The Spectral Method For The Cahn-Hilliard Equation With Varying Mobility

In this paper, we study the Cahn-Hilliard equation, which is a typical class of nonlinear fourth order diffusion equations, by using the spectral method. Diffusion phenomena is widespread in the nature. Therefore, the study of the diffusion equation which describing the diffusion phenomena caught wide concern. Cahn-Hilliard equation was propounded 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 [17], moving process of river basin [28] and diffusion of oil film over a solid surface [45]. Because of the important application in chemistry and material science, there were many investigation on the Cahn-Hilliard equations, and abundant results were brought about.The systematic study for Cahn-Hilliard equations began from the 1980s. It was Elliott and Zheng [22] who first study the Cahn-Hilliard equation in mathematics. They considered the following so-called standard Cahn-Hilliard equation with constant mobilityBasing on global energy estimates, they proved the global existence and uniqueness of classical solution of the initial boundary problem under the condition that the coefficient of the leading term of the interior chemical potential is a positive constant. They also discussed the blow-up property of classical solutions. After that, there were many remarkable studies on the Cahn-Hilliard equations, for example the asymptotic behavior of solutions [15, 38, 46, 56], perturbation of solutions [16, 43], stability of solutions [9, 13], and the properties of the solutions for the Cahn-Hilliard equations with dynamic boundary conditions [36, 41, 42, 49] which appeared recently, etc. Since the solutions of the Cahn-Hilliard equations can not be expressed by analytic formula, we should consider the approximate solution by numerical technique. Therefore, a number of the numerical techniques for Cahn-Hilliard equations were produced, these techniques include the finite element method [14, 22, 20, 19, 21, 4, 5, 6, 7], the finite difference method [26, 44, 23, 33, 32], the spectral method and the pseudo-spectral method [50, 51, 52, 3, 29, 30]. We know the key of the numerical analysis of Cahn-Hilliard equations is to construct a suitable discrete schemes in the finite element method. For example, in [37, 19], the equation (7.0.1) was discreted by conforming finite schemes with an implicit time discretization. In [20], semi-discrete schemes which can define a Lyapunov functional and remains mass constant were used for a mixed formulation of the governing equation. In [18], a mixed finite element formulation with an implicit time discretization was presented for the Cahn-Hilliard equation (7.0.1) with Direchlet boundary conditions. The conventional strategy to obtain numerical solutions by the finite difference method is to choose appropriate mesh size based on the linear stability analysis for difference schemes. However, this conventional strategy does not work well for the Cahn-Hilliard equation. In fact, there are not many-successful studies on the finite difference method for this equation. Therefore an alternative strategy is proposed for general problems, for example, in [25, 26] the strategy was to design such schemes that inherit the energy dissipation property and the mass conservation by Furihata. In [33, 32], conservative multi-grid method was developed by Kim.Moreover, the spectral and pseudo-spectral method which have a long history became one of the important method after the fast Fourier transform appeared . In 1970s, some work about the computation , the application and the stability of the spectral method had been done, see the result of Oliger,Orszag[39, 40]. After that, especially in 1980s, it was Canuto, Maday and Guo who studied the spectral method systematically in theory, which giving the error estimate of projection operator and interpolation operator under the different norms, and apply them to a series of the important linear and nonlinear partial difference equations and get satisfying result. Meanwhile, a many of computation proves that the spectral method is a effectively technique. At present the spectral method is applied in hydrodynamics, meteorology, computational physics etc. The spectral method is the technique that takes the trigonometric function or other orthogonal polynomial as the basis function. The advantage of the spectral method is the infinite order convergence,that is, if the exact solution of the Cahn-Hilliard equation is infinite smooth, then the approximate solution will be convergent to the exact solution with any power for exponent N^-1,where N 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 experiment prove the validity of the spectral method. Furthermore, the quick algorithm reduces computation largely. Certainly, the spectral method have the shortage. It requires both well regularity of the solution and product-type region. In addition, the singularity of the weighted function in Chebyshev spectral method leads to the instability. In last years, many people managed to overcome this problem. For instance, in [2], the author introduced the negative norm when studying the solution with non-smooth initial value. A large amount of authors have studied the solution of the Cahn-Hilliard equation which has constant mobility by using spectral method. For example, in [50, 51, 52], Ye Xinde studied the solution of the Cahn-Hilliard equation by Fourier collocation spectral method and Legendre collocation spectral method under different boundary conditions. In [3], the author studied a class of the Cahn-Hilliard equation