A two-grid mixed ﬁnite volume element method for nonlinear time fractional reaction-di ﬀ usion equations

: In this paper, a two-grid mixed ﬁnite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-di ﬀ usion equations, where the Caputo fractional derivative is approximated by the classical L 1-formula. The coarse and ﬁne grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the ﬁne grid are given. By using the Browder ﬁxed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and e ﬀ ectiveness of the proposed algorithm.

The two-grid method is proposed and developed by Xu [24,25] to solve nonlinear elliptic partial differential equations based on FE methods. Because of the advantage of saving computing time, many scholars have extended and applied it to integer order partial differential equations. Dawson et al. [26] presented a two-grid mixed finite element (MFE) method for nonlinear parabolic equations which arises in flow through porous media, and gave the error analysis. Yan et al. [27] proposed a two-grid FVE method for the nonlinear Sobolev equations, and obtained optimal H 1 -norm error estimate. Hou et al. [28] applied a two-grid expanded MFE method to solve semi-linear parabolic integro-differential equations, and gave the convergence analysis and some numerical results. Liu [29] presented a two-grid FVE method for semi-linear reaction-diffusion system of the solutes in the groundwater flow, and obtained the error estimates in L 2 -norm and H 1 -norm. In recent years, the twogrid method was also applied to solve fractional partial differential equations. Liu et al. [30] proposed a two-grid MFE algorithm for a nonlinear fourth-order reaction-diffusion model with the Caputo time fractional derivative, and obtained the unconditional stability and error estimates. Liu et al. [31] presented a two-grid FE algorithm for a time fractional Cable equation, in which the Riemann-Liouville fractional derivative was approximated by the second-order weighted and shifted Grünwald difference (WSGD) scheme. Li et al. [32] constructed a two-grid expanded MFE scheme to solve a semilinear time fractional reaction-diffusion equation, in which the Caputo fractional derivative was approximated by the L1-formula. Li et al. [33] proposed a two-grid FE method for a nonlinear time fractional diffusion equation, and gave some numerical results to confirm the theoretical results. Chen et al. [34] studied a two-grid modified method of characteristics scheme to solve nonlinear variableorder time fractional advection-diffusion equations, and obtained the optimal L 2 -norm error estimates. Liu et al. [35] presented a two-grid FE fast algorithm to solve a nonlinear space-time fractional diffusion equation, and gave the stability and convergence analysis. From the current literatures, we find that there is no report about the two-grid fast algorithm based on the mixed finite volume element (MFVE) method [36][37][38][39] for solving the FDEs.
In this paper, we will construct a two-grid MFVE algorithm to solve the nonlinear time fractional reaction-diffusion equations. In temporal discretization, we select the classical L1-formula to approximate the Caputo time fractional derivative. In spatial discretization, we construct coarse and fine grids (containing primal and dual grids), and establish a two-grid MFVE scheme by introducing an auxiliary variable λ and using the transfer operator. The calculation process is divided into two steps: firstly, the coarse solution is computed iteratively by using the nonlinear MFVE scheme on the space coarse grid, then a linearized scheme is constructed by using the coarse solution, and finally solution on the space fine grid is obtained. In our theoretical analysis, we give the existence and uniqueness results of the fully discrete solutions for the two-grid MFVE scheme by applying the Browder fixed point theorem and the matrix theory, and obtain unconditional stability results and error estimates in L 2 (Ω)-norm for the variable u. Moreover, we derive the conditional stability results and error estimates in (L 2 (Ω)) 2 -norm and H(div)-norm for the variable λ by using a special analytical technique. Finally, we give some numerical results to verify the feasibility and effectiveness, and find that the proposed two-grid MFVE algorithm can greatly save the computing time.
The layout of this paper is as follows: By constructing coarse and fine grids (primal and dual) and introducing the transfer operator, a two-grid MFVE algorithm for the nonlinear time fractional reaction-diffusion equation is proposed in Section 2. Some properties of the transfer operator γ and the fractional Gronwall inequality are given, and the existence and uniqueness results are obtained in Section 3. In Sections 4 and 5, the stability and error estimates are derived in detailed. In Section 6, two numerical examples are given to verify the feasibility and effectiveness.
In order to get the MFVE scheme, by introducing an auxiliary variable λ(x, t) = −A(x)∇u(x, t), we can rewrite the primal problem (1.1) as x ∈Ω.
