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Superconvergence analysis of a two-grid method for an energy-stable Ciarlet-Raviart type scheme of Cahn-Hilliard equation

  • Qian Liu
  • Dongyang ShiEmail author
Original Paper
  • 32 Downloads

Abstract

In this paper, superconvergence analysis of a mixed finite element method (MFEM) combined with the two-grid method (TGM) is presented for the Cahn-Hilliard (CH) equation for the first time. In particular, the discrete energy-stable Ciarlet-Raviart scheme is constructed with the bilinear element. By use of the high accuracy character of the element, the superclose estimates are deduced for both of the traditional MFEM and of the TGM. Crucially, the main difficulty brought by the coupling of the unknowns is dealt with by some technical methods. Furthermore, the global superconvergent results are achieved by interpolation postprocessing skill. Numerical results illustrate that the proposed TGM is very effective and its computing cost is almost one-third of that of the traditional FEM without loss of accuracy.

Keywords

CH equation MFEM TGM Superconvergent estimates 

Notes

Funding information

This research is supported by National Natural Science Foundation of China (Grant No. 11671369).

References

  1. 1.
    Cahn, J.W.: On spinodal decomposition. Acta. Metall. 9, 795 (1961)Google Scholar
  2. 2.
    Cahn, J.W., Hilliard, JE: Free energy of a nonuniform system I, Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)Google Scholar
  3. 3.
    Hilliard, J.E.: Spinodal decomposition. Phase transformations in ASM Cleveland, 497–560 (1970)Google Scholar
  4. 4.
    Novick, C.A.: The Cahn-Hilliard equation:mathematical and modeling perspectives. Adv. Math. Sci. Appl. 8(2), 965–985 (1998)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cahn, J.W., Elliot, C.M., Novick-Cohen, A.: The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. Eur. J. Appl. Math. 7(3), 287–301 (1996)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cahn, J.W., Novick, C.A.: Limiting motion for anAllen-Cahn/Cahn-Hilliard system. In: Free Boundary Problems Theory and Applications, pp 89–97, Harlow (1996)Google Scholar
  7. 7.
    Bates, P.W., Fife, P.C.: The dynamics of nucleation for the Cahn-Hilliard equation. SIAM J. Appl. Math. 53(4), 990–1008 (1993)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Miranville, A., Zelik, S.: Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions. Math. Methods Appl. Sci. 28, 709–735 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Zheng, S., Milani, A.: Global attractors for singular perturbations of the Cahn-Hilliard equations. J. Diff. Equations 209, 101–139 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Willie, R.: The Cahn-Hilliard equation with boundary nonlinearity and high viscosity. Methods Appl. Anal. 10, 589–602 (2003)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Alikakos, N.D., Fusco, G., Karali, G.: Motion of bubbles towards the boundary for the Cahn-Hilliard equation. Euro. J. Appl. Math. 15, 103–124 (2004)MathSciNetzbMATHGoogle Scholar
  12. 12.
    He, Y., Liu, Y.: Stability and convergence of the spectral Galerkin method for the Cahn-Hilliard equation. Numer. Methods Partial Differ. Eq. 24(6), 1485–1500 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Li, D., Qiao, Z.: On second order semi-implicit fourier spectral methods for 2D Cahn-Hilliard equations. J. Sci. Comput. 70(1), 301–34 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Khiari, N., Achouri, T., Ben Mohamed, M.L., Omrani, K.: Finite difference approximate solutions for the Cahn-Hilliard equation. Numer. Methods Partial Differ. Eq. 23(2), 437–455 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Chen, K., Feng, W., Wang, C., Wise, S.M.: An energy stable fourth order finite difference scheme for the Cahn-Hilliard equation. J. Comput. Appl. Math. 362 (15), 574–595 (2019)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Tang, P., Qiu, F., Zhang, H., Yang, Y.: Phase separation patterns for diblock copolymers on spherical surfaces: a finite volume method. Phys. Rev. E. 72, 016710 (2005)Google Scholar
  17. 17.
    Elliott, C.M., French, D.A.: A nonconforming finite-element method for the two dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 26(4), 884–903 (1989)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Liu, Q., Chen, L., Zhou, J.: Two-level methods for the Cahn-Hilliard equation. Math. Comput. in Simula. 126(C), 89–103 (2016)MathSciNetGoogle Scholar
  19. 19.