with pseudo-spectral 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. It was professor Yin [53, 54] who first study the Cahn-Hilliard equation with concentration dependent mobility in one dimension and get the existence and uniqueness of the classical solution. In the last year, professor Yin Jingxue and Liu Changchun [34, 55] investigate the regularity for the solution in two dimension. As far as we know, the main numerical technique for the Cahn-Hilliard equation with concentration dependent mobility are the finite element method [33] and finite difference method [31], there is no study on the Cahn-Hilliard equation with varying mobility by the spectral method.This monograph includes two chapters. We will study the solution of the Cahn-Hilliard equation with varying mobility by using the spectral method. In the first chapter, we consider the Cahn-Hilliard equation with given mobility and present the existence and uniqueness of the solution for the initial value and boundary condition problem. Furthermore, we prove the existence of the periodic solution about time variable. By constructing the semi-discrete scheme and full-discrete scheme, we study the existence, boundedness and convergence of the approximate solutions of the semi-discrete scheme and full-discrete. In the second chapter, we investigate the Cahn-Hilliard equation with concentration dependent mobility by the spectral method. Through building the semi- discrete scheme and full-discrete scheme, we study boundedness and convergence of the approximate solution and verify them with the numerical exercise.In the first chapter, by utilizing the spectral method, we study the numerical solution for the Cahn-Hilliard equation with non-constant mobility. We consider the following equationsatisfies the initial value conditionand boundary conditionwhere D = (?),A(u) is the interior chemical potential with a typical form A(s)= -s +γ₁s² +γ₂s³(γ₂ > 0).We know from the reference [22] that the sign of the coefficientγ₂ of the leading term of the interior chemical potential is important to the properties of the solutions. Exactly speaking, ifγ₂ > 0 and the initial datum is suitably small, then the solution exists globally in time. In this chapter we think about the same thing, but the mobility is not a constant. Firstly, we study the existence and uniqueness for the weak solution of the Cahn-Hilliard equation (7.0.2)-(7.0.4) by Leray-Schauder fixed point theorem, and investigate the regularity for the weak solution, then we get the classical solution of the equation (7.0.2)-(7.0.4) under the initial value which is suitable smoothness.Secondly, we consider the time periodic problem of the Cahn-Hilliard equation with non-constant mobility. We consider the following equationsatisfies the boundary condition (7.0.3) and periodic conditionwhere f(x) is the source function. If the mobility m(x,t) is a nontrival periodic function with respect to t, then we get the existence of the periodic solution by some a priori estimates and Leray-Schauder fixed point theorem. Thirdly, from the weak form of the equation (7.0.2)-(7.0.4), we discrete the space variable by the spectral method and get the semi-discrete scheme for the Cahn-Hilliard equation. We study the existence, uniqueness, boundedness and convergence of the approximate solution. Based on the semi-discrete scheme of the Cahn-Hilliard equation, a full-discrete scheme which inherit the energy dissipation property is constructed by discrete time variable.We investigate the boundedness and convergence of the numerical solution. It is worth to give our attention, the error estimate between the numerical solution of the full-discretezation scheme and the exact solution of the Cahn-Hilliard equation (7.0.2)-(7.0.4) is O(h²),where h is a step of time variable. That scheme is more precise than the standard backward Eular difference. At last, we practice the exercise in one dimension and examine the result of the conclusion. Moreover, we give the numerical example in two and three dimension.In the second chapter, we study the solution of the Cahn-Hilliard equation with concentration dependent mobility by the spectral method. We consider the following equationwith boundary condition (7.0.3) and initial value condition (7.0.4). In [54], we know that there is a unique classical solution u∈C^1+(?),4+α((?)_T) of the problem (7.0.7),(7.0.3), (7.0.4) when m(u),u₀ are smooth enough. It provides the theoretical basis for the spectral method. Firstly, from the weak form of the solution of equation (7.0.7), we construct the semi-discrete scheme by discreting the space variable with the spectral method, and analysis the existence, uniqueness, boundedness and convergence for the approximate solution. Secondly, based on the semi-discrete equation, a full-discrete scheme which inherit the energy dissipation property is constructed. We investigate the boundedness and convergence of the numerical solution. It is worth to give our attention, the error estimate between the numerical solution of the full-discretezation scheme and the exact solution of the Cahn-Hilliard equation (7.0.7),(7.0.3),(7.0.4) is O(h²),that is more precise than the standard backward Eular difference. Since the mobility is depending on the solution u(x, t),so it is difficult to analysis the error estimate because of the item m'(u)Du in (7.0.7). We introduce the Nirenberg inequality to get the boundedness of the infinite norm for Du_N.At last, we practice the exercise in one dimension and examine the result of the conclusion.Moreover, we give the numerical example in two and three dimension.