Skip to main content

Advertisement

The Convergence Analysis of a Class of Stabilized Semi-Implicit Isogeometric Methods for the Cahn-Hilliard Equation

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

Isogeometric analysis (IGA) has been widely used as a spatial discretization method for phase field models since the seminal work of Gómez et al. (Comput. Methods Appl. Mech. Engrg. 197(49), pp. 4333–4352, 2008), and the first numerical convergence study of IGA for the Cahn-Hilliard equation was presented by Kästner et al. (J. Comput. Phys. 305(15), pp. 360–371, 2016). However, to the best of our knowledge, the theoretical convergence analysis of IGA for the Cahn-Hilliard equation is still missing in the literature. In this paper, we provide the convergence analysis of IGA for the multi-dimensional Cahn-Hilliard equation for the first time. The two important steps to carry out the convergence analysis are (1) we rigorously prove that the \(L^{\infty }\) norm of IGA solution is uniformly bounded for all mesh sizes, and (2) we construct an appropriate Ritz projection operator for the bi-Laplacian term in the Cahn-Hilliard equation. The first- and second-order stabilized semi-implicit schemes are used to obtain the fully discrete schemes. The energy stability analyses are rigorously proved for the resulting fully discrete schemes. Finally, several two- and three-dimensional numerical examples are presented to verify the theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Data Availability

Data sets generated during the current study are available from the corresponding author on reasonable request.

References

  1. Antonietti, P.F., Beirão da Veiga, L., Scacchi, S., Verani, M.: A \(C^1\) virtual element method for the Cahn-Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54(1), 34–56 (2016)

    MathSciNet  MATH  Google Scholar 

  2. Atroshchenko, E., Tomar, S., Xu, G., Bordas, S.P.A.: Weakening the tight coupling between geometry and simulation in isogeometric analysis: from sub- and super-geometric analysis to Geometry-Independent Field approximation (GIFT). Int. J. Numer. Meth. Eng. 114(10), 1131–1159 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Bartezzaghi, A., Dedè, L., Quarteroni, A.: Isogeometric analysis of high order partial differential equations on surfaces. Comput. Methods Appl. Mech. Eng. 295, 446–469 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Baskaran, A., Lowengrub, J.S., Wang, C., Wise, S.M.: Convergence analysis of a second order convex splitting scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 51(5), 2851–2873 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Bazilevs, Y., Beirão da Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes. Math. Models Methods Appl. Sci. 16(07), 1031–1090 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Bazilevs, Y., Calo, V.M., Cottrell, J.A., Evans, J.A., Hughes, T.J.R., Lipton, S., Scott, M.A., Sederberg, T.W.: Isogeometric analysis using T-splines. Comput. Methods Appl. Mech. Eng. 199(5), 229–263 (2010)

    MathSciNet  MATH  Google Scholar 

  7. Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods,. Springer Science & Business Media, Cham (2012)

    MATH  Google Scholar 

  8. Dedè, L., Borden, M.J., Hughes, T.J.R.: Isogeometric analysis for topology optimization with a phase field model. Arch. Comput. Methods Eng. 19, 427–465 (2012)

    MathSciNet  MATH  Google Scholar 

  9. Caffarelli, L.A., Muler, N.E.: An \(L^{\infty }\) bound for solutions of the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 133, 129–144 (1995)

    MathSciNet  MATH  Google Scholar 

  10. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958)

    MATH  Google Scholar 

  11. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 31(3), 688–699 (1959)

    MATH  Google Scholar 

  12. Chen, L., Shen, J.: Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Commun. 108(2–3), 147–158 (1998)

    MATH  Google Scholar 

  13. Cheng, 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, 574–595 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Ciarlet, P.G.: The finite element method for elliptic problems. SIAM, (2002)

  15. Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric analysis: toward integration of CAD and FEA. John Wiley & Sons, Amsterdam (2009)

    MATH  Google Scholar 

  16. 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(4), 1867–1897 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Du, Q., Ju, L., Li, X., Qiao, Z.: Stabilized linear semi-implicit schemes for the nonlocal Cahn-Hilliard equation. J. Comput. Phys. 363, 39–54 (2018)

    MathSciNet  MATH  Google Scholar 

  18. 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)

    MathSciNet  MATH  Google Scholar 

  19. 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)