    Wang, W., Chen, L., Zhou, J.: Postprocessing mixed finite element methods for solving Cahn-Hilliard equation: methods and error analysis. J. Sci. Comput. 67, 724–746 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Elliott, C.M., French, D.A.: Numerical studies of the Cahn-Hilliard equation for phase separation. IMA J. Appl. Math. 38(2), 97–128 (1987)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Feng, X., Prohl, A.: Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99, 47–84 (2004)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kay, D., Styles, V., Süli, E.: Discontinuous Galerkin finite element approximation of the Cahn-Hilliard equation with convection. SIAM J. Numer. Anal. 47(4), 2660–2685 (2009)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Brenner, S.C., Diegel, A.E., Li-Yeng, S.: A robust solver for a mixed finite element method for the Cahn-Hilliard equation. J. Sci. Comput. 77, 1234–1249 (2018)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Wu, X., van Zwieten, G.J., van der Zee, K.G.: Stabilized second-order convex splitting schemes for Cahn-Hilliard models with application to diffuse-interface tumor-growth models. Int. J. Numer. Methods Biomed. Eng. 30, 180–203 (2014)MathSciNetGoogle Scholar
  25. 25.
    Diegel, A.E., Wang, C., Wise, S.M.: Stability and convergence of a second order mixed finite element method for the Cahn-Hilliard equation. IMA J. Numer. AnaL. 36, 1867–1897 (2016)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Yan, Y., Chen, W., Wang, C., Wise, S.M.: A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation. Commun. Comput. Phys. 23 (2), 572–602 (2018)MathSciNetGoogle Scholar
  27. 27.
    Xu, J.: A novel two-grid method for semilinear equations. SIAM J. Sci. Comput. 15, 231–237 (1994)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Xu, J.: Two-grid discretization techniques for linear and non-linear pdes. SIAM J. Numer. Anal. 33, 1759–1777 (1996)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Dawson, C., Wheeler, M., Woodward, C.: A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35, 435–452 (1998)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Chen, Y., Huang, Y., Yu, D.: A two-grid method for expanded mixed finite-element solution of semilinear reaction diffusion equations. Int. J. Numer. Mech. Eng. 57, 193–209 (2003)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Liu, W., Yin, Z., Li, J.: A two-grid algorithm based on expanded mixed element discretizations for strongly nonlinear elliptic equations. Numer. Algor. 70(1), 93–111 (2015)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Shi, D., Yang, H.: Unconditional optimal error estimates of a two-grid method for semilinear parabolic equation. Appl. Math. Comput. 310, 40–47 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Chen, Y., Chen, L., Zhang, X.: Two-grid method for nonlinear parabolic equations by expanded mixed finite element methods. Numer. Methods Partial Differ. Eq. 29(4), 1238–1256 (2013)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Zuo, L., Du, G.: A parallel two-grid linearized method for the coupled Navier-Stokes-Darcy problem. Numer. Algor. 77(1), 151–165 (2018)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Wang, Y., Chen, Y.: A two-grid method for incompressible miscible displacement problems by mixed finite element and Eulerian-Lagrangian localized adjoint methods. J. Math. Anal. Appl. 468, 406–422 (2018)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Shi, D., Mu, P., Yang, H.: Superconvergence analysis of a two-grid method for semilinear parabolic equations. Appl. Math. Lett. 84, 34–41 (2018)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Zhou, J., Chen, L., Huang, Y., Wang, W.: An efficient two-grid scheme for the Cahn-Hilliard equation. Commun. Comput. Phys. 17(1), 127–145 (2015)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Lin, Q., Lin, J.: Finite Element Methods: Accuracy and Improvement. Science Press, Beijing (2006)Google Scholar
  39. 39.
    Diegel, A., Feng, X., Wise, S.M.: Analysis of a mixed finite element method for a Cahn-Hilliard-Darcy-Stokes system. SIAM J. Numer. Anal. 53(1), 127–152 (2015)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Lin, Q., Tobiska, L., Zhou, A.: Superconvergence and extrapolation of nonconforming low order finite elements applied to the Poisson equation. IMA J. Numer. Anal. 25(1), 160–181 (2005)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Shi, D., Pei, L.: Nonconforming quadrilateral finite element method for a class of nonlinear sine-Gordon equations. Appl. Math. Comput. 219(17), 9447–9460 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouChina

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