^{1}

^{2}

In this paper, an alternating direction Galerkin finite element method is presented for solving 2D time fractional reaction sub-diffusion equation with nonlinear source term. Firstly, one order implicit-explicit method is used for time discretization, then Galerkin finite element method is adopted for spatial discretization and obtain a fully discrete linear system. Secondly, Galerkin alternating direction procedure for the system is derived by adding an extra term. Finally, the stability and convergence of the method are analyzed rigorously. Numerical results confirm the accuracy and efficiency of the proposed method.

In this paper, we consider the following two-dimensional nonlinear fractional reaction- subdiffusion equation

with boundary and initial conditions

where

where

In addition, we assume that the nonlinear source term

Problem (1) can be considered as a model for reaction-diffusion phenomena with anomalous diffusion, which has been widely applied in various fields of science and engineering. Generally, solutions of (1) can’t be obtained by analytical approach. So, there are various numerical methods developed for solving (1). Li and Ding [

Alternating direction implicit (ADI) method was proposed by Peaceman, Rachford and Douglas [

where

In the case of_{2} error estimate are analyzed. ADI orthogonal spline collocation (OSC) scheme was developed by Fairweather, Yang and Xu, et al. [

is also often studied. For instances, Cui [

There are also lots of ADI based numerical methods for multidimensional space fractional differential equations. Fast iterative ADI FD schemes [

Most of the above mentioned works contribute on linear fractional differential equations and finite difference method combined with ADI technique. A few work consider ADI FEM [

The outline of this paper is organized as follows. In Section 2, we introduce some preliminaries and notations which will be used later. The formulation of ADI finite element method for nonlinear time fractional reaction-subdiffusion equation is presented in Section 3. The stability and error estimates of the proposed method are discussed in Section 4. In convenience of computation, we give the matrix form of ADI finite element scheme in Section 5. Some numerical experiments are displayed in Section 6. It aims to confirm our theoretical results. In the end, some concluding remarks are given in Section 5.

In the following,

Let

Recall that the Sobolev space

Denote by

If

and

Denote

and

In convenience, the following notations will be used. For a positvie integer

Define linear operator

and its variational form

and corresponding energy norm by

Obviously, we have

Some useful lemmas are given as follows.

Lemma 1. [

then

1)

2)

Lemma 2. [

where

We state here for convenience the discrete version of Gronwall’s inequality.

Lemma 3. [

where

Integrating both sides of (1) with respect to the time variable

Applying Lemma 2 and the following integration formula

Equation (8) is equivalent to

where the remainder term

The weak form of Equation (9) is: find

Then the finite element approximation to Equation (11) is: find

or equivalently

The choice of the initial values

Let

so that the alternating-direction Galerkin scheme of (1) can be defined as, for

where

Remark 1. Numerical experiments in Section 6 demonstrate that the ADI Galerkin finite element scheme (14) has bad numerical performance for

in left hand side of (14) is extra added. Its effect on temporal accuracy cannot be ignored when

on the right hand side of (14). By the way, the similar remedy was also adopted by [

In conclusion, when

where

In the following, we will focus our attention on the ADI Galerkin finite element scheme (14).

Firstly, we introduce some notation and lemmas. Given a smooth function

or equivalently

The operator defined in (18) has the following approximate properties:

Lemma 4. [

Lemma 5. [

where

Lemma 6. Let

then it holds that

Proof. When

and

By direct algebraic calculation, and noticing the properties of

The proof is completed.

Next, we consider the stability of the ADI Galerkin method (14). Define the following problem dependent norm for any

which will be used in stability analysis.