    MathSciNet  MATH  Google Scholar 

  20. Elliott, C.M., French, D.A., Milner, F.A.: A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54, 575–590 (1989)

    MathSciNet  MATH  Google Scholar 

  21. Elliott, C.M., Songmu, Z.: On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96(4), 339–357 (1986)

    MATH  Google Scholar 

  22. Elliott, C.M., Stuart, A.M.: The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal. 30(6), 1622–1663 (1993)

    MathSciNet  MATH  Google Scholar 

  23. Eyre, D.J.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. MRS Online Proc. Libr. 529, 39–46 (1998)

    MathSciNet  MATH  Google Scholar 

  24. Feng, X., Prohl, A.: Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99, 47–84 (2004)

    MathSciNet  MATH  Google Scholar 

  25. Fu, Z., Tang, T., Yang, J.: Energy diminishing implicit-explicit Runge–Kutta methods for gradient flows. Mathematics of Computation, (2024)

  26. Furihata, D.: A stable and conservative finite difference scheme for the Cahn-Hilliard equation. Numer. Math. 87(4), 675–699 (2001)

    MathSciNet  MATH  Google Scholar 

  27. Gerasimov, T., Stylianou, A., Sweers, G.: Corners give problems when decoupling fourth order equations into second order systems. SIAM J. Numer. Anal. 50(3), 1604–1623 (2012)

    MathSciNet  MATH  Google Scholar 

  28. Gómez, H., Calo, V.M., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of the Cahn-Hilliard phase-field model. Comput. Methods Appl. Mech. Eng. 197(49), 4333–4352 (2008)

    MathSciNet  MATH  Google Scholar 

  29. Gómez, H., Nogueira, X.: A new space-time discretization for the Swift-Hohenberg equation that strictly respects the Lyapunov functional. Commun. Nonlinear Sci. Numer. Simul. 17(12), 4930–4946 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Gómez, H., Nogueira, X.: An unconditionally energy-stable method for the phase field crystal equation. Comput. Methods Appl. Mech. Eng. 249–252, 52–61 (2012)

    MathSciNet  MATH  Google Scholar 

  31. Gómez, H., van der Zee, K.G.: Computational Phase-Field Modeling, pp. 1–35. John Wiley & Sons Ltd, Amsterdam (2017)

    MATH  Google Scholar 

  32. He, Y., Liu, Y., Tang, T.: On large time-stepping methods for the Cahn-Hilliard equation. Appl. Numer. Math. 57(5), 616–628 (2007)

    MathSciNet  MATH  Google Scholar 

  33. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)

    MathSciNet  MATH  Google Scholar 

  34. Hu, X., Zhu, S.: Isogeometric analysis for time-fractional partial differential equations. Numer. Algorithms 85(3), 909–930 (2020)

    MathSciNet  MATH  Google Scholar 

  35. Jansen, K.E., Whiting, C.H., Hulbert, G.M.: A generalized-\(\alpha \) method for integrating the filtered Navier-Stokes equations with a stabilized finite element method. Comput. Methods Appl. Mech. Eng. 190(3), 305–319 (2000)

    MathSciNet  MATH  Google Scholar 

  36. Kaessmair, S., Steinmann, P.: Comparative computational analysis of the Cahn-Hilliard equation with emphasis on \(C^1\)-continuous methods. J. Comput. Phys. 322, 783–803 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Kästner, M., Metsch, P., de Borst, R.: Isogeometric analysis of the Cahn-Hilliard equation - a convergence study. J. Comput. Phys. 305, 360–371 (2016)

    MathSciNet  MATH  Google Scholar 

  38. 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)

    MathSciNet  MATH  Google Scholar 

  39. Li, D.: Why large time-stepping methods for the Cahn-Hilliard equation is stable. Math. Comput. 91(338), 2501–2515 (2022)

    MathSciNet  MATH  Google Scholar 

  40. Li, D., Qiao, Z.: On second order semi-implicit Fourier spectral methods for 2D Cahn-Hilliard equations. J. Sci. Comput. 70, 301–341 (2017)

    MathSciNet  MATH  Google Scholar 

  41. Li, R., Wu, Q., Zhu, S.: Proper orthogonal decomposition with SUPG-stabilized isogeometric analysis for reduced order modelling of unsteady convection-dominated convection-diffusion-reaction problems. J. Comput. Phys. 387, 280–302 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Li, Y., Lee, H.G., Xia, B., Kim, J.: A compact fourth-order finite difference scheme for the three-dimensional Cahn-Hilliard equation. Comput. Phys. Commun. 200, 108–116 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Liu, J., Dedè, L., Evans, J.A., Borden, M.J., Hughes, T.J.R.: Isogeometric analysis of the advective Cahn-Hilliard equation: Spinodal decomposition under shear flow. J. Comput. Phys. 242, 321–350 (2013)

