Journal of Applied Mathematics

Volume 2012 (2012), Article ID 808216, 35 pages

doi:10.1155/2012/808216

Research Article

The Spectral Method for the Cahn-Hilliard Equation with Concentration-Dependent Mobility

Shimin Chai and Yongkui Zou

Department of Mathematics, Jilin University, Changchun 130012, China

Received 14 April 2012; Accepted 9 July 2012

Academic Editor: Ram N. Mohapatra

Copyright © 2012 Shimin Chai and Yongkui Zou. 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 the numerical approximations of the Cahn-Hilliard-type equation with concentration-dependent mobility. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method for the space and the implicit Euler method for the time. Numerical experiments are carried out to illustrate the theoretical analysis.

1. Introduction

In this paper, we apply the spectral method to approximate the solutions of Cahn-Hilliard equation, which is a typical class of nonlinear fourth-order diffusion equations. Diffusion phenomena is widespread in the nature. Therefore, the study of the diffusion equation caught wide concern. Cahn-Hilliard equation was proposed by Cahn and Hilliard in 1958 as a mathematical model describing the diffusion phenomena of phase transition in thermodynamics. Later, such equations were suggested as mathematical models of physical problems in many fields such as competition and exclusion of biological groups [1], moving process of river basin [2], and diffusion of oil film over a solid surface [3]. Due to the important application in chemistry, material science, and other fields, there were many investigations on the Cahn-Hilliard equations, and abundant results are already brought about.

The systematic study of Cahn-Hilliard equations started from the 1980s. It was Elliott and Zheng [4] who first study the following so-called standard Cahn-Hilliard equation with constant mobility:

Basing on global energy estimates, they proved the global existence and uniqueness of classical solution of the initial boundary problem. They also discussed the blow-up property of classical solutions. Since then, there were many remarkable studies on the Cahn-Hilliard equations, for example, the asymptotic behavior of solutions [5–8], perturbation of solutions [9, 10], stability of solutions [11, 12], and the properties of the solutions for the Cahn-Hilliard equations with dynamic boundary conditions [13–16]. In the mean time, a number of the numerical techniques for Cahn-Hilliard equations were produced and developed. These techniques include the finite element method [4, 17–24], the finite difference method [25–29], the spectral method, and the pseudospectral method [30–35]. The finite element method for the Cahn-Hilliard equation is well investigated by many researchers. For example, in [23, 36], (1.1) was discreted by conforming finite element method with an implicit time discretization. In [22], semidiscrete schemes which can define a Lyapunov functional and remain mass constant were used for a mixed formulation of the governing equation. In [37], a mixed finite element formulation with an implicit time discretization was presented for the Cahn-Hilliard equation (1.1) with Dirichlet boundary conditions. The conventional strategy to obtain numerical solutions by the finite differece method is to choose appropriate mesh size based on the linear stability analysis for different schemes. However, this conventional strategy does not work well for the Cahn-Hilliard equation due to the bad numerical stability. Therefore, an alternative strategy is proposed for general problems, for example, in [26, 38] the strategy was to design such that schemes inherit the energy dissipation property and the mass conservation by Furihata. In [27, 28], a conservative multigrid method was developed by Kim.

The advantage of the spectral method is the infinite order convergence; that is, if the exact solution of the Cahn-Hilliard equation is smooth, the approximate solution will be convergent to the exact solution with power for exponent , where is the number of the basis function. This method is superior to the finite element methods and finite difference methods, and a lot of practice and experiments convince the validity of the spectral method [39]. Many authors have studied the solution of the Cahn-Hilliard equation which has constant mobility by using spectral method. For example, in [33–35], Ye studied the solution of the Cahn-Hilliard equation by Fourier collocation spectral method and Legendre collocation spectral method under different boundary conditions. In [30], the author studied a class of the Cahn-Hilliard equation with pseudospectral method. However, the Cahn-Hilliard equation with varying mobility can depict the physical phenomena more accurately; therefore, there is practical meaning to study the numerical solution for the Cahn-Hilliard equation with varying mobility. Yin [40, 41] studied the Cahn-Hilliard equation with concentration-dependent mobility in one dimension and obtained the existence and uniqueness of the classical solution. Recently, Yin and Liu [42, 43] investigate the regularity for the solution in two dimensions. Some numerical techniques for the Cahn-Hilliard equation with concentration-dependent mobility are already studied with the finite element method [28] and with finite difference method [44].

