Skip to main content
SpringerLink
Log in

A Fully Discrete Mixed Finite Element Method for the Stochastic Cahn–Hilliard Equation with Gradient-Type Multiplicative Noise

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

Abstract

This paper develops and analyzes a fully discrete mixed finite element method for the stochastic Cahn–Hilliard equation with gradient-type multiplicative noise that is white in time and correlated in space. The stochastic Cahn–Hilliard equation is formally derived as a phase field formulation of the stochastically perturbed Hele–Shaw flow. The main result of this paper is to prove strong convergence with optimal rates for the proposed mixed finite element method. To overcome the difficulty caused by the low regularity in time of the solution to the stochastic Cahn–Hilliard equation, the Hölder continuity in time with respect to various norms for the stochastic PDE solution is established, and it plays a crucial role in the error analysis. Numerical experiments are also provided to validate the theoretical results and to study the impact of noise on the Hele–Shaw flow as well as the interplay of the geometric evolution and gradient-type noise.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Adams, R., Fournier, J.: Sobolev Spaces, vol. 140. Academic Press, Cambridge (2003)

    MATH  Google Scholar 

  2. Alikakos, N.D., Bates, P.W., Chen, X.: Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Ration. Mech. Anal. 128(2), 165–205 (1994)

    Article  MathSciNet  Google Scholar 

  3. Aristotelous, A.C., Karakashian, O.A., Wise, S.M.: A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation. Disc. Cont. Dyn. Syst. Ser. B. 18(9), 2211–2238 (2013)

    MATH  Google Scholar 

  4. Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1998)

    Article  MathSciNet  Google Scholar 

  5. Blömker, D., Maiker-Paape, S., Wanner, T.: Spinodal decomposition for the stochastic Cahn-Hilliard equation. Trans. Am. Math. Soc. 360, 449–489 (2008)

    Article  Google Scholar 

  6. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)

    Book  Google Scholar 

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

    Article  Google Scholar 

  8. Chen, X.: Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differ. Geom. 44(2), 262–311 (1996)

    Article  MathSciNet  Google Scholar 

  9. Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  10. Cook, H.E.: Brownian motion in spinodal decomposition. Acta Metallurgica 18, 297–306 (1970)

    Article  Google Scholar 

  11. Du, Q, Feng, X.: The phase field method for geometric moving interfaces and their numerical approximations. arXiv:1902.04924 [math.NA] (2019)

  12. Eyre, D.: Unconditionally gradient stable time marching the Cahn-Hilliard equation. In: Bullard, J.W., Kalia, R., Stoneham, M., Chen, L.Q. (eds.) Computational and Mathematical Models of Microstructural Evolution, vol. 53, pp. 1686–1712. Materials Research Society, Warrendale (1998)

    Google Scholar 

  13. Feng, X., Li, Y., Xing, Y.: Analysis of mixed interior penalty discontinuous Galerkin methods for the Cahn-Hilliard equation and the Hele-Shaw flow. SIAM J. Numer. Anal. 54(2), 825–847 (2016)

    Article  MathSciNet  Google Scholar 

  14. Feng, X., Prohl, A.: Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 74, 47–84 (2004)

    Article  MathSciNet  Google Scholar 

  15. Feng, X., Prohl, A.: Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem. Inter. Free Bound. 7, 1–28 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Feng, X., Li, Y., Zhang, Y.: Finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noise. SIAM J. Numer. Anal. 55, 194–216 (2017)

    Article  MathSciNet  Google Scholar 

  17. Feng, X., Li, Y., Zhang, Y.: Strong convergence of a fully discrete finite element method for a class of semilinear stochastic partial differential equations with multiplicative noise. arXiv:1811.05028 [math.NA] (2018)

  18. Funaki, T.: The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Relat. Fields 102, 221–288 (1995)

    Article  MathSciNet  Google Scholar 

  19. Furihata, D., Kovács, M., Larsson, S., Lindgren, F.: Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. SIAM J. Numer. Anal. 56, 708–731 (2018)

    Article  MathSciNet  Google Scholar 

  20. Girault, V., Raviart, P.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. vol. 5 of Springer Series in Computational Mathematics. Springer, Berlin (1986)

  21. Katsoulakis, M., Kossioris, G., Lakkis, O.: Noise regularization and computations for the \(1\)-dimensional stochastic Allen-Cahn problem. Interfaces Free Bound. 9, 1–30 (2007)

    Article  MathSciNet  Google Scholar 

  22. Kawasaki, K., Ohta, T.: Kinetic drumhead model of interface. I. Progr. Theor. Phys. 67, 147–163 (1982)

    Article  Google Scholar 

  23. Kovács, M., Larsson, S., Lindgren, F.: On the backward Euler approximation of the stochastic Allen-Cahn equation. J. Appl. Probab. 52, 323–338 (2015)

