A nonconforming finite element method for the Cahn-Hilliard equation

This paper reports a fully discretized scheme for the Cahn-Hilliard equation. The method uses a convexity-splitting scheme to discretize in the temporal variable and a nonconforming finite element method to discretize in the spatial variable. And, the scheme can preserve the mass conservation and energy dissipation properties of the original problem. Some typical phase transition phenomena are also observed through the numerical examples.

[1]  S. M. Choo,et al.  Conservative nonlinear difference scheme for the Cahn-Hilliard equation—II , 1998 .

[2]  Donald A. French,et al.  Continuous finite element methods which preserve energy properties for nonlinear problems , 1990 .

[3]  Charles M. Elliott,et al.  Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy , 1992 .

[4]  Zhang,et al.  A POSTERIORI ESTIMATOR OF NONCONFORMING FINITE ELEMENT METHOD FOR FOURTH ORDER ELLIPTIC PERTURBATION PROBLEMS , 2008 .

[5]  Jie Shen,et al.  Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier spectral method. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  R. Nicolaides,et al.  Numerical analysis of a continuum model of phase transition , 1991 .

[7]  Xiaobing Feng,et al.  Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition , 2007, Math. Comput..

[8]  C. M. Elliott,et al.  Numerical Studies of the Cahn-Hilliard Equation for Phase Separation , 1987 .

[9]  Charles M. Elliott,et al.  Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation , 1992 .

[10]  John W. Cahn,et al.  On Spinodal Decomposition , 1961 .

[11]  Xingde Ye,et al.  The Fourier spectral method for the Cahn-Hilliard equation , 2005, Appl. Math. Comput..

[12]  Charles M. Elliott,et al.  The Cahn-Hilliard Model for the Kinetics of Phase Separation , 1989 .

[13]  Xingde Ye,et al.  The Fourier collocation method for the Cahn-Hilliard equation☆ , 2002 .

[14]  B. Vollmayr-Lee,et al.  Fast and accurate coarsening simulation with an unconditionally stable time step. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  M ChooS,et al.  Cahn‐Hilliad方程式に関する保存型非線形差分スキーム‐II , 2000 .

[16]  Charles M. Elliott,et al.  A second order splitting method for the Cahn-Hilliard equation , 1989 .

[17]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[18]  Andreas Prohl,et al.  Error analysis of a mixed finite element method for the Cahn-Hilliard equation , 2004, Numerische Mathematik.

[19]  Charles M. Elliott,et al.  On the Cahn-Hilliard equation , 1986 .

[20]  J. Lowengrub,et al.  Conservative multigrid methods for Cahn-Hilliard fluids , 2004 .

[21]  Harald Garcke,et al.  A Phase Field Model for Continuous Clustering on Vector Fields , 2001, IEEE Trans. Vis. Comput. Graph..

[22]  John W. Barrett,et al.  Finite Element Approximation of a Degenerate Allen-Cahn/Cahn-Hilliard System , 2002, SIAM J. Numer. Anal..

[23]  S. M. Choo,et al.  A discontinuous Galerkin method for the Cahn-Hilliard equation , 2005 .

[24]  Xue-Cheng Tai,et al.  A robust nonconforming H2-element , 2001, Math. Comput..

[25]  E. Mello,et al.  Numerical study of the Cahn–Hilliard equation in one, two and three dimensions , 2004, cond-mat/0410772.

[26]  Donald A. French,et al.  Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems , 1994 .

[27]  Formation of Liesegang patterns: Simulations using a kinetic Ising model , 2000, cond-mat/0009467.

[28]  Krishna Garikipati,et al.  A discontinuous Galerkin method for the Cahn-Hilliard equation , 2006, J. Comput. Phys..

[29]  P. Sheng,et al.  Continuum Modelling of Nanoscale Hydrodynamics , 2008 .

[30]  Yan Xu,et al.  Local discontinuous Galerkin methods for the Cahn-Hilliard type equations , 2007, J. Comput. Phys..

[31]  J. Langer,et al.  New computational method in the theory of spinodal decomposition , 1975 .

[32]  Andreas Prohl,et al.  Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem, Part II: Error analysis and convergence of the interface , 2001 .

[33]  Yinnian He,et al.  On large time-stepping methods for the Cahn--Hilliard equation , 2007 .

[34]  Zhi-zhong Sun,et al.  A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation , 1995 .

[35]  J. E. Hilliard,et al.  Early stages of spinodal decomposition in an aluminum-zinc alloy , 1967 .

[36]  Andrea L. Bertozzi,et al.  Inpainting of Binary Images Using the Cahn–Hilliard Equation , 2007, IEEE Transactions on Image Processing.

[37]  Haijun Wu,et al.  A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF THE CAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW * , 2007, 0708.2116.

[38]  Daisuke Furihata,et al.  A stable and conservative finite difference scheme for the Cahn-Hilliard equation , 2001, Numerische Mathematik.

[39]  Jie Shen,et al.  Applications of semi-implicit Fourier-spectral method to phase field equations , 1998 .

[40]  D. Furihata,et al.  Finite Difference Schemes for ∂u∂t=(∂∂x)αδGδu That Inherit Energy Conservation or Dissipation Property , 1999 .

[41]  Tao Tang,et al.  Stability Analysis of Large Time-Stepping Methods for Epitaxial Growth Models , 2006, SIAM J. Numer. Anal..

[42]  Ming Wang,et al.  A new class of Zienkiewicz-type non-conforming element in any dimensions , 2007, Numerische Mathematik.

[43]  C. M. Elliott,et al.  A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation , 1989 .

[44]  J. Barrett,et al.  Finite element approximation of an Allen-Cahn/Cahn-Hilliard system , 2002 .

[45]  Xingde Ye,et al.  The Legendre collocation method for the Cahn-Hilliard equation , 2003 .