In this paper, we consider an initial-boundary value problem for Cahn-Hilliard equation of the following form:

where and

Here, represents a relative concentration of one component in binary mixture. The function is the mobility which depends on the unknown function , which restricts diffusion of both components to the interfacial region only. Denote . Throughout this paper, we assume that

where , , and are positive constants. The existence and uniqueness of the classical solution of the problems (1.2)–(1.4) were proved by Yin [41].

In this paper, we will apply the spectral method to discretize the spatial variables of (1.2) to construct a semidiscrete system. We prove the existence and boundedness of the solutions of this semidiscrete system. Then, we apply implicit midpoint Euler scheme to discretize the time variable and obtain a full-discrete scheme, which inherits the energy dissipation property. The property of the mobility depending on the solution of (1.2) causes much troubles for the numerical analysis. Furthermore, with the aid of Nirenberg inequality we investigate the boundedness and convergence of the numerical solutions of the full-discrete equations. We also obtain the error estimation for the numerical solutions to the exact ones.

This paper is organized as follows. In Section 2, we study the spectral method for (1.2)–(1.4) and obtain the error estimate between the exact solution and the spectral approximate solution . In Section 3, we use the implicit Euler method to discretize the time variable and obtain the error estimate between the exact solution and the full-discrete approximate solution . Finally in Section 4, we present a numerical computation to illustrate the theoretical analysis.

2. Semidiscretization with Spectral Method

In this section, we apply the spectral method to discretize (1.2)–(1.4) and study the error estimate between the exact solution and the semidiscretization solution.

Denote by and the norm and seminorm of the Sobolev spaces , respectively. Let be the standard inner product over . Define

A function is said to be a weak solution of the problems (1.2)–(1.4), if and satisfies the following equations:

For any integer , let . Define a projection operator by

We collect some properties of this projection in the following lemma (see [39]).

Lemma 2.1. (i)  commutes with the second derivation on , that is,

(ii) For any , there exists a positive constant such that

The following Nirenberg inequality is a key tool for our theoretical estimates.

Lemma 2.2. Assume that is a bounded domain, , then we have

where

By [41], we have the following.

Lemma 2.3. Assume that , , , then there exists a unique solution of the problems (1.2)–(1.4) such that

The spectral approximation to (2.2) is to find an element such that

Now we study the norm estimates of the function and for .

Theorem 2.4. Assume (1.6) and . Then there is a unique solution of (2.10) and (2.11) such that

where is a positive constant.

Proof. From (2.11) it follows that . The existence and uniqueness of the initial problem follow from the standard ODE theory. Now we study the estimate.

Define an energy function:

where . Direct computation gives

Noticing that and setting in (2.10), applying integration by part, we obtain

Hence,

Applying Young inequality, we obtain

where and are positive constants. Letting , then for all ,

where is a positive constant depending only on and . Therefore, we get

Thus,

where is a constant. By Hölder inequality, we obtain

Therefore,

From the embedding theorem it follows that

Theorem 2.5. Assume (1.6) and let be the solution of (2.10) and (2.11). Then there is a positive constant such that

Proof. Setting in (2.10) and integrating by parts, we get

Consequently,

where

Noticing (1.6), we have

In terms of the Nirenberg inequality (2.7), there is a constant such that

Noticing the definition of the function and the estimates in (2.12), we have

for some constant . Applying Hölder inequality and Young inequality, for any ,

where is a positive constant. Similarly, we obtain

Hence,

Taking , we have

where is a positive constant. Therefore,

From Gronwall inequality it follows that

where is a positive constant. According to the embedding theorem, we have

where is a positive constant.

