Δ𝑡_𝑘ℎ^𝑘𝑟𝑢⋅,𝑡_{𝑘−1𝐻𝑟(Ω)}, (37) which is verified numerically by examples in the next section. In addition subtracting (17) by (9) gives that 𝑒⁰,𝑣=0,∀𝑣∈𝒱⁰. (38) Taking 𝑣=𝜎⁰ into (38) we derive that 𝜎⁰²_𝐿2(Ω)=𝜂⁰,𝜎⁰≤𝜂⁰_𝐿2(Ω)𝜎⁰_𝐿2(Ω). (39) Consequently, it follows from Lemma 2 that 𝜎⁰_𝐿2(Ω)≤‖𝜂⁰‖_𝐿²_(Ω)≲ℎ^0𝑟‖𝑢(𝑥,0)‖_𝐻^𝑟_(Ω). (40) Combining (26), (34), (35), (37), and (40) into (33) gives that 𝜎^𝑛_𝐿2(Ω)≲ℎ^𝑟_𝑛max_{0≤𝑡≤𝑇}‖𝑢(𝑥,𝑡)‖_𝐻^𝑟_(Ω)+max_{0≤𝑡≤𝑇}𝑢_𝑡(𝑥,𝑡)_𝐻𝑟(Ω)+(Δ𝑡)^2−𝛼max_{0≤𝑡≤𝑇}𝑢_𝑡𝑡(𝑥,𝑡)_𝐿2(Ω). (41) Finally, by applying Lemma 2 and (41) to 𝑒^𝑛_𝐿2(Ω)≤𝜂^𝑛_𝐿2(Ω)+𝜎^𝑛_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 𝑡_𝑘−1 and 𝑡_𝑘 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, 𝑅_𝑘𝑢⋅,𝑡_𝑘−1−𝑅_𝑘−1𝑢⋅,𝑡_{𝑘−1𝐿2(Ω)}=0. (43) For moving spatial mesh, the finite element spaces for time level 𝑡_𝑘−1 and 𝑡_𝑘 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.

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)=𝑥^𝜇1−𝑥^𝜇2, 𝜇₁, 𝜇₂>0, the moving meshes {𝑥^𝑛_𝑘}^𝑁_𝑘=0 be generated by de Boor' algorithm [19] based on equidistribution principle and satisfy ℎ_𝑘^𝑛+1−ℎ^𝑛_𝑘≲Δ𝑡_𝑛+1minℎ_𝑘^𝑛+1,ℎ^𝑛_𝑘, (44) which is often used in the analysis (see, e.g., [17, 18]). The bilinear form is given by 𝐵()1𝑢,𝑣∶=2⁰𝐷^𝛾_𝑥𝑢,_𝑥𝐷^𝛾₁𝑣+12^𝑥𝐷^𝛾₁𝑢,₀𝐷^𝛾_𝑥𝑣. (45) We calculate the fractional Ritz projection 𝑅_𝑛𝑔 on the 1st-order FEM spaces 𝒱^𝑛 and verify the error estimation (37).

On meshes {𝑥^𝑛_𝑘}^𝑁_𝑘=0, we construct piecewise linear finite element spaces 𝒱^𝑛=span𝜙^𝑛_𝑘(𝑥),𝑘=1,…,𝑁−1, (46) where 𝜙^𝑛_𝑘(𝑥),𝑘=1,…,𝑁−1 are hat functions. So the fractional Ritz projection of function 𝑔 on the finite element spaces 𝒱^𝑛 can be written as 𝑅_𝑛𝑔(𝑥)=^𝑁−1_𝑘=1𝑔^𝑛_𝑘𝜙^𝑛_𝑘(𝑥). (47) Inserting (47) into (20) gives that 𝐵𝑔−^𝑁−1_𝑘=1𝑔^𝑛_𝑘𝜙^𝑛_𝑘(𝑥),𝑣=0,∀𝑣∈𝒱^𝑛. (48) Thus, we may obtain a system of algebraic equations by taking 𝑣=𝜙^𝑛_𝑖(𝑥), 𝑖=1,…,𝑁−1: 𝐀𝐠^𝑛=𝐛, (49) for unknown vector 𝐠^𝑛∶=𝑔^𝑛₁,…,𝑔^𝑛_𝑁−1^𝑇, (50) with matrix 𝐀 and 𝐛 given by 𝐵𝜙𝐀=^𝑛1,𝜙^𝑛₁⋯𝐵𝜙^𝑛_𝑁−1,𝜙^𝑛₁𝐵𝜙^𝑛₁,𝜙^𝑛₂⋯𝐵𝜙^𝑛_𝑁−1,𝜙^𝑛₂𝐵𝜙⋮⋮⋮_𝑛1,𝜙^𝑛_𝑁−1⋯𝐵𝜙^𝑛_𝑁−1,𝜙^𝑛_𝑁−1,𝐛=𝐵𝑔,𝜙^𝑛₁,𝐵𝑔,𝜙^𝑛₂,…,𝐵𝑔,𝜙^𝑛_𝑁−1^𝑇. (51)

We check the rate (37) in the following way: For fixed time meshsize Δ𝑡_𝑛≡1/𝑀 (𝑀 fixed), calculate the space rate for varying 𝑁, Rateforspace(∶=logError(𝑁)/Error(𝑁−1)))),log((𝑁−1/𝑁 (52) for fixed 𝑁, calculate the time rate for varying time meshsize Δ𝑡_𝑛=1/𝑀, Ratefortime(∶=logError(𝑀)/Error(𝑀−1)))).log((𝑀−1/𝑀 (53) 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.

