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`Advances in Mathematical PhysicsVolume 2013 (2013), Article ID 293706, 10 pageshttp://dx.doi.org/10.1155/2013/293706`
Research Article

## Maximum Norm Error Estimates of ADI Methods for a Two-Dimensional Fractional Subdiffusion Equation

1Department of Mathematics, East China Normal University, Shanghai 200241, China
2Scientific Computing Key Laboratory of Shanghai Universities, Division of Computational Science, E-Institute of Shanghai Universities, Shanghai Normal University, Shanghai 200234, China

Received 25 June 2013; Accepted 8 July 2013

Copyright © 2013 Yuan-Ming Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with two alternating direction implicit (ADI) finite difference methods for solving a two-dimensional fractional subdiffusion equation. An explicit error estimate for each of the two methods is provided in the discrete maximum norm. It is shown that the methods have the same order as their truncation errors with respect to the discrete maximum norm. Numerical results are given to confirm the theoretical analysis results.

#### 1. Introduction

Fractional differential equations and fractional calculus arise in various application problems in science and engineering [116]. Various numerical methods have been developed for the computation of fractional differential equations [1734]. Fractional subdiffusion equations describe a special type of anomalous diffusion [35], and it is a more difficult task to solve this kind of equation numerically.

Numerical works for fractional subdiffusion equations are mostly focused on one-dimensional problems due to the memory effect in fractional derivatives; see, for example, [19, 20, 2026, 3133, 3642]. A two-dimensional anomalous subdiffusion equation was numerically treated in [43, 44], where explicit and implicit finite difference schemes were proposed. Chen et al. [28] extended their work in [43] to a variable-order subdiffusion equation. Liu et al. [45] developed an implicit meshless approach based on the radial basis function for the numerical simulation of a two-dimensional subdiffusion problem. Chen and Liu [18] considered an implicit difference scheme for a three-dimensional fractional advection-diffusion equation, and a Richardson extrapolation was applied to improve the accuracy.

The main concern in that paper is the construction of the schemes and error estimates in the discrete -norm. Since -norm error estimates do not provide immediate insight on the phase error occurring during the time evolution, it is more preferable to give error estimates in the discrete maximum norm when we measure computation errors in practice. In this paper, we continue the investigation of the paper [59], by establishing a maximum norm error estimate for the ADI discretizations. It is known that an -norm error estimate does not imply a maximum norm error estimate for two-dimensional problems. We here present a technique of discrete energy analysis in order to obtain an explicit maximum norm error estimate.

The outline of the paper is as follows. In Section 2, we derive ADI finite difference schemes for (1) and present our main results of the maximum norm error estimates. The proof of the main error results is given in Section 3. In Section 4, we give some numerical results demonstrating the accuracy of the schemes in the discrete maximum norm. Section 5 contains some concluding remarks.

#### 2. ADI Schemes and Maximum Norm Error Estimates

We partition with nonisotropic uniform mesh sizes and in the and directions, respectively. The integers and . The mesh points (). Let and be the sets of mesh points lying in and on , respectively, and let . For any grid function , we denote

For a positive integer , we let be the time step. Define and

For the temporal approximation, we introduce the operators where and .

Using the Taylor expansion and the approximation of (see [25, 58, 59]), we have where There exists a positive constant independent of , , and the time level such that Substituting (6) into (1), we obtain

In order to construct an ADI scheme, we add the term to (9). This yields where

It was shown in [59] that for a positive constant independent of , , and the time level . Thus, there exists a positive constant independent of , , and the time level such that

Denote by the finite difference approximation to , and let . After multiplying (10) by and then dropping the term , we derive a finite difference scheme as follows: By introducing the intermediate variable , we obtain the following -ADI scheme (see [59]):

Adding the term to (9), we have where

Since for a positive constant independent of , , and the time level (see [59]), there exists a positive constant independent of , , and the time level such that

By (15), we obtain the following finite difference scheme:

It is equivalent to the following BD-ADI scheme (see [59]):

##### 2.2. Maximum Norm Error Estimates

For any grid function , we define its maximum norm by

Let be the value of the solution of (1) at the mesh point , and let be the solution of the -ADI scheme (14) or the BD-ADI scheme (19). We now present our main results of the maximum norm estimate for the error in the following two theorems. Their proofs will be given in the next section.

Theorem 1. Assume that the solution of (1) is sufficiently smooth, and let be the solution of the -ADI scheme (14). Then where and .