Then, we can obtain the weak formulation of (2.1): x ∈Ω, where V = H(div, Ω) and W = L 2 (Ω). Now, we use K h to denote a quasiuniform triangulation partition of the domain Ω, that is K h = ∪K B , where K B stands for the triangle with the barycenter B, referring to Figure 1. Let h = max{h K B }, where h K B is the diameter of the triangle K B . Moreover, we should define the nodes of a triangular element to be its midpoints of three sides, and mark P 1 , P 2 , ..., P M S as the inner nodes and P M S +1 , P M S +2 , ..., P M as the boundary nodes. We select the lowest order Raviart-Thomas space V h and piecewise constant function space W h as the trial function spaces for λ and u, respectively, where Based on the primal partition K h , we construct the dual partition K * h . Referring to Figure 1, the interior node P 3 belongs to the common side of two adjacent triangles K B 1 = ∆A 1 A 2 A 3 and K B 2 = ∆A 1 A 3 A 5 , then we define the quadrilateral A 1 B 1 A 3 B 2 to be the dual element for P 3 . In general, for an interior node P, the dual element K * P is the union of two triangles K L (with ∆A 1 B 1 A 3 ) and K R (with ∆A 1 A 3 B 2 ). For a boundary node such as P 6 , the associated dual element is a triangle K I (with ∆A 5 B 3 A 4 ).
Integrating (2.1) on all the primal and dual partitions, respectively, we obtain We define the transfer operator γ h : V h → (L 2 (Ω)) 2 as follows where χ * K is characteristic function of a set K. We useȲ h = γ h V h as the test function space, and rewrite (2.3) as (2.4) Similar to [37], making use of the operator γ h and the Green theorem, we have ( (2.5) In order to approximate the Caputo time fractional derivative and give the fully discrete scheme, we should give the grid points t n = nτ (n = 0, 1, · · · , N) in time interval [0, T ], where N is a positive integer and τ = T/N. We denote ϕ n = ϕ(·, t n ) for a function ϕ. Following [4,5], we will approximate the fractional derivative ∂ α u(x,t) ∂t α at t = t n by using the L1-formula as follows . Following [4,5], we can get that if u ∈ C 2 (J, L 2 (Ω)), then there exist a constant C > 0 independent of τ such that R n t (x) ≤ Cτ 2−α . Let λ n h and u n h be the numerical solutions of λ and u at t = t n , respectively. Then, we can obtain the nonlinear fully discrete MFVE scheme for the problem (1.1): For the properly selected (λ 0 For improving the nonlinear fully discrete MFVE scheme (2.7), we consider the following two-grid MFVE system based on the coarse grid K H and the fine grid K h with the corresponding dual grids K * H and K * h , where h H. STEP I. On the coarse primal and dual grids (K H and K * H ), solve the following nonlinear system where (λ 0 H , u 0 H ) ∈ V H × W H is defined in Section 5. STEP II. On the fine primal and dual grids (K h and K * h ), solve the following linearized system for where (λ 0 h ,û 0 h ) ∈ V h × W h is defined in Section 5.
Remark 2.1. In the actual numerical calculation of the two-grid systems (2.8) and (2.9), we can find a solution (λ n H , u n H ) ∈ V H × W H on the coarse primal and dual grids (K H and K * H ) by calculating the nonlinear implicit system (2.8), then obtain the final solution (λ n h ,û n h ) ∈ V h × W h on the fine primal and dual grids (K h and K * h ) by calculating the linearized system (2.9). This calculation method will be more efficient than the standard nonlinear implicit system (2.7), and we will see this advantage from the numerical results.
Furthermore, the above result can be further written as Lemma 3.7.
We first give the existence and uniqueness results for the nonlinear MFVE scheme (2.8) by using Lemma 3.9.
There exists a constant τ 0 > 0 such that, if τ < τ 0 , then there exists a unique solution (λ n H , u n H ) ∈ V H × W H for the nonlinear MFVE scheme (2.8) on the coarse primal and dual grids.
The map G is obviously continuous. Furthermore, setting v H =λ H , w H =ū H in (3.1), and applying Lemma 3.3, we have Noting that Thus, there exists a constant τ 0,1 > 0 such that, if τ < τ 0,1 , then Because of the norm equivalence in finite dimensional normed linear space, there exists a constant Next, we prove the uniqueness of the solution. Let (Λ n H , U n H ) ∈ V H ×W H be another solution of (2.8), and (Λ 0 Applying Lemma 3.3, we have There exits a constant τ 0,2 > 0 such that, if τ ≤ τ 0,2 , then g 1,∞ τ α ≤ 1 2Γ(2−α) , and It follows that p n H = 0 and q n H = 0. Setting τ 0 = min{τ 0,1 , τ 0,2 }, we have completed the proof of the theorem. Now, we give the existence and uniqueness results for the linearized scheme (2.9).
) on the fine primal and dual grids.
be the basis functions of V h and W h , respectively. Then (λ n h ,û n h ) can be expressed asλ whereΛ n = (r n 1 , r n 2 , · · · , r n M 1 ) T ,Û n = (u n 1 , u n 2 , · · · , u n M 2 ) T , Noting that B 1 and B 2 are invertible, and applying the multiplication of partitioned matrices, we can get According to the property of continuous function, there exists a constant τ 1 So the coefficient matrix of (3.11) is invertible, then there exists a unique solution for the linearized scheme (2.9).