    Article  MathSciNet  Google Scholar 

  24. Kovács, M., Larsson, S., Lindgren, F.: On the discretisation in time of the stochastic Allen-Cahn equation. Math. Nachr. 291, 966–995 (2018)

    Article  MathSciNet  Google Scholar 

  25. Kovács, M., Larsson, S., Mesforush, A.: Finite element approximation of the Cahn-Hilliard-Cook equation. SIAM J. Numer. Anal. 49, 2407–2429 (2011)

    Article  MathSciNet  Google Scholar 

  26. Kovács, M., Larsson, S., Mesforush, A.: Erratum: finite element approximation of the Cahn-Hilliard-Cook equation. SIAM J. Numer. Anal. 52, 2594–2597 (2014)

    Article  MathSciNet  Google Scholar 

  27. Krylov, N.V., Rozovskii, B.L.: Stochastic Evolution Equations. Stochastic Differential Equations: Theory and Applications: Interdisciplinary Math and Science. World Science Publication, Hackensack 2, 1–69 (2007)

  28. Larsson, S., Mesforush, A.: Finite-element approximation of the linearized Cahn-Hilliard-Cook equation. IMA J. Numer. Anal. 31, 1315–1333 (2011)

    Article  MathSciNet  Google Scholar 

  29. Li, Y.: Error analysis of a fully discrete Morley finite element approximation for the Cahn–Hilliard equation. J. Sci. Comput. 78, 1862–1892 (2019)

    Article  MathSciNet  Google Scholar 

  30. Liu, Z., Qiao, Z.: Wong–Zakai approximations of stochastic Allen–Cahn equation. arXiv:1710.0953 [math.NA] (2017)

  31. Lord, G., Powell, C., Shardlow, T.: An Introduction to Computational Stochastic PDEs. Cambridge Texts in Applied Mathematics. CUP, Cambridge (2014)

    Book  Google Scholar 

  32. Majee, A.K., Prohl, A.: Optimal strong rates of convergence for a space-time discretization of the stochastic Allen-Cahn equation with multiplicative noise. Comput. Methods Appl. Math. 18, 297–311 (2018)

    Article  MathSciNet  Google Scholar 

  33. Nochetto, R.H., Verdi, C.: Convergence past singularities for a fully discrete approximation of curvature-driven interfaces. SIAM J. Numer. Anal. 34(2), 490–512 (1997)

    Article  MathSciNet  Google Scholar 

  34. Da Prato, G., Debussche, A.: Stochastic Cahn-Hilliard equation. Nonlinear Anal. 26, 241–263 (1996)

    Article  MathSciNet  Google Scholar 

  35. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    Book  Google Scholar 

  36. Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Springer, Berlin (2007)

    MATH  Google Scholar 

  37. Pego, R.L.: Front migration in the nonlinear Cahn-Hilliard equation. Proc. R. Soc. Lond. Ser. A 422, 261–278 (1989)

    MathSciNet  MATH  Google Scholar 

  38. Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Frontiers in Applied Mathematics. SIAM, Philadelphia (2008)

    Book  Google Scholar 

  39. Röger, M., Weber, H.: Tightness for a stochastic Allen-Cahn equation. Stoch. PDE: Anal. Comp. 1, 175–203 (2013)

    Article  MathSciNet  Google Scholar 

  40. Stoth, B.: Convergence of the Cahn-Hilliard equation to the Mullins-Sekerka problem in spherical symmetry. J. Differ. Eqs. 125, 154–183 (1996)

    Article  MathSciNet  Google Scholar 

  41. Wu, S., Li, Y.: Analysis of the Morley element for the Cahn–Hilliard equation and the Hele–Shaw flow. arXiv:1808.08581 [math.NA] (2018)

  42. Yip, N.K.: Stochastic curvature driven flows. In: Da Prato, G., Tubaro, L. (eds.) Stochastic Partial Differential Equations and Applications, Lecture Notes in Pure and Applied Mathematics, vol. 227, pp. 443–460. Marcel Dekker, New York (2002)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Tennessee, Knoxville, TN, 37996, USA

    Xiaobing Feng

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Yukun Li

  3. Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC, 27402, USA

    Yi Zhang

Authors
  1. Xiaobing Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yukun Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi Zhang.

Additional information

Publisher's Note

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

The work of the Xiaobing Feng was partially supported by the NSF Grant DMS-1620168.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, X., Li, Y. & Zhang, Y. A Fully Discrete Mixed Finite Element Method for the Stochastic Cahn–Hilliard Equation with Gradient-Type Multiplicative Noise. J Sci Comput 83, 23 (2020). https://doi.org/10.1007/s10915-020-01202-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01202-3

Keywords

Mathematics Subject Classification

Search

Navigation