Theorem 2. Assume that the solution of (1) is sufficiently smooth, and let be the solution of the BD-ADI scheme (19). Then where and .

Theorems 1 and 2 show that the ADI difference solution from (14) or (19) converges to the analytical solution of (1) in the discrete maximum norm. We also see from the estimates (12), (17), (21), and (22) that the -ADI scheme (14) and the BD-ADI scheme (19) have the same order as their truncation error with respect to the discrete maximum norm.

#### 3. Proof of the Main Results

Let be the set of all grid functions defined in and vanishing on . For arbitrary , we define the following inner products:

For any , we introduce the following norms:

Using a simple calculation, we have that for arbitrary ,

Before proving Theorems 1 and 2, we first introduce the following embedding theorem from [60, page 281].

Lemma 3. For any , one has

Proof of Theorem 1. Let . Then by (10) and (13), where
This implies that for each . Taking the inner product of the first equation in (27) and , we get
It follows from (25) that
Similarly,
Since we have from Cauchy-Schwarz inequality that
Substituting (30) and (33) into (29) gives the following: where
This implies that
Since and by (12), , the estimate (21) follows from (36) and (26) immediately.

Proof of Theorem 2. The proof follows from the similar argument as that in the proof of Theorem 1 and we give a sketch. Let . By (15) and (18), where is defined by (28). Taking the inner product of the first equation in (37) and , we get
Since we obtain that where
Therefore
The estimate (22) follows from the previous inequality, (17), and (26) immediately.

#### 4. Numerical Results

In this section, we give some numerical results to demonstrate the accuracy of the -ADI scheme (14) and the BD-ADI scheme (19) with respect to the discrete maximum norm. Some detailed numerical comparisons of these two ADI schemes with the implicit scheme proposed in [43] can be found in [59]. The dependence of the solution of (1) on the anomalous diffusion exponent was also exhibited in [59] through some numerical results.

In our numerical computations, we take an equal mesh size in each of the space directions; that is, . We compute the discrete maximum norm error of the numerical solution by and its convergence orders by where represents the value of the exact analytic solution of (1) at .

Example 4. We consider the problem (1) in the domain , and let . Assume that the solution of this problem is . It can be checked that the corresponding known functions are given by

In Table 1, we present the maximum norm error and the temporal convergence order of the numerical solution by the -ADI scheme (14) and the BD-ADI scheme (19). We see that the -ADI scheme has the temporal accuracy of in the discrete maximum norm, and the best temporal accuracy is attained at . It is also seen that the BD-ADI scheme generates the temporal accuracy of in the discrete maximum norm, and it gets the best temporal accuracy when . These observations coincide well with the theoretical analysis.

Table 1: The maximum norm error and the temporal accuracy of for Example 4.

Table 2 gives the maximum norm error and the spatial convergence order of the numerical solution by the -ADI scheme (14) and the BD-ADI scheme (19). As expected from the theoretical analysis, these two schemes have the second-order spatial accuracy.

Table 2: The maximum norm error and the spatial accuracy of for Example 4.

Example 5. We consider the subdiffusion equation with boundary and initial conditions where and denotes the Riemann-Liouville fractional derivative operator defined as

Operating Riemann-Liouville fractional derivative operator on both sides of (46), we obtain the equivalent problem of the subdiffusion equation with the temporal Caputo fractional derivative [40, 42, 59] as follows:

We now solve the aforementioned problem by the -ADI scheme (14) and the BD-ADI scheme (19). Tables 3 and 4 give the maximum norm error and the convergence orders and of the numerical solution . It is easily seen that the numerical results confirm the theoretical analysis results.

Table 3: The maximum norm error and the temporal accuracy of for Example 5.
Table 4: The maximum norm error and the spatial accuracy of for Example 5.

#### 5. Conclusions

We have studied two ADI finite difference methods for a two-dimensional fractional subdiffusion equation. An explicit error estimate for each of the two methods has been provided in the discrete maximum norm. It has been shown that the methods have the same order as their truncation errors with respect to the discrete maximum norm. The maximum norm error estimates presented here are more preferable for measuring computation errors in practice, compared to the -norm error estimates in [59]. Numerical results have confirmed the theoretical analysis results.

#### Acknowledgments

This work was supported in part by E-Institutes of Shanghai Municipal Education Commission no. E03004 and Shanghai Leading Academic Discipline Project no. B407.

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