Stability and Convergence of an Effective Finite Element Method for Multiterm Fractional Partial Differential Equations

A finite element method (FEM) for multiterm fractional partial differential equations (MT-FPDEs) is studied for obtaining a numerical solution effectively. The weak formulation for MT-FPDEs and the existence and uniqueness of the weak solutions are obtained by the well-known Lax-Milgram theorem. The Diethelm fractional backward difference method (DFBDM), based on quadrature for the time discretization, and FEM for the spatial discretization have been applied to MT-FPDEs. The stability and convergence for numerical methods are discussed. The numerical examples are given to match well with the main conclusions.


Introduction
In recent years, the numerical treatment and supporting analysis of fractional order differential equations has become an important research topic that offers great potential.The FEMs for fractional partial differential equations have been studied by many authors (see [1][2][3]).All of these papers only considered single-term fractional equations, where they only had one fractional differential operator.In this paper, we consider the MT-FPDEs, which include more than one fractional derivative.Some authors also considered solving linear problems with multiterm fractional derivatives (see [4,5]).This motivates us to consider their effective numerical solutions for such MT-FPDEs, which have been proposed in [6,7].

Stability of the Numerical Method
In this section, we analyze the stability of the FEM for MT-FPEDs (1)-( 3).Now we do some preparations before proving the stability of the method.Based on the definition of coefficients  ()   in Section 3, we can obtain the following lemma easily.Lemma 7.For 0 <  < 1, the coefficients  ()   , ( = 1, . . ., ) satisfy the following properties: (i)  ()  0 > 0 and  ()  < 0 for  = 1, 2, . . ., , Now we report the stability theorem of this FEM for MT-FPDEs in this section as follows.
Theorem 8.The FEM defined as in (38) is unconditionally stable.

Numerical Experiments
In this section, we present the numerical examples of MT-FPDEs to demonstrate the effectiveness of our theoretical analysis.The main purpose is to check the convergence behavior of numerical solutions with respect to Δ and Δ, which have been shown in Theorem 4 and Theorem 6.It is noted that the method in [29] is a special case of the method in our paper for fractional partial differential equation with single fractional order.So, we just need to compare FEM in our paper with other existing methods in [8,28].We use this example to check the convergence rate (c.rate) and CPU time (CPUT) of numerical solutions with respect to the fractional orders  and .
In the first test, we fix  = 1,  = 0.9 and  = 0.5 and choose Δ = 0.001 which is small enough such that the space discretization errors are negligible as compared with the time errors.Choosing Δ = 1/2  ( = 2, 4, . . ., 7), we report that the convergence rate of FDM in time is nearly 1.15 in Table 1, which matches well with the result of Theorem 4. On the other hand, Table 2 shows that an approximate convergence rate is 2, by fixing Δ = 0.001 and choosing Δ = 1/2  ( = 2, . . ., 6), which matches well with the result of Theorem 6.In the second test, we give the convergence rate when  = 0.5,  = 0.25 for Δ in Table 3, and Δ in Table 4, respectively.We also report the  2 -norm and  1 -norm of errors in Figures 1  and 2, respectively.
Fixing Δ = 0.001,  = 0.9, and  = 0.3 in (46), we compare the error and CPUT calculated by the FEM in this paper with the FDM in [8] and the FPCM in [8].From Table 5, it can be seen that the FEM in this paper is computationally effective.
For the problem (47), our method in this paper is just the DFBDM in Section 3. Therefore, we only need to compare M1 with the FEM in [28] (FEM2).In Table 6, although the convergence rate of FEM2 is higher than that of DFBDM, the error and CPUT of DFBDM are smaller than those of FEM2.