Assume the initial value

Theorem 1. The ADI Galerkin method (14) is stable with respect to initial value

holds for any

Proof. The equivalent form of (14) is:

Then the perturbation equation of (23) can be written as

where

Taking

Further, by Lipschitz property of

Summing for

For sufficiently small

Using discrete Gronwall inequality, we have

Let

Theorem 2. Let

Assume that

provided the initial value

Proof. Denote

Subtracting (28) from (23) leads to

Taking

Now, by Young inequality and Schwartz inequality, we can write

Substitute the above three inequalities into the right hand side of (30), and sum for

By discrete Gronwall inequality, if

Consequently, we obtain

Using the triangle inequality and (31), we get

It remains to estimate terms on the right-hand side of (32). Firstly, by Lemma 3, we can conclude

By (10) and

Secondly, using calculus equality

and Hölder inequality, we have

Further, combined with Lemma 3, we can write

Since

similar as (36), we can prove

Additionally, according to Lemma 3 and Lemma 4, we have

As a result, we obtain

Similar as (37), we can write

Combining (33)-(38), and recalling

provided

Remark 2. Although in our theoretic analysis, we only obtain

Note that we may choose the initial approximation as

This involves an elliptic problem to be solved. With this choice,

In practical computations, it is often sufficient to take

Equations (14) define the ADI finite element method in inner product form. To describe the algebraic problem to which these equations lead, suppose

where

be a tensor product basis for

so that

For convenience, we denote

If in (14) we choose

We define the matrices

and let

with

where, from (14), the components of

and

which is equivalent to

where

in the

in the

In this section, two numerical examples are given to demonstrate the effectiveness and accuracy of the ADI Galerkin finite element methods. In all numerical examples, we take the linear tensor product basis

where

In our numerical simulation, we present the errors in

and numerical convergence orders are computed by

Example 1. Consider the following problem

where

with initial and boundary conditions

where

^{2} norm errors and the temporal convergence orders for

To investigate the necessary of the correction term (16) in ADI scheme (17) if

for ^{2} norm errors and its corresponding temporal convergence orders using ADI scheme (14). Computational results demonstrate the temporal

τ | α = 0.6 | α = 0.7 | α = 0.8 | α = 0.9 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

err(τ, h) | γ_{t} | err(τ, h) | γ_{t} | err(τ, h) | γ_{t} | err(τ, h) | γ_{t} | ||||

1/8 | 7.54e−02 | -- | 8.59e−02 | -- | 9.52e−02 | -- | 1.03e−01 | -- | |||

1/16 | 3.71e−02 | 1.02 | 4.45e−02 | 0.95 | 5.00e−02 | 0.93 | 5.43e−02 | 0.92 | |||

1/32 | 1.83e−02 | 1.02 | 2.27e−02 | 0.97 | 2.56e−02 | 0.97 | 2.76e−02 | 0.98 | |||

1/64 | 8.89e−03 | 1.04 | 1.13e−02 | 1.01 | 1.27e−02 | 1.01 | 1.36e−02 | 1.02 | |||

1/128 | 4.13e−03 | 1.11 | 5.37e−03 | 1.07 | 6.02e−03 | 1.08 | 6.44e−03 | 1.08 | |||

1/256 | 1.70e−03 | 1.28 | 2.31e−03 | 1.22 | 2.61e−03 | 1.21 | 2.81e−03 | 1.20 | |||

ADI Scheme (14) | ADI Scheme (17) | |||||
---|---|---|---|---|---|---|

τ | err(τ, h) | γ_{t} | τ | err(τ, h) | γ_{t} | |

1/64 | 8.48e−02 | -- | 1/16 | 6.96e−02 | -- | |

1/128 | 8.45e−02 | 0.01 | 1/32 | 3.37e−02 | 1.05 | |

1/256 | 7.99e−02 | 0.08 | 1/64 | 1.61e−02 | 1.07 | |

1/512 | 7.37e−02 | 0.12 | 1/128 | 7.43e−03 | 1.12 | |

1/1024 | 6.71e−02 | 0.14 | 1/256 | 3.19e−03 | 1.22 | |

convergence order is 0.1, which also suggest our theoretical order ^{2} norm errors and its corresponding temporal convergence orders using ADI scheme (17). As we can see, the experiment convergence order is approximately one now. It indicates that the correction term (16) in ADI scheme (17) is beneficial to keep the temporal accuracy. In a word,

scheme (14) if

^{2} norm errors and the spatial convergence orders for

Example 2. Consider the following problem

where

with initial and boundary conditions

where

The nonlinearity of Example 2 is stronger than Example 1. In the following simulation, we have to take much smaller temporal step ^{2} norm errors and the temporal convergence orders for

h | α = 0.6 | α = 0.7 | α = 0.8 | α = 0.9 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

err(τ, h) | γ_{h} | err(τ, h) | γ_{h} | err(τ, h) | γ_{h} | err(τ, h) | γ_{h} | ||||

