Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

LARGE TIME-STEPPING METHOD BASED ON THE FINITE ELEMENT DISCRETIZATION FOR THE CAHN-HILLIARD EQUATION

  • Yang, Yanfang (Faculty of Science, Xi'an Jiaotong University) ;
  • Feng, Xinlong (College of Mathematics and Systems Science, Xinjiang University) ;
  • He, Yinnian (Faculty of Science, Xi'an Jiaotong University)
  • Received : 2011.01.08
  • Accepted : 2011.03.17
  • Published : 2011.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, a class of large time-stepping method based on the finite element discretization for the Cahn-Hilliard equation with the Neumann boundary conditions is developed. The equation is discretized by finite element method in space and semi-implicit schemes in time. For the first order fully discrete scheme, convergence property is investigated by using finite element analysis. Numerical experiment is presented, which demonstrates the effectiveness of the large time-stepping approaches.

Keywords

References

  1. P.W. Bates and P.C. Fife, The dynamics of nucleation for the Cahn-Hilliard equation, SIAM J. Appl. Math., 53(4) (1993), 990-1008. https://doi.org/10.1137/0153049
  2. J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. https://doi.org/10.1063/1.1744102
  3. X.F. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equaiton, J. Diff. Geom., 44(2) (1996), 262-311.
  4. Z.X. Chen, Finite element methods and their applications, Spring-Verlag, Heidelberg, 2005.
  5. P.G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1987.
  6. M. Dehghan and D. Mirzaei, A numerical method based on the boundary integral equa- tion and dual reciprocity methods for one-dimensional Cahn-Hilliard equation, Eng. Anal. Bound. Elem., 33 (2009) 522-528. https://doi.org/10.1016/j.enganabound.2008.08.008
  7. C.M. Elliot and D.A. French, A nonconforming finite element method for the two- dimensional Cahn-Hilliard equation, SIAM J. Numer. Anal., 26 (1989) 884-903. https://doi.org/10.1137/0726049
  8. C.M. Elliot and D.A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38(1987) 97-128. https://doi.org/10.1093/imamat/38.2.97
  9. C.M. Elliot, D.A. French and D.A. Milner, A second order splitting method for the Cahn- Hilliard equation, Numer. Math., 54(1989) 575-590. https://doi.org/10.1007/BF01396363
  10. X.B. Feng and A. Prohl, Error analysis of a mixed finite element method for the Cahn- Hilliard equation, Numer. Math., 99 (2004) 47-84. https://doi.org/10.1007/s00211-004-0546-5
  11. M. Fernandino and C.A. Dorao, The least squares spectral element method for the Cahn- Hilliard equation, Appl. Math. Model., 35 (2011) 797-806. https://doi.org/10.1016/j.apm.2010.07.034
  12. D. Furihata, A stable and conservative finite difference scheme for the Cahn-Hilliard equation, Numer. Math., 87 (2001) 675-699. https://doi.org/10.1007/PL00005429
  13. Y.N. He, Y.X. Liu and T. Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math., 57 (2007) 616-628. https://doi.org/10.1016/j.apnum.2006.07.026
  14. M. Jobelin, C. Lapuerta, J.C. Latche, Ph. Angot and B. Piar, A finite element penalty- projection method for incompressible flows, J. Comput. Phys., 217(2)(2006) 502-518. https://doi.org/10.1016/j.jcp.2006.01.019
  15. D. Kay and R. Welford, A multigrid finite element solver for the Cahn-Hilliard equation, J. Comput. Phys., 212 (2006) 288-304. https://doi.org/10.1016/j.jcp.2005.07.004
  16. A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8(2) (1998) 965-985.
  17. R.L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. Lon- don Ser. A, 422(1863) (1989) 261-278. https://doi.org/10.1098/rspa.1989.0027
  18. Y.H. Xia, Y. Xu and C.W. Shu, Local discontinuous Galerkin methods for the Cahn- Hilliard type equations, J. Comput. Phys., 227 (2007) 472-491. https://doi.org/10.1016/j.jcp.2007.08.001
  19. C.J. Xu and T. Tang, Stability analysis of large time-stepping methods for epitaxial growth models, SIAM J. Numer. Anal., 44 (2007) 1759-1779.
  20. X.D. Ye, The Fourier collocation mehtod for the Cahn-Hilliard equation, Comput. Math. Appl., 44 (2002) 213-229. https://doi.org/10.1016/S0898-1221(02)00142-6
  21. X.D. Ye, The Legendre collocation method for the Cahn-Hilliard equation, J. Comput. Appl. Math., 150 (2003) 87-108. https://doi.org/10.1016/S0377-0427(02)00566-6
  22. X.D. Ye and X.L. Cheng, The Fourier spectral method for the Cahn-Hilliard equation, Appl. Math. Comput., 171 (2005) 345-357. https://doi.org/10.1016/j.amc.2005.01.050
  23. T. Zhang, Finite element analysis for Cahn-Hillard equation, Mathematica Numerica Sinica, 28(3) (2006) 281-292. https://doi.org/10.3321/j.issn:0254-7791.2006.03.006