Now, we study the error estimates between the exact solution and the semidiscrete spectral approximation solution . Set the following decomposition:

From the inequality (2.6) it follows that

Hence, it remains to obtain the approximate bounds of .

Theorem 2.6. Assume that is the solution of (1.2)–(1.4), is the solution of (2.10) and (2.11), and is smooth and satisfies (1.6), then there exists a constant such that

Before we prove this theorem, we study some useful approximation properties.

Lemma 2.7. For any , we have

where is a positive constant.

Proof. Direct computation gives

where is a positive constant.

Lemma 2.8. For any , we have

where is a positive constant.

Proof. Noticing that

where

From the boundedness of in Theorem 2.4 and the property of in Lemma 2.3, it follows that

Then we obtain

where is a positive constant. By Cauchy inequality, for any , we have

where is a positive constant.

Lemma 2.9. Assume that is the solution of (1.2)–(1.4), there exists a positive constant such that

where is a positive constant.

Proof. By Lemma 2.3, we have

In the other hand, it follows that

By the Young inequality,

Choosing in the previous inequality, we obtain (2.49).

Proof of Theorem 2.6. Setting in (2.2), we obtain

Setting in (2.10), we get

(2.53) minus (2.54) gives

According to Lemmas 2.7, 2.8 and 2.9, we have

Set , then there exists a positive constant such that

By Gronwall inequality, we have

where is a constant.

Summig up the properties above, we obtain the following.

Theorem 2.10. Assume is sufficiently smooth and satisfies (1.6), is the solution of (1.2)–(1.4), and is the solution of (2.10) and (2.11). Then there exists a positive constant such that

3. Full-Discretization Spectral Scheme

In this section, we apply implicit midpoint Euler scheme to discretize time variable and get a full-discrete form. Furthermore, we investigate the boundedness of numerical solution and the convergence of the numerical solutions of the full-discrete system. We also obtain the error estimates between the numerical solution and the exact ones.

Firstly, we introduce a partition of . Let , where and is time-step size. Then the full-discretization spectral method for (1.2)–(1.4) reads: , find such that

where and

The solution has the following property.

Lemma 3.1. Assume that is the solution of (3.1)-(3.2). Then there exists a constant such that

Proof. Define a discrete energy function at time by

Notice that

Setting in (3.1), we obtain

which implies

By (2.18), we have

Then

So we obtain

where and are positive constants. By Hölder inequality, we get

Therefore,

By the embedding theorem, we obtain

where is a constant.

Lemma 3.2. Assume that is the solution of the full-discretization scheme (3.1)-(3.2), then there is a constant such that

Proof. Setting in (3.1), we have

Therefore,

By Nirenberg inequality (2.7), we have

According to (3.14), we obtain

where is a positive constant. By Young inequality, for any positive constant , it follows that

where is a constant. Therefore,

Setting , there is a positive constant such that

Denoted by , if is sufficiently small such that , we have

where is a positive constant. By the embedding theorem, the estimate (3.15) holds.

Next, we investigate the error estimates for the numerical solution to the exact solution . Our analysis is based on the error decomposition denoted by

The boundedness estimate of follows from the inequality (2.6), that is, for any , there is a positive constant such that

Hence, it remains to obtain the approximate bounds of . If no confusion occurs, we denote the average of the two instant errors and by :

For later use, we give some estimates in the next lemmas.

Lemma 3.3. Assume that the solution of (1.2)–(1.4) is such that , then

Proof. Applying Taylor expansion about , we have

Then

From Hölder inequality it follows that

Noticing that for any , we have

Taking in (3.31), we obtain

Taking in (2.2) and (3.1), respectively, we have

Comparing (3.33) and (3.34), we have

Now we investigate the error estimates of the three items in the right-hand side of the previous equation.