π/8 | 5.63e−02 | -- | 5.60e−02 | -- | 5.57e−02 | -- | 5.54e−02 | -- | |||

π/16 | 1.41e−02 | 2.00 | 1.40e−02 | 2.00 | 1.39e−02 | 2.00 | 1.38e−02 | 2.01 | |||

π/32 | 3.33e−03 | 2.04 | 3.39e−03 | 2.05 | 3.36e−03 | 2.05 | 3.33e−03 | 2.05 | |||

π/64 | 7.50e−04 | 2.19 | 7.21e−04 | 2.23 | 7.06e−04 | 2.25 | 6.93e−04 | 2.26 | |||

τ | α = 0.6 | α = 0.7 | α = 0.8 | α = 0.9 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

err(τ, h) | γ_{t} | err(τ, h) | γ_{t} | err(τ, h) | γ_{t} | err(τ, h) | γ_{t} | ||||

1/32 | 5.78e−01 | -- | 5.84e−01 | -- | 5.89e−01 | -- | 5.93e−01 | -- | |||

1/64 | 2.83e−01 | 1.03 | 2.86e−01 | 1.03 | 2.89e−01 | 1.03 | 2.91e−01 | 1.03 | |||

1/128 | 1.38e−01 | 1.04 | 1.40e−01 | 1.03 | 1.41e−01 | 1.04 | 1.42e−01 | 1.04 | |||

1/256 | 6.62e−02 | 1.06 | 6.71e−02 | 1.06 | 6.77e−02 | 1.06 | 6.81e−02 | 1.06 | |||

1/512 | 3.06e−02 | 1.11 | 3.10e−02 | 1.11 | 3.13e−02 | 1.11 | 3.15e−02 | 1.11 | |||

ADI Scheme (14) | ADI Scheme (17) | |||||
---|---|---|---|---|---|---|

τ | err(τ, h) | γ_{t} | τ | err(τ, h) | γ_{t} | |

1/512 | 4.12e−02 | -- | 1/128 | 1.42e−01 | -- | |

1/1024 | 3.57e−02 | 0.21 | 1/256 | 6.99e−02 | 1.02 | |

1/2048 | 3.08e−02 | 0.21 | 1/512 | 3.42e−02 | 1.03 | |

1/4096 | 2.58e−02 | 0.26 | 1/1024 | 1.64e−02 | 1.06 | |

h | α = 0.6 | α = 0.7 | α = 0.8 | α = 0.9 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

err(τ, h) | γ_{h} | err(τ, h) | γ_{h} | err(τ, h) | γ_{h} | err(τ, h) | γ_{h} | ||||

π/8 | 3.85e−01 | -- | 3.82e−01 | -- | 3.79e−01 | -- | 3.76e−01 | -- | |||

π/16 | 9.93e−02 | 1.95 | 9.85e−02 | 1.96 | 9.77e−02 | 1.96 | 9.70e−02 | 1.95 | |||

π/32 | 2.35e−02 | 2.08 | 2.32e−02 | 2.09 | 2.30e−02 | 2.09 | 2.28e−02 | 2.09 | |||

π/64 | 4.29e−03 | 2.45 | 4.20e−03 | 2.47 | 4.13e−03 | 2.48 | 4.06e−03 | 2.49 | |||

^{2} norm errors and the spatial convergence orders for

In this work, we proposed an alternating direction Galerkin finite element method for 2D nonlinear time fractional reaction sub-diffusion equation in the Riemann-Liouville type. The stability and convergence of the method are proved for

We thank the Editor and the referee for their comments. Research of P. Zhu is funded by the Natural Science Foundation of Zhejiang province, China (Grant No. LY15A010018). This support is greatly appreciated.

Zhu, P. and Xie, S.L. (2016) ADI Finite Element Method for 2D Nonlinear Time Fractional Reaction- Subdiffusion Equation. American Journal of Computational Mathematics, 6, 336-356. http://dx.doi.org/10.4236/ajcm.2016.64034