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In this article, we consider a two-dimensional symmetric space-fractional diffusion equation in which the space fractional derivatives are defined in Riesz potential sense. The well-posed feature is guaranteed by energy inequality. To solve the diffusion equation, a fully discrete form is established by employing Crank-Nicolson technique in time and Galerkin finite element method in space. The stability and convergence are proved and the stiffness matrix is given analytically. Three numerical examples are given to confirm our theoretical analysis in which we find that even with the same initial condition, the classical and fractional diffusion equations perform differently but tend to be uniform diffusion at last.

Fractional convection-diffusion equations are generalizations of classical convection-diffusion equations, which have come to be applied in Physics [

The symmetric space-fractional convection-diffusion equation (including both left and right derivatives) was firstly proposed by Chaves [

Recently, numerical methods for multi-dimensional problems of fractional differential equational are studied. For example, in [

In this paper, we consider the following two-dimensional symmetric space-fractional diffusion equation (SSFDE)

where

Remark: In this paper, the default fractional derivative is Riemann-Liouville derivative.

This article is organized as follows. In Section 2, we introduce some functional spaces. In Section 3 and Section 4, we prove existence and uniqueness of the variational solution. The full discretization of SSFDE is given in Section 5, where we apply Crank-Nicolson technique in time and Galerkin finite element method in space. Moreover, a detailed stability and convergence analysis is carried out. In section 6, we present the imple- mentation of how to get the stiffness matrix. Finally, some numerical examples are given in Section 7 to confirm our theretical analysis and to compare the difference between fractional diffusion and integer order diffusion system.

Ervin and Roop [

Definition 2.1 (Directional Integral [

Definition 2.2 ([

Definition 2.3 (Directional Derivative [

Definition 2.4 ([

and norm

and let

Definition 2.5 ([

and norm

and let

Theorem 2.1 ([

Theorem 2.2 ([

Definition 2.6 ([

and norm

where

In the following, a semi-norm is defined by integral

holds independent of the value of

Remark: The condition holds if

which is positive for all such

Definition 2.7 ([

and norm

and let

Theorem 2.3 ([

Theorem 2.4 (Fractional Poincarà Friedrichs Inequality [

The definitions and theorems above are basic frame of multi-dimensional fractional derivative spaces. In terms of Equation (1), we let M be atomic with atoms

Definition 2.8 Let

or

and norm

It is easy to derive that (12) is equivalent to (13) with using theorem 2.1 and Parseval equality.

Lemma 2.1 (The relationship between R-L and Caputo fractional order derivatives [

where

So when

And if

Lemma 2.2 ([

So, if

Lemma 2.3 (Adjoint Property). The left and right Riemann-Liouville fractional integral operator are adjoints in the sense

Theorem 2.5 Let

Proof. Let

From Lemma 2.1, we know if

For convenience, we denote

then Equation (1) can be written in the following form

where

To derive the variational form of (23), we introduce two properties of

Property 1 (Fourier Transform of

where

Proof. In view of Theorem 2.1, we can derive the Fourier Transform

Therefore, we have

Remark: Here, we use

Property 2 If

where

In fact, when

Proof. Using Theorem 2.5 and taking notice that

In order to derive the variational form of (23), we assume u is a sufficiently smooth solution of (23), and multiply by arbitrary

The weak formulation of the equation is to find the

With using property 2, the above formula could be written as

Thus we define the associated bilinear form

Theorem 3.1 The form

Proof. According to the definition of

Using Cauchy-Schwarz inequality we can obtain

Associating the definition of the semi-norm of

So we have

Combining the equivalence of

i.e., the form

According to the equivalence of

i.e., the form

Theorem 3.2 (Energy Inequality). If

i.e. the solution of (23) is well posed.

Proof. Multiply the first formula of (23) by u and integrate both sides of the equation in

As the coercivity of the form

Take

Corollary. The solution of variational formulation (28) exists and is unique.

