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

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.

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][13][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  ≲  and  ≳  to denote  ≤  and  ≥ , respectively, where  is a generic positive constant independent of any functions and numerical discretization parameters.
where  ∈ R or  = −∞, and right Riemann-Liouville fractional integral as where  ∈ R or  = +∞.The Caputo left and right fractional derivatives are defined by, respectively, Define a functional space   0 (Ω),  > 0 as the closure of  ∞ 0 (Ω) under the norm where F(ũ) denotes the Fourier transform of ũ, and ũ is the extension of  by zero outside of Ω.

Convergence Analysis of Moving Finite Element Method
Define a temporal mesh Define spatial mesh (moving mesh) at time   , Define a finite element space V  ⊂   0 (Ω) on the above moving mesh as where  −1 denotes the space of polynomials of degree less than or equal to  − 1.
Then, the moving finite element method for the proposed problems is defined as follows: where   (V) := ⟨(⋅,   ), V⟩ and In the scheme (16), (  , V) is the discretization of (, V), and is the discretization of the time-fractional derivative ( 0    , V) in (8).To do the convergence analysis, we introduce a fractional Ritz projection operator (an analog of the standard one in [16]),   : 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  ∈   0 (Ω) ∩   (Ω) ( ≤  ≤ ), one has the following estimation: Proof.The proof of this lemma can be obtained by simply modifying the proof for Theorem 4.4 in [15].
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  −1 and   are equal.Therefore, the Ritz-projections of  on the finite element spaces remain unchanged and, thus, the left-hand side of (37) is zero, that is, For moving spatial mesh, the finite element spaces for time level  −1 and   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][13][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.

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 ().
On meshes {   }  =0 , we construct piecewise linear finite element spaces where    (),  = 1, . . .,  − 1 are hat functions.So the fractional Ritz projection of function  on the finite element spaces V  can be written as Inserting (47) into (20) gives that Thus, we may obtain a system of algebraic equations by taking V =    (),  = 1, . . .,  − 1: for unknown vector with matrix A and b given by 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  = 2 for the use of linear finite element methods.

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.