Stability analysis
We will give the stability results for the nonlinear MFVE scheme (2.8) and linearized MFVE scheme (2.9) on the coarse and fine grids, respectively.
, then there exist a constant C > 0 independent of h and τ such that Moreover, there exists a constant C > 0 independent of h, τ and c 0 such that, if h ≤ c 0 τ ≤ c 0 min{τ 2 , τ 1 } and h < 0 , then where τ 1 is defined in Theorem 3.2, c 0 , 0 and τ 2 are defined in Theorem 4.1.
Moreover, for j = 0, 1, the following superconvergence result holds Now, let β n = u n −ũ n H , σ n =ũ n H − u n H , ζ n = λ n −λ n H , δ n =λ n H − λ n H , where (λ n H ,ũ n H ) ∈ V H × W H is the generalized MFVE projection of (λ, u), then we can obtain the error equations as follows where τ 0 is defined in Theorem 3.1, c 0 , 0 and τ 2 are defined in Theorem 4.1.
(II) Similar to Remark 4.1, when the coefficient A(x) is a symmetry and positive definite constant matrix, we can remove the conditions H ≤ c 0 τ and h ≤ c 0 τ in the analysis and results of Theorems 5.1 and 5.2, respectively.
Example 6.1. By choosing T = 1, g(u) = sin(u), and = 2, we carry out numerical simulation for some different fractional parameters α = 0.2, 0.4, 0.6.0.8 and grid sizes. In Tables 1 and 2, we take τ = 1/5, 1/8, 1/10, h ≈ √ 2τ 2−α , H 2 ≈ 2τ 2−α (in two-grid MFVE algorithm), and h ≈ √ 2τ 2−α (in MFVE algorithm (2.7)), and obtain that the convergence rates in time direction are close to 2 − α for u in L 2 (Ω)-norm and λ in (L 2 (Ω)) 2 and H(div, Ω)-norms, which is consistent with the theoretical results in Theorems 5.1 and 5.2. For testing convergence rates in space direction, we fix the time step length τ = 1/100, select the coarse and fine grid sizes to satisfy h = H 2 / √ 2 =  Table 5, and the corresponding numerical results for the MFVE algorithm (2.7) in Table 6. Then we obtain the same conclusions as that discussed in Tables 3 and 4. Furthermore, for the time parameter t = 1, we show the graphs of the exact solutions for u and λ with h = √ 2/32 in Figures 2 and 4, respectively, also show the graphs of the numerical solutions based on the two-grid MFVE algorithm with h = Figures 3 and 5. We find that the numerical solutions and the exact solutions have the same numerical behaviors.         Example 6.2. In this example, we take T = 1, g(u) = u 3 − u, and = 2 + α, then obtain the exact solution u(x, t) = t 2+α sin(2πx 1 ) sin(2πx 2 ), x = (x 1 , x 2 ) ∈ [0, 1] 2 , t ∈ [0, 1], the auxiliary variable λ(x, t) = −∇u(x, t). For some different fractional parameters α = 0.2, 0.4, 0.6.0.8 and grid sizes, we conduct numerical experiments as in Example 6.1. For the two-grid MFVE algorithm and MFVE algorithm (2.7), we can see that the convergence rates in time direction are close to 2 − α (in Tables 7 and 8), and the convergence rates in space direction are close to 1 (in Tables 9 and 10). Moreover, in Tables 11 and 12, we choose h = 2τ (in MFVE algorithm), then obtain the same convergence rates as in Tables 9 and 10.     Base on the above the numerical results in Tables 1-6 and Figures 2-5 for Example 6.1 and  Tables 7-12 for Example 6.2, we can know that the convergence rates are consistent with the theoretical results in Theorems 5.1 and 5.2. We also find that the two-grid MFVE algorithm can save the computing time compared with the MFVE algorithm while maintaining the same convergence rates. Finally, numerical results and the figures show that the proposed two-grid MFVE algorithm for the nonlinear time fractional reaction-diffusion equations is feasible and effective.

Conclusions
In this paper, we construct the two-grid MFVE fast algorithm to solve the nonlinear time fractional reaction-diffusion equations with the Caputo time fractional derivative. We obtain the stability results and the optimal error estimates for u (in L 2 (Ω)-norm) and λ (in (L 2 (Ω)) 2 -norm), and the sub-optimal error estimates for λ (in H(div, Ω)-norm). Furthermore, we also give two numerical examples to verify that the proposed algorithm can greatly save the computing time. In future works, for the Caputo fractional derivative (1.2) with α ∈ (0, 1), we will try to use other approximation methods (such as L1-2, L2-1 σ , L1-2-3 formulas [17][18][19][20]) and the two-grid MFVE method to solve more fractional partial differential equations in scientific and engineering fields.