Table 1 

Rate for space for 𝑔(𝑥)=𝑥^𝜇1−𝑥^𝜇2 with 𝜇₁=7/2,𝜇₂=7/3 for Example 6 using 1st-order FEM.

Table 2 

Rate for time for 𝑔(𝑥)=𝑥^𝜇1−𝑥^𝜇2 with 𝜇₁=7/2, 𝜇₂=7/3 for Example 6 using 1st-order FEM.

Example 7. We calculate the fractional Ritz projection for function 𝑔(𝑥)=𝑥^𝜇1−𝑥^𝜇2, 𝜇₁, 𝜇₂>0 on the following constructed 3rd-order FEM spaces 𝒱^𝑛. Also, like Example 6, we restrict the moving meshes {𝑥^𝑛_𝑘}^𝑁_𝑘=0 to satisfy (44) and we use the bilinear form 𝐵()1𝑢,𝑣∶=2⁰𝐷^𝛾_𝑥𝑢,_𝑥𝐷^𝛾₁𝑣+12^𝑥𝐷^𝛾₁𝑢,₀𝐷^𝛾_𝑥𝑣. (54)

On meshes {𝑥^𝑛_𝑘}^𝑁_𝑘=0, we construct finite element spaces (3rd-order piecewise polynomials) 𝒱^𝑛=span𝜙^𝑛_𝑘/3(𝑥),𝑘=1,…,3𝑁−1, (55) where 𝜙^𝑛_𝑘/3(𝑥), 𝑘=1,…,3𝑁−1 are basis functions defined by 𝜙^𝑛_𝑘(𝑥)=ℓ^𝑛_𝑘−1,3(𝑥),𝑥∈𝑥^𝑛_𝑘−1,𝑥^𝑛_𝑘;ℓ^𝑛_𝑘,0(𝑥),𝑥∈𝑥^𝑛_𝑘,𝑥^𝑛_𝑘+1;0,otherwise,𝜙𝑘=1,…,𝑁−1,_{𝑛𝑘+𝑗/3}=ℓ^𝑛_𝑘,𝑗(𝑥),𝑥∈𝑥^𝑛_𝑘,𝑥^𝑛_𝑘+1;0,otherwise,𝑗=1,2,𝑘=0,1,…,𝑁−1, (56) where ℓ^𝑛_𝑘,𝑗(𝑥), 𝑗=0,1,2,3, are the cubic Lagrange basis functions with respect to local mesh points 𝑥^𝑛_𝑘, 𝑥^𝑛_𝑘+1/3, 𝑥^𝑛_𝑘+2/3, 𝑥^𝑛_𝑘+1, where 𝑥^𝑛_𝑘+1/3=∶𝑥^𝑛_𝑘+𝑥^𝑛_𝑘+1−𝑥^𝑛_𝑘3,𝑥^𝑛_𝑘+2/3=∶𝑥^𝑛_𝑘+2𝑥^𝑛_𝑘+1−𝑥^𝑛_𝑘3. (57)

So, the fractional Ritz projection of function 𝑔 on the finite element spaces 𝒱^𝑛 can be written as 𝑅_𝑛𝑔(𝑥)=^3𝑁−1_𝑘=1𝑔^𝑛_𝑘/3𝜙^𝑛_𝑘/3(𝑥). (58) Inserting (58) into (20) gives that 𝐵𝑔−^3𝑁−1_𝑘=1𝑔^𝑛_𝑘/3𝜙^𝑛_𝑘/3(𝑥),𝑣=0,∀𝑣∈𝒱^𝑛. (59)

Thus, we may obtain a system of algebraic equations by taking 𝑣=𝜙^𝑛_𝑖/3(𝑥), 𝑖=1,…,3𝑁−1: 𝐀𝐠^𝑛=𝐛, (60) for unknown vector 𝐠^𝑛∶=𝑔^𝑛_1/3,𝑔^𝑛_2/3,…,𝑔^𝑛_{()3𝑁−1/3}^𝑇, (61) with matrix 𝐀 and 𝐛 given by 𝐵𝜙𝐀=^𝑛1/3,𝜙^𝑛_1/3⋯𝐵𝜙^𝑛_{(3𝑁−1)/3},𝜙^𝑛_1/3𝐵𝜙^𝑛_1/3,𝜙^𝑛_2/3⋯𝐵𝜙^𝑛_{(3𝑁−1)/3},𝜙^𝑛_2/3𝐵𝜙⋮⋮⋮_𝑛1/3,𝜙^𝑛_{(3𝑁−1)/3}⋯𝐵𝜙^𝑛_{(3𝑁−1)/3},𝜙^𝑛_{(3𝑁−1)/3},𝐛=𝐵𝑔,𝜙^𝑛_1/3,𝐵𝑔,𝜙^𝑛_2/3,…,𝐵𝑔,𝜙^𝑛_{(3𝑁−1)/3}^𝑇. (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.

Table 3 

Rate for space for 𝑔(𝑥)=𝑥^𝜇1−𝑥^𝜇2 with 𝜇₁=7/2, 𝜇₂=7/3 for Example 7 using 3rd-order FEM.

Table 4 

Rate for time for 𝑔(𝑥)=𝑥^𝜇1−𝑥^𝜇2 with 𝜇₁=7/2, 𝜇₂=7/3 for Example 7 using 3rd-order FEM.

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.

Acknowledgment

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