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Evolving surface finite element method for the Cahn–Hilliard equation

Abstract

We use the evolving surface finite element method to solve a Cahn–Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport formulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes according to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subsequence, of the finite element scheme. We conclude the paper by deriving error estimates and present various numerical examples.

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References

  1. 1.

    Aubin, T.: Nonlinear Analysis on Manifolds Monge–Ampère Equations. Springer, New York (1982)

  2. 2.

    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2002)

  3. 3.

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

  4. 4.

    Clarenz, U., Diewald, U., Dziuk, G., Rumpf, M.: A finite element method for surface restoration with smooth boundary conditions. Comput. Aided Geom. Des. 21(5), 427–455 (2004)

  5. 5.

    Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139–232 (2005)

  6. 6.

    Du, Q., Ju, L., Tian, L.: Finite element approximation of the Cahn–Hilliard equation on surfaces. Comput. Methods Appl. Mech. Eng. 200(29–32), 2458–2470 (2011)

  7. 7.

    Dziuk, G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: Hildebrandt, S., Leis, R. (eds.) Partial Differential Equations and Calculus of Variations. Lecture Notes in Mathematics, vol. 1357, pp. 142–155. Springer, Berlin (1988)

  8. 8.

    Dziuk, G., Elliott, C.M.: Finite elements on evolving surfaces. IMA J. Numer. Anal. 27(2), 262–292 (2007)

  9. 9.

    Dziuk, G., Elliott, C.M.: Surface finite elements for parabolic equations. J. Comput. Math. 25(4), 385–407 (2007)

  10. 10.

    Dziuk, G., Elliott, C.M.: A fully discrete evolving surface finite element method. SIAM J. Numer. Anal. 50(5), 2677–2694 (2012)

  11. 11.

    Dziuk, G., Elliott, C.M.: Finite element methods for surface PDEs. Acta Numer. 22, 289–396 (2013)

  12. 12.

    Dziuk, G., Elliott, C.M.: \(L^2\)-Estimates for the evolving surface finite element method. Math. Comput. 82, 1–24 (2013)

  13. 13.

    Dziuk, G., Lubich, C., Mansor, D.: Runga–Kutta time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal. 32(2), 394–416 (2012)

  14. 14.

    Eilks, C., Elliott, C.M.: Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method. J. Comput. Phys. 227(23), 9727–9741 (2008)

  15. 15.

    Elliott, C.M.: The Cahn–Hilliard model for the kinetics of phase separation. In: Rodrigues, J.F. (ed.) Mathematical Models for Phase Change Problems, International Series of Numerical Mathematics, vol. 88, pp. 35–73. Birkhäuser, Basel (1989)

  16. 16.

    Elliott, C.M., French, D.A., Milner, F.A.: A second order splitting method for the Cahn–Hilliard equation. Numer. Math. 54(5), 575–590 (1989)

  17. 17.

    Elliott, C.M., Stinner, B.: A surface phase field model for two-phase biological membranes. SIAM J. Appl. Math. 70(8), 2904–2928 (2010)

  18. 18.

    Elliott, C.M., Stinner, B.: Modeling and computation of two phase geometric biomembranes using surface finite elements. J. Comput. Phys. 229(18), 6585–6612 (2010)

  19. 19.

    Elliott, C.M., Stinner, B.: Computation of two-phase biomembranes with phase dependent material parameters using surface finite elements. Commun. Comput. Phys. 13, 325–360 (2013)

  20. 20.

    Elliott, C.M., Styles, V.: An ALE ESFEM for solving PDEs on evolving surfaces. Milan J. Math. 80(2), 469–501 (2012)

  21. 21.

    Erlebacher, J., Aziz, M.J., Karma, A., Dimitrov, N., Sieradzki, K.: Evolution of nanoporosity in delloying. Nature 410, 450–453 (2001)

  22. 22.

    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press Inc., Boca Raton (1992)

  23. 23.

    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)

  24. 24.

    Hartman, P.: Ordinary Differential Equations. SIAM, Philadelphia (2002)

  25. 25.

    Hebey, E.: Nonlinear Analysis on Manifolds: Soblev Spaces and Inequalities. Courant Institute of Mathematical Sciences, New York (2000)

  26. 26.

    Lubich, C., Mansour, D., Venkataraman, C.: Backward difference time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal. (2013). doi:10.1093/imanum/drs044

  27. 27.

    Mercker, M., Ptashnyk, M., Kühnle, J., Hartmann, D., Weiss, M., Jäger, W.: A multiscale approach to curvature modulated sorting in biological membranes. J. Theor. Biol. 301, 67–82 (2012)

  28. 28.

    Olshanskii, M.A., Reusken, A., Xu, X.: An Eulerian space-time finite element method for diffusion problems on evolving surfaces. (2013). arXiv: 1304.6155

  29. 29.

    Ranner, T.: Computational surface partial differential equations. Ph.D. thesis, University of Warwick (2013)

  30. 30.

    Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001)

  31. 31.

    Schmidt, A., Siebert, K.G., Köster, D., Heine, C.J.: Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA. Springer, Berlin, Heidelberg (2005)

  32. 32.

    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (2006)

  33. 33.

    Vierling, M.: Control-constrained parabolic optimal control problems on evolving surfaces—theory and variational discretization. (2011). arXiv: 1106.0622v4

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Acknowledgments

The authors would like to thank Andrew Stuart and Endre Sülli for thoughtful comments and discussion which have improved this work greatly.

Author information

Correspondence to Thomas Ranner.

Additional information

The work of C. M. Elliott was supported by the UK Engineering and Physical Sciences Research Council EPSRC Grant EP/G010404 and the work of T. Ranner was supported by a EPSRC Ph.D. studentship (Grant EP/P504333/1 and EP/P50516X/1) and the Warwick Impact Fund.

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Elliott, C.M., Ranner, T. Evolving surface finite element method for the Cahn–Hilliard equation. Numer. Math. 129, 483–534 (2015). https://doi.org/10.1007/s00211-014-0644-y

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Mathematics Subject Classification

  • Primary 65M12
  • Secondary 35R01
  • 25D30
  • 35K55
  • 65M60
  • 65M15