    MathSciNet  MATH  Google Scholar 

  44. Liu, X., He, Z., Chen, Z.: A fully discrete virtual element scheme for the Cahn-Hilliard equation in mixed form. Comput. Phys. Commun. 246, 106870 (2020)

    MathSciNet  MATH  Google Scholar 

  45. Medina, E.Y., Toledo, E.M., Igreja, I., Rocha, B.M.: A stabilized hybrid discontinuous Galerkin method for the Cahn-Hilliard equation. J. Comput. Appl. Math. 406, 114025 (2022)

    MathSciNet  MATH  Google Scholar 

  46. Mu, L., Ye, X., Zhang, S.: Development of a \(P_2\) element with optimal \(L^2\) convergence for biharmonic equation. Numer. Methods Partial Differ. Equ. 35(4), 1497–1508 (2019)

    MATH  Google Scholar 

  47. Pan, Q., Chen, C., Rabczuk, T., Zhang, J., Yang, X.: The subdivision-based IGA-EIEQ numerical scheme for the binary surfactant Cahn-Hilliard phase-field model on complex curved surfaces. Comput. Methods Appl. Mech. Eng. 406, 115905 (2023)

    MathSciNet  MATH  Google Scholar 

  48. Pan, Q., Chen, C., Zhang, Y., Yang, X.: A novel hybrid IGA-EIEQ numerical method for the Allen-Cahn/Cahn-Hilliard equations on complex curved surfaces. Comput. Methods Appl. Mech. Eng. 404(1), 115767 (2023)

    MathSciNet  MATH  Google Scholar 

  49. Philip, K.C., Alejandro, D.R.: A numerical method for the nonlinear Cahn-Hilliard equation with nonperiodic boundary conditions. Comput. Mater. Sci. 3(3), 377–392 (1995)

    MATH  Google Scholar 

  50. Piegl, L., Tiller, W.: The NURBS Book (Monographs in Visual Communication), 2nd edn. Springer-Verlag, New York (1997)

    MATH  Google Scholar 

  51. Roache, P.J.: Code verification by the method of manufactured solutions. J. Fluids Eng. 124(1), 4–10 (2002)

    MATH  Google Scholar 

  52. Shen, J., Xu, J., Yang, J.: The scalar auxiliary variable (SAV) approach for gradient flows. J. Comput. Phys. 353(15), 407–416 (2018)

    MathSciNet  MATH  Google Scholar 

  53. Shen, J., Xu, J., Yang, J.: A new class of efficient and robust energy stable schemes for gradient flows. SIAM Rev. 61(3), 474–506 (2019)

    MathSciNet  MATH  Google Scholar 

  54. Shen, J., Yang, X.: Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. 28(4), 1669–1691 (2010)

    MathSciNet  MATH  Google Scholar 

  55. Stefan, J.: \(\ddot{\text{ U }}\)ber die theorie der eisbildung. Monatshefte für Mathematik 1(1), 1–6 (1890)

    MathSciNet  MATH  Google Scholar 

  56. Steinbach, I.: Phase-field models in materials science. Modell. Simul. Mater. Sci. Eng. 17(7), 073001 (2009)

    MATH  Google Scholar 

  57. Stogner, R.H., Carey, G.F., Murray, B.T.: Approximation of Cahn-Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with \(C^1\) elements. Int. J. Numer. Meth. Eng. 76(5), 636–661 (2008)

    MATH  Google Scholar 

  58. Sun, Z.: A second-order accurate linearized difference scheme for the two-dimensional Cahn-Hilliard equation. Math. Comput. 64(212), 1463–1471 (1995)

    MathSciNet  MATH  Google Scholar 

  59. Tagliabue, A., Dedè, L., Quarteroni, A.: Isogeometric analysis and error estimates for high order partial differential equations in fluid dynamics. Comput. & Fluids 102, 277–303 (2014)

    MathSciNet  MATH  Google Scholar 

  60. Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics, 2nd edn. Springer, New York (1997)

    MATH  Google Scholar 

  61. Valizadeh, N., Rabczuk, T.: Isogeometric analysis for phase-field models of geometric PDEs and high-order PDEs on stationary and evolving surfaces. Comput. Methods Appl. Mech. Eng. 351, 599–642 (2019)

    MathSciNet  MATH  Google Scholar 

  62. Wang, J., Zhai, Q., Zhang, R., Zhang, S.: A weak Galerkin finite element scheme for the Cahn-Hilliard equation. Math. Comput. 88(315), 211–235 (2019)