Lemma 3.4. Assume that is the solution of (1.2)–(1.4) such that , then there exists a positive constant such that

Proof. By Taylor expansion and Hölder inequality, we obtain

Therefore,

By Hölder inequality, we have

Direct computation gives

Then

By Cauchy inequality, it follows that

Then we obtain

where , and are positive constants. Choosing , and terms of the properties of the projection operator , we complete the proof of the estimate (3.36).

Lemma 3.5. Assume that is the solution of (1.2)–(1.4) such that and . Then for any positive constant , there exists a constant , such that

Proof. Firstly, we consider

Direct computation gives

Then

Direct computation yields

where

Applying Lemma 2.3 and Theorem 3.7, we obtain that and . Taking in (3.47),we have

where is a constant.

Lemma 3.6. Assume that is the solution of (1.2)–(1.4). Then there exists a positive constant such that

Proof. By (2.9), we have

In the other hand,

By Young inequality, we obtain

Choosing in the previous inequality leads to (3.51).

Finally, we obtain the main theorem of this paper.

Theorem 3.7. Assume that is the solution of (1.2)–(1.4) and satisfies that

is the solution of the full-discretization (3.1) and (3.2). If is sufficiently small, there exists a positive constant such that, for any ,

where .

Proof. By (3.27), (3.36), (3.44), and (3.51), we have

where and are constants. For sufficiently small such that , denoting , we obtain

where

By the Gronwall inequality of the discrete form, we obtain

Direct computation gives

Then we complete the conclusion (3.56).

Furthermore, we get the following theorem.

Theorem 3.8. Assume is the solution of (1.2)–(1.4) and satisfies that

is the solution of the full-discretization scheme (3.1)-(3.2), and satisfies . If is sufficiently small, there exists a constant such that

4. Numerical Experiments

In this section, we apply the spectral method described in (3.1) and (3.2) to carry out numerical computations to illustrate theoretical estimations in the previous section. Consider (2.2) with settings:

where is a constant. The full-discretization spectral method of (2.2) and (2.3) reads: find such that

In our computations we fix and choose five different time-step sizes . Let be the integer with . Since we have no exact solution of (2.2) and (2.3), we take and to compute an approximating solution with and regard this as an exact solution. we also choose five different time-step sizes with to obtain five approximating solutions and compute the error estimation. Define an error function:

This function characterizes the estimations with respect to time-step size.

4.1. Example 1

Take and . We also take two different initial functions and to carry out numerical computations. Figure 1 shows the development of the solutions for time from to with fixed step-size .

(a)

(b)

Figure 1: The development of the solution of the full-discrete scheme when initial value is (a) and (b).

We also choose five different time-step sizes to carry out numerical computations and apply the error function in (4.3) to illustrate the estimation and convergence order in time variable , see Table 1.

Table 1: The error and the convergence order.

—

—

1.9466

1.9651

2.4159

2.2799

2.6142

2.5515

2.9518

2.9368

4.2. Example 2

Take and . We also take two different initial functions and to carry out numerical computations. Figure 2 shows the development of the solutions for time from to with fixed step-sizes .

(a)

(b)

Figure 2: The development of the solution of the full-discrete scheme with initial value (a) and (b).

We also choose five different time-step sizes to carry out numerical computations and apply the error function in (4.3) to illustrate the estimation and convergence order in time variable , see Table 2.

Table 2: The error and the convergence order.

—

—

2.5077

2.4907

2.6069

2.6922

2.3467

2.4525

2.0741

2.0736

4.3. Example 3

Take and . We also take two different initial functions and to carry out numerical computations. Figure 3 shows the development of the solutions for time from to with fixed step-sizes .

(a)

(b)

Figure 3: The development of the solution of the full-discrete scheme with initial value (a) and (b).

We also choose five different time-step sizes to carry out numerical computations and apply the error function in (4.3) to illustrate the estimation and convergence order in time variable , see Table 3.

Table 3: The error and the convergence order.

Order₁

Order₂

—

—

2.0013

2.0019

2.0043

2.0044

2.0170

2.0171

2.0704

2.0704

Acknowledgment

This work is supported by NSFC no. 11071102.

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