Proof. The existence can be derived directly from Theorem 3.6 with Lax-Milgram theorem and Theorem 3.7 ensure the uniqueness.

Let

with the piecewise polynomials

where

hold for each

Theorem 4.1 For

Proof. Assume that

In view of Theorem 4.1, we have

and

Therefore, the bilinear

i.e., the right side of (33) is continuous. According to Lax-Milgram theorem, the fully discrete approximating system (32) has unique solution

Theorem 4.2 (Energy Inequality). If

Proof. Taking

Then we can obtain

So the result is valid.

Lemma 4.1 (Approximation Property [

where

Theorem 4.3 (Convergence). Assume that

Proof. Let

where

Define

Looking back to the first formula of (32), we can derive

Noting that

holds for

Taking

So we have

In the following we will estimate the three parts of the above inequality respectively. The first part

The second part

Hence we can obtain a recursive inequality

Summing up from 1 to

Take

From (38) and (39), we can derive the following error estimate

Finally, the formula (40) leads to (36).

Since the fractional derivative is a non-local operator, the implementation of finite element method for fractional differential equations is very complex. The main problem is how to obtain the stiffness matrix. In [

First of all, we consider the problem of finding the fractional derivative of each of the basis function

where

If

If

If

For

Secondly, we consider the problem of calculating the inner product

then the coordinate of the jth node is

It is easy to know when

Case 1:

Case 2:

Case 3:

Case 4:

Case 5:

Case 6:

Case 7:

Case 8:

Finally, we consider the problem of calculating the stiffness matrix A via the inner product obtained. Form Equation (29) we can see that A can be decomposed into four parts

then it is obvious that

In fact, for

where

which means

Example 1. Consider the following problem:

Which has exact solution

Obviously,

Remark: The trial function in all of the numerical experiments is bilinear function.

We can see that the results support our error estimate and ensure the numerical approximation is effective. In the following, we take fixed initial value and source term independent of

Example 2. Consider the following problem

Cvge. rate | Cvge. rate | |||
---|---|---|---|---|

1/8 | ||||

1/16 | 1.88 | 1.86 | ||

1/32 | 1.93 | 1.90 | ||

1/64 | 1.97 | 1.94 |

Let

Example 3. In order to compare the difference between fractional diffusion and classical diffusion, consider the following equation with homogeneous boundary condition:

where u represents concentration and the diffusion coefficient is

which means the initial concentration concentrates in a rhombus. We take

We note that the initial condition in the fractional system affect wider area than integer order in a short period of time by comparing the first two contour maps. Moreover, the diffusion under the influence of initial condition last longer in the fractional system. So at

Many different numerical methods for fractional convection-diffusion equation have been discussed by researchers in recent 10 years. In this paper, we discussed one kind of space-fractional diffusion equation which could be derived through replacing the second order derivative of x and y by corresponding Riesz fractional derivative in the classical diffusion equation. A numerical approximation for the equation was presented by using C-N tech- nique in time direction and Galerkin finite method in space. Furthermore, a detailed stability and convergence analysis was carried out for the fully discrete system. Then, some numerical examples were given and the dif- ferences between fractional and classical diffusion were presented. It is known that the stiffness matrix of frac- tional differential equation is rather complex, so to make the approach applicatory. We give the implementation of computational aspect. However, because of the non-local property of fractional derivative, the stiffness matrix is not sparse (almost dense) which challenges the computational resources.

The authors were supported by the National Natural Science Foundation of China under Project 51174236, and the National Basic Research Program of China under Project 2011CB606306.

Here, we give the computational details of case 5 to case 8. It is analogous for case 1 to case 4. To begin with, we introduce one formula which is used frequently in the procedure of computing the inner product and can be derived directly from the definition of beta function by integral transformation:

where

In the following analysis, we always denote

Case 5:

It is obvious that

Case 6:

In this cas

Because the two basis functions are symmetrical about the straight lines

Case 7:

It is easy to see

Case 8:

First, we consider the case of

By induction, we can conclude that for