Stability and convergence of the spectral Galerkin method for the Cahn‐Hilliard equation

A spectral Galerkin method in the spatial discretization is analyzed to solve the Cahn‐Hilliard equation. Existence, uniqueness, and stabilities for both the exact solution and the approximate solution are given. Using the theory and technique of a priori estimate for the partial differential equation, we obtained the convergence of the spectral Galerkin method and the error estimate between the approximate solution u_N(t) and the exact solution u(t). © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Finite difference approximate solutions for the Cahn‐Hilliard equation

In this article, we analyze a Crank‐Nicolson‐type finite difference scheme for the nonlinear evolutionary Cahn‐Hilliard equation. We prove existence, uniqueness and convergence of the difference solution. An iterative algorithm for the difference scheme is given and its convergence is proved. A linearized difference scheme is presented, which is also second‐order convergent. Finally a new difference method possess a Lyapunov function is presented. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 437–455, 2007     

Slow dynamics for the hyperbolic Cahn‐Hilliard equation in one‐space dimension

The aim of this paper is to study the metastable properties of the solutions to a hyperbolic relaxation of the classic Cahn‐Hilliard equation in one‐space dimension, subject to either Neumann or Dirichlet boundary conditions. To perform this goal, we make use of an “energy approach," already proposed for various evolution PDEs, including the Allen‐Cahn and the Cahn‐Hilliard equations. In particular, we shall prove that certain solutions maintain a N‐transition layer structure for a very long time, thus proving their metastable dynamics. More precisely, we will show that, for an exponentially long time, such solutions are very close to piecewise constant functions assuming only the minimal points of the potential, with a finitely number of transition layers, which move with an exponentially small velocity.     

Exponential attractors for a singularly perturbed Cahn‐Hilliard system

Our aim in this article is to give a construction of exponential attractors that are continuous under perturbations of the underlying semigroup. We note that the continuity is obtained without time shifts as it was the case in previous studies. Moreover, we obtain an explicit estimate for the symmetric distance between the perturbed and unperturbed exponential attractors in terms of the perturbation parameter. As an application, we prove the continuity of exponential attractors for a viscous Cahn‐Hilliard system to an exponential attractor for the limit Cahn‐Hilliard system. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical scheme for the solution of viscous Cahn–Hilliard equation

In this paper, we present a numerical scheme for the solution of viscous Cahn–Hilliard equation. The scheme is based on Adomian's decomposition approach and the solutions are calculated in the form of a convergent series with easily computable components. Some numerical examples are presented. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Convergence analysis for second‐order accurate schemes for the periodic nonlocal Allen‐Cahn and Cahn‐Hilliard equations

In this paper, we provide a detailed convergence analysis for fully discrete second‐order (in both time and space) numerical schemes for nonlocal Allen‐Cahn and nonlocal Cahn‐Hilliard equations. The unconditional unique solvability and energy stability ensures ? ⁴ stability. The convergence analysis for the nonlocal Allen‐Cahn equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nonlocal Cahn‐Hilliard equation, because of the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken, and an H ^?1 inner‐product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a priori bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (eg, 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an O (s ³+h ⁴) convergence in norm, in which s and h denote the time step and spatial mesh sizes, respectively. This in turn leads to the necessary bound under a standard constraint s ≤C h . Here, we also prove convergence of the scheme in the maximum norm under the same constraint.     

Robust exponential attractors for Cahn‐Hilliard type equations with singular potentials

Our aim in this article is to study the long time behaviour of a family of singularly perturbed Cahn‐Hilliard equations with singular (and, in particular, logarithmic) potentials. In particular, we are able to construct a continuous family of exponential attractors (as the perturbation parameter goes to 0). Furthermore, using these exponential attractors, we are able to prove the existence of the finite dimensional global attractor which attracts the bounded sets of initial data for all the possible values of the spatial average of the order parameter, hence improving previous results which required strong restrictions on the size of the spatial domain and to work on spaces on which the average of the order parameter is prescribed. Finally, we are able, in one and two space dimensions, to separate the solutions from the singular values of the potential, which allows us to reduce the problem to one with a regular potential. Unfortunately, for the unperturbed problem in three space dimensions, we need additional assumptions on the potential, which prevents us from proving such a result for logarithmic potentials. Copyright © 2004 John Wiley & Sons, Ltd.     

