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

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.


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: where D ∂/∂x and A s −s γ 1 s 2 γ 2 s 3 , γ 2 > 0. 1.5 Here, u x, t represents a relative concentration of one component in binary mixture. The function m u is the mobility which depends on the unknown function u, which restricts diffusion of both components to the interfacial region only. Denote Q T 0, 1 × 0, T . Throughout this paper, we assume that where m 0 , M 0 , and M 1 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 fulldiscrete 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 u and the spectral approximate solution u N . In Section 3, we use the implicit Euler method to discretize the time variable and obtain the error estimate between the exact solution u and the full-discrete approximate solution U j N . Finally in Section 4, we present a numerical computation to illustrate the theoretical analysis.

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 · k and |·| k the norm and seminorm of the Sobolev spaces H k 0, 1 k ∈ N , respectively. Let ·, · be the standard L 2 inner product over 0, 1 . Define A function u is said to be a weak solution of the problems 1.2 -1.4 , if u ∈ L ∞ 0, T; H 2 E 0, 1 and satisfies the following equations:

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For any integer N > 0, let S N span{cos kπx, k 0, 1, 2, . . . , N}. Define a projection operator P N : We collect some properties of this projection P N in the following lemma see 39 .
Lemma 2.1. i P N commutes with the second derivation on H 2 E I , that is, The following Nirenberg inequality is a key tool for our theoretical estimates.
By 41 , we have the following.

Lemma 2.3.
Assume that m s ∈ C 1 α R , u 0 ∈ C 4 α I , D i u 0 0 D i u 0 1 0 i 0, 1, 2, 3, 4 , m s > 0, 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 u N ·, t ∈ S N such that Now we study the L ∞ norm estimates of the function u N ·, t and Du N ·, t for 0 ≤ t ≤ T .
Proof. From 2.11 it follows that u N ·, 0 P N u 0 · . The existence and uniqueness of the initial problem follow from the standard ODE theory. Now we study the estimate.
Define an energy function: where H s s

2.14
Noticing that P N A u N − D 2 u N ∈ S N and setting v N P N A u N − D 2 u N in 2.10 , applying integration by part, we obtain

2.15
Hence, Applying Young inequality, we obtain where C 1ε and C 2ε are positive constants. Letting ε 3γ 2 / 8|γ 1 | 12 , then for all 0 ≤ t ≤ T , where K 1 is a positive constant depending only on γ 1 and γ 2 . Therefore, we get

2.21
Therefore, From the embedding theorem it follows that

2.23
Theorem 2.5. Assume 1.6 and let u N ·, t be the solution of 2.10 and 2.11 . Then there is a positive constant C C u 0 , m, T, γ 1 , γ 2 > 0 such that Proof. Setting v N D 4 u N in 2.10 and integrating by parts, we get

2.25
Journal of Applied Mathematics 7 Consequently, where

2.28
In terms of the Nirenberg inequality 2.7 , there is a constant C > 0 such that

2.29
Noticing the definition of the function A and the estimates in 2.12 , we have Journal of Applied Mathematics for some constant C C u 0 , T, γ 1 , γ 2 > 0. Applying Hölder inequality and Young inequality, for any ε > 0, where C C u 0 , m, T, γ 1 , γ 2 , ε > 0 is a positive constant. Similarly, we obtain

2.32
Hence, Taking ε m 0 /8, we have From Gronwall inequality it follows that where C C u 0 , m, T, γ 1 , γ 2 > 0 is a positive constant. According to the embedding theorem, we have Now, we study the error estimates between the exact solution u and the semidiscrete spectral approximation solution u N . Set the following decomposition: From the inequality 2.6 it follows that Hence, it remains to obtain the approximate bounds of e.
Proof. Noticing that From the boundedness of u N in Theorem 2.4 and the property of u in Lemma 2.3, it follows that Then we obtain
Proof. By Lemma 2.3, we have Journal of Applied Mathematics

13
In the other hand, it follows that By the Young inequality,

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Set ε m 0 / 4 3M 2 0 , then there exists a positive constant C C u 0 , m, γ 1 , γ 2 such that By Gronwall inequality, we have Summig up the properties above, we obtain the following.