    MathSciNet  MATH  Google Scholar 

  63. Wang, L., Yu, H.: On efficient second order stabilized semi-implicit schemes for the Cahn-Hilliard phase-field equation. J. Sci. Comput. 77(2), 1185–1209 (2018)

    MathSciNet  MATH  Google Scholar 

  64. Wells, G.N., Kuhl, E., Garikipati, K.: A discontinuous Galerkin method for the Cahn-Hilliard equation. J. Comput. Phys. 218(2), 860–877 (2006)

    MathSciNet  MATH  Google Scholar 

  65. Wise, S.M., Wang, C., Lowengrub, J.S.: An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47(3), 2269–2288 (2009)

    MathSciNet  MATH  Google Scholar 

  66. Xia, Y., Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for the Cahn-Hilliard type equations. J. Comput. Phys. 227(1), 472–491 (2007)

    MathSciNet  MATH  Google Scholar 

  67. Xu, C., Tang, T.: Stability analysis of large time-stepping methods for epitaxial growth models. SIAM J. Numer. Anal. 44(4), 1759–1779 (2006)

    MathSciNet  MATH  Google Scholar 

  68. 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)

    MathSciNet  MATH  Google Scholar 

  69. Yang, X.: Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 327(15), 294–316 (2016)

    MathSciNet  MATH  Google Scholar 

  70. Zhang, L., Tonks, M.R., Gaston, D., Peterson, J.W., Andrs, D., Millett, P.C., Biner, B.S.: A quantitative comparison between \(C^0\) and \(C^1\) elements for solving the Cahn-Hilliard equation. J. Comput. Phys. 236, 74–80 (2013)

    MathSciNet  Google Scholar 

  71. Zhang, R., Qian, X.: Triangulation-based isogeometric analysis of the Cahn-Hilliard phase-field model. Comput. Methods Appl. Mech. Eng. 357, 112569 (2019)

    MathSciNet  MATH  Google Scholar 

  72. Zhang, S., Wang, M.: A nonconforming finite element method for the Cahn-Hilliard equation. J. Comput. Phys. 229(19), 7361–7372 (2010)

    MathSciNet  MATH  Google Scholar 

  73. Zhang, S., Zhang, Z.: Invalidity of decoupling a biharmonic equation to two poisson equations on non-convex polygons. Int. J. Numer. Anal. Model. 5(1), 73–76 (2008)

    MathSciNet  MATH  Google Scholar 

  74. Zhao, W., Guan, Q.: Numerical analysis of energy stable weak Galerkin schemes for the Cahn-Hilliard equation. Commun. Nonlinear Sci. Numer. Simul. 118, 106999 (2023)

    MathSciNet  MATH  Google Scholar 

  75. Zhu, J., Chen, L., Shen, J., Tikare, V.: Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier spectral method. Phys. Rev. E 60(4), 3564 (1999)

    MATH  Google Scholar 

  76. Zhu, S., Dedè, L., Quarteroni, A.: Isogeometric analysis and proper orthogonal decomposition for parabolic problems. Numer. Math. 135(2), 333–370 (2017)

    MathSciNet  MATH  Google Scholar 

  77. Zhu, S., Dedè, L., Quarteroni, A.: Isogeometric analysis and proper orthogonal decomposition for the acoustic wave equation. ESAIM: Math. Modell. Numer. Anal. 51(4), 1197–1221 (2017)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their careful reading and for their critical questions, which helped to improve the paper.

Funding

The research of X. Meng was partially supported by the National Natural Science Foundation of China (Grant No. 12101057), the Scientific Research Fund of Beijing Normal University (Grant No. 28704-111032105), and the Guangdong Higher Education Upgrading Plan (UIC-R0400024-21). The research of Y. Qin was partially supported by the National Natural Science Foundation of China (Grant No. 12201369), and the Natural Science Foundation of Shanxi Province (Grant No. 202303021211004). G. Hu would like to thank the support from the National Natural Science Foundation of China (Grant No. 11922120), the Science and Technology Development Fund of Macao SAR (File/Project Nos. 001/2024/SKL and 0068/2024/RIA1), and the MYRG of University of Macau (Grant No. MYRG-GRG2023-00157-FST-UMDF).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yuzhe Qin.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meng, X., Qin, Y. & Hu, G. The Convergence Analysis of a Class of Stabilized Semi-Implicit Isogeometric Methods for the Cahn-Hilliard Equation. J Sci Comput 102, 26 (2025). https://doi.org/10.1007/s10915-024-02753-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-024-02753-5

Keywords

Mathematics Subject Classification