Infinite‐energy solutions for the Cahn–Hilliard equation in cylindrical domains

We give a detailed study of the infinite‐energy solutions of the Cahn–Hilliard equation in the 3D cylindrical domains in uniformly local phase space. In particular, we establish the well‐posedness and dissipativity for the case of regular potentials of arbitrary polynomial growth as well as for the case of sufficiently strong singular potentials. For these cases, we prove the further regularity of solutions and the existence of a global attractor. For the cases where we have failed to prove the uniqueness (e.g., for the logarithmic potentials), we establish the existence of the trajectory attractor and study its properties. Copyright © 2013 John Wiley & Sons, Ltd.     

Periodic solutions to the Cahn–Hilliard equation with constraint

This paper is concerned with the multidimensional Cahn–Hilliard equation with a constraint. The existence of periodic solutions of the problem is mainly proved under consideration by the viscosity approach. More precisely, with the help of the subdifferential operator theory and Schauder fixed point theorem, the existence of solutions to the approximation of the original problem is shown, and then the solution is obtained by using a passage‐to‐limit procedure based on a prior estimate. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical approximation of the conservative Allen–Cahn equation by operator splitting method

In this paper, a second‐order fast explicit operator splitting method is proposed to solve the mass‐conserving Allen–Cahn equation with a space–time‐dependent Lagrange multiplier. The space–time‐dependent Lagrange multiplier can preserve the volume of the system and keep small features. Moreover, we analyze the discrete maximum principle and the convergence rate of the fast explicit operator splitting method. The proposed numerical scheme is of spectral accuracy in space and of second‐order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, efficiency, mass conservation, and stability of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.     

Error estimates of fully discrete finite element solutions for the 2D Cahn–Hilliard equation with infinite time horizon

In this article, we deal with a rigorous error analysis for the finite element solutions of the two‐dimensional Cahn–Hilliard equation with infinite time. The error estimates with respect to are proven for the fully discrete conforming piecewise linear element solution under Assumption (A1) on the initial value and Assumption (A2) on the discrete spectrum estimate in the finite element space. The analysis is based on sharp a‐priori estimates for the solutions, particularly reflecting their behavior as . Numerical experiments are carried out to support the theoretical analysis and demonstrate the efficiency of the fully discrete mixed finite element methods. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 742–762, 2017     

Stabilized semi‐implicit spectral deferred correction methods for Allen–Cahn and Cahn–Hilliard equations

Stabilized semi‐implicit spectral deferred correction methods are constructed for the time discretization of Allen–Cahn and Cahn–Hilliard equations. These methods are unconditionally stable, lead to simple linear system to solve at each iteration, and can achieve high‐order time accuracy with a few iterations in each time step. Ample numerical results are presented to demonstrate the effectiveness of the stabilized semi‐implicit spectral deferred correction methods for solving the Allen–Cahn and Cahn–Hilliard equations. Copyright © 2013 John Wiley & Sons, Ltd.     

A numerical method for the Cahn–Hilliard equation with a variable mobility

We consider a conservative nonlinear multigrid method for the Cahn–Hilliard equation with a variable mobility of a model for phase separation in a binary mixture. The method uses the standard finite difference approximation in spatial discretization and the Crank–Nicholson semi-implicit scheme in temporal discretization. And the resulting discretized equations are solved by an efficient nonlinear multigrid method. The continuous problem has the conservation of mass and the decrease of the total energy. It is proved that these properties hold for the discrete problem. Also, we show the proposed scheme has a second-order convergence in space and time numerically. For numerical experiments, we investigate the effects of a variable mobility.     

Practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy

We propose a practical estimation of a splitting parameter for a spectral method for the ternary Cahn–Hilliard system with a logarithmic free energy. We use Eyre's convex splitting scheme for the time discretization and a Fourier spectral method for the space variables. Given an absolute temperature, we find composition values that make the total free energy be minimum. Then, we find the splitting parameter value that makes the two split homogeneous free energies be convex on the neighborhood of the local minimum concentrations. For general use, we also propose a sixth‐order polynomial approximation of the minimum concentration and derive a useful formula for the practical estimation of the splitting parameter in terms of the absolute temperature. The numerical tests are phase separation and total energy decrease with different temperature values. The linear stability analysis shows a good agreement between the exact and numerical solutions with an optimal value s. Various computational experiments confirm that the proposed splitting parameter estimation gives stable numerical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Adaptive operator splitting finite element method for Allen–Cahn equation

In this paper, a new numerical method is proposed and analyzed for the Allen–Cahn (AC) equation. We divide the AC equation into linear section and nonlinear section based on the idea of operator splitting. For the linear part, it is discretized by using the Crank–Nicolson scheme and solved by finite element method. The nonlinear part is solved accurately. In addition, a posteriori error estimator of AC equation is constructed in adaptive computation based on superconvergent cluster recovery. According to the proposed a posteriori error estimator, we design an adaptive algorithm for the AC equation. Numerical examples are also presented to illustrate the effectiveness of our adaptive procedure.     

Asymptotic behavior of higher‐order Navier‐Stokes‐Cahn‐Hilliard systems

Our aim in this paper is to study the asymptotic behavior, in terms of finite‐dimensional attractors, for higher‐order Navier‐Stokes‐Cahn‐Hilliard systems. Such equations describe the evolution of a mixture of 2 immiscible incompressible fluids. We also give several numerical simulations.     

On the convergence of operator splitting for the Rosenau–Burgers equation

We present convergence analysis of operator splitting methods applied to the nonlinear Rosenau–Burgers equation. The equation is first splitted into an unbounded linear part and a bounded nonlinear part and then operator splitting methods of Lie‐Trotter and Strang type are applied to the equation. The local error bounds are obtained by using an approach based on the differential theory of operators in Banach space and error terms of one and two‐dimensional numerical quadratures via Lie commutator bounds. The global error estimates are obtained via a Lady Windermere's fan argument. Lastly, a numerical example is studied to confirm the expected convergence order.     

A Cahn‐Hilliard–type equation with application to tumor growth dynamics

We consider a Cahn‐Hilliard–type equation with degenerate mobility and single‐well potential of Lennard‐Jones type. This equation models the evolution and growth of biological cells such as solid tumors. The degeneracy set of the mobility and the singularity set of the cellular potential do not coincide, and the absence of cells is an unstable equilibrium configuration of the potential. This feature introduces a nontrivial difference with respect to the Cahn‐Hilliard equation analyzed in the literature. We give existence results for different classes of weak solutions. Moreover, we formulate a continuous finite element approximation of the problem, where the positivity of the solution is enforced through a discrete variational inequality. We prove the existence and uniqueness of the discrete solution for any spatial dimension together with the convergence to the weak solution for spatial dimension d=1. We present simulation results in 1 and 2 space dimensions. We also study the dynamics of the spinodal decomposition and the growth and scaling laws of phase ordering dynamics. In this case, we find similar results to the ones obtained in standard phase ordering dynamics and we highlight the fact that the asymptotic behavior of the solution is dominated by the mechanism of growth by bulk diffusion.     

Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial‐boundary problem P for the nonlinear diffusion equation in an unbounded domain ( ), written as which represents the porous media, the fast diffusion equations, etc, where β is a single‐valued maximal monotone function on , and T>0. In Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) existence and uniqueness of solutions for P were directly proved under a growth condition for β even though the Stefan problem was excluded from examples of P . This paper completely removes the growth condition for β by confirming Cauchy's criterion for solutions of the following approximate problem _ε with approximate parameter ε>0: which is called the Cahn‐Hilliard system, even if ( ) is an unbounded domain. Moreover, it can be seen that the Stefan problem excluded from Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) is covered in the framework of this paper.     

Exponential attractors for the Cahn–Hilliard equation with dynamic boundary conditions

We consider in this article the Cahn–Hilliard equation endowed with dynamic boundary conditions. By interpreting these boundary conditions as a parabolic equation on the boundary and by considering a regularized problem, we obtain, by the Leray–Schauder principle, the existence and uniqueness of solutions. We then construct a robust family of exponential attractors. Copyright © 2005 John Wiley & Sons, Ltd.