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 0, T . Let 0 t 0 < t 1 < · · · < t Λ , where t j jh and h T/Λ is time-step size. Then the full-discretization spectral method for 1.2 -1.4 reads: The solution U j N has the following property.
Proof. Define a discrete energy function at time t j by Notice that

3.10
So we obtain

3.17
By Nirenberg inequality 2.7 , we have

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According to 3.14 , we obtain where C C u 0 , m, γ 1 , γ 2 is a positive constant. By Young inequality, for any positive constant ε > 0, it follows that

3.22
Denoted by C C/ 1 − Ch , if h is sufficiently small such that Ch ≤ 1/2, we have Journal of Applied Mathematics where C C u 0 , m, T, γ 1 , γ 2 > 0 is a positive constant. By the embedding theorem, the estimate 3.15 holds.
Next, we investigate the error estimates for the numerical solution U j N to the exact solution u t j . Our analysis is based on the error decomposition denoted by The boundedness estimate of η j follows from the inequality 2.6 , that is, for any 0 ≤ j ≤ Λ, there is a positive constant C C u 0 , m, γ 1 , γ 2 such that

3.25
Hence, it remains to obtain the approximate bounds of e j . If no confusion occurs, we denote the average of the two instant errors e j and e j 1 by e j 1/2 : e j 1/2 e j e j 1 2 , η j 1/2 η j η j 1 2 .

3.26
For later use, we give some estimates in the next lemmas.

3.27
Proof. Applying Taylor expansion about t j 1/2 , we have

3.28
Then From Hölder inequality it follows that

3.30
Noticing that for any v ∈ S N , we have Taking v 2e j 1/2 in 3.31 , we obtain e j 1 2 e j 2 2h

3.35
Now we investigate the error estimates of the three items in the right-hand side of the previous equation.

3.36
Proof. By Taylor expansion and Hölder inequality, we obtain

3.42
Then we obtain

3.43
where C 1ε , C 2ε , and C 3ε are positive constants. Choosing ε m 0 /8, and terms of the properties of the projection operator P N , we complete the proof of the estimate 3.36 .

Lemma 3.5.
Assume that u is the solution of 1.2 -1.4 such that u tt ∈ L 2 Q T and u t ∈ L ∞ Q T . Then for any positive constant ε > 0, there exists a constant C 2 C 2 m, u 0 , T, γ 1 , γ 2 > 0, such that

3.47
Direct computation yields

3.52
In the other hand,

3.53
By Young inequality, we obtain

3.54
Choosing ε m 0 /4 in the previous inequality leads to 3.51 .
Finally, we obtain the main theorem of this paper.
3.55 U j N ∈ S N j 1, 2, . . . , Λ is the solution of the full-discretization 3.1 and 3.2 . If h is sufficiently small, there exists a positive constant C such that, for any j 0, 1, 2, . . . , Λ,

3.60
Direct computation gives Then we complete the conclusion 3.56 .
Furthermore, we get the following theorem.

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. where m 0 > 0 is a constant. The full-discretization spectral method of 2.2 and 2.3 reads: find U j N N l 0 α jl cos lπx j 1, 2, . . . , Λ such that

4.2
In our computations we fix N 32 and choose five different time-step sizes h k k 1, 2, . . . , 5 . Let Λ k be the integer with h k Λ k T . Since we have no exact solution of 2.2 and 2.3 , we take N 32 and h 0 0.15625 × 10 −4 to compute an approximating solution U Λ 0 N with h 0 Λ 0 T and regard this as an exact solution. we also choose five different time-step sizes h k k 1, 2, . . . , 5 with h k Λ k T to obtain five approximating solutions U Λ k N k 1, 2, . . . , 5 and compute the error estimation. Define an error function:

4.3
This function characterizes the estimations with respect to time-step size.

Example 1
Take m 0 1 and T 0.1. We also take two different initial functions u 1 0 x x 5 1 − x 5 and u 2 0 x 5 1 − x 5 e x to carry out numerical computations. Figure 1 shows the development of the solutions for time t from t 0 to t 0.1 with fixed step-size h 0 0.15625 × 10 −4 .    We also choose five different time-step sizes h k to carry out numerical computations and apply the error function in 4.3 to illustrate the estimation and convergence order in time variable t, see Table 1.

Example 2
Take m 0 0.05 and T 0.1. We also take two different initial functions u 1 0 x x 5 1 − x 5 and u 2 0 x 5 1 − x 5 e x to carry out numerical computations. Figure 2 shows the development of the solutions for time t from t 0 to t 0.1 with fixed step-sizes h 0 0.15625 × 10 −4 .
We also choose five different time-step sizes h k to carry out numerical computations and apply the error function in 4.3 to illustrate the estimation and convergence order in time variable t, see Table 2.

Example 3
Take m 0 0.005 and T 0.1. We also take two different initial functions u 1 0 x x 5 1 − x 5 and u 2 0 x 5 1 − x 5 e x to carry out numerical computations. Figure 3 shows the development of the solutions for time t from t 0 to t 0.1 with fixed step-sizes h 0 0.15625 × 10 −4 .
We also choose five different time-step sizes h k to carry out numerical computations and apply the error function in 4.3 to illustrate the estimation and convergence order in time variable t, see Table 3.