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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 808216, 35 pages
http://dx.doi.org/10.1155/2012/808216
Research Article

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

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 [58], 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 [1316]. 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, 1724], the finite difference method [2529], the spectral method, and the pseudospectral method [3035]. 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 [3335], 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