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Error analysis of a mixed finite element method for the Cahn-Hilliard equation

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Summary.

We propose and analyze a semi-discrete and a fully discrete mixed finite element method for the Cahn-Hilliard equation u t + Δ(ɛΔu−ɛ−1f(u)) = 0, where ɛ > 0 is a small parameter. Error estimates which are quasi-optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain. In particular, it is shown that all error bounds depend on only in some lower polynomial order for small ɛ. The cruxes of our analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [2], and Chen [15] to prove a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on a stretched time grid. The ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem as ɛ → 0 in [29].

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References

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

    MATH  Google Scholar 

  2. Alikakos, N.D., Fusco, G.: The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions. Indiana Univ. Math. J. 42(2), 637–674 (1993)

    MATH  Google Scholar 

  3. Allen, S., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1084–1095 (1979)

    Google Scholar 

  4. Barrett, J.W., Blowey, J.F.: An error bound for the finite element approximation of a model for phase separation of a multi-component alloy. IMA J. Numer. Anal. 16(2), 257–287 (1996)

    MATH  Google Scholar 

  5. Barrett, J.W., Blowey, J.F.: Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy. Numer. Math. 77(1), 1–34 (1997)

    Article  MATH  Google Scholar 

  6. Barrett, J.W., Blowey, J.F., Garcke H.: On fully practical finite element approximations of degenerate Cahn-Hilliard systems. M2AN Math. Model. Numer. Anal. 35(4), 713–748 (2001)

    Google Scholar 

  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)

    MATH  Google Scholar 

  8. Blowey, J.F., Elliott, C.M.: The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy, II. Numer. Anal. Eur. J. Appl. Math. 3(2), 147–179 (1992)

    MATH  Google Scholar 

  9. Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Meth. Appl. Sci. 2(4), 556–581 (1980)

    MATH  Google Scholar 

  10. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. Springer-Verlag, New York, 1994

  11. Cahn, J.W., Elliott, 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)

    MATH  Google Scholar 

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

    Google Scholar 

  13. Cahn, J.W., Novick-Cohen, A.: Limiting motion for an Allen-Cahn/Cahn-Hilliard system. In: Free boundary problems theory and applications (Zakopane 1995), Longman, Harlow, pp. 89–97, 1996,

  14. Carstensen, C.: Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomée criterion for H1-stability of the L2-projection onto finite element spaces. Math. Comp. 71, pp. 157–163 (2002)

  15. Chen, X.: Spectrum for the Allen-Cahn Cahn-Hilliard and phase-field equations for generic interfaces. Comm. Partial Diff. Eqs. 19(7–8), 1371–1395 (1994)

    Google Scholar 

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

    MATH  Google Scholar 

  17. Chen, X., Elliott, C.M., Gardiner A., Zhao J.J.: Convergence of numerical solutions to the Allen-Cahn equation. Appl. Anal. 69(1), 47–56 (1998)

    MATH  Google Scholar 

  18. Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland Publishing Co. Amsterdam, 1978, Studies in Mathematics and its Applications Vol. 4

  19. Copetti, M.I.M., Elliott, C.M.: Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer. Math. 63(1), 39–65 (1992)

    MATH  Google Scholar 

  20. de Mottoni, P., Schatzman, M.: Geometrical evolution of developed interfaces. Trans. Am. Math. Soc. 347(5), 1533–1589 (1995)

    MATH  Google Scholar 

  21. Du, Q., Nicolaides, R.A.: Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal. 28(5) 1310–1322 (1991)

    Google Scholar 

  22. Dupont, T.: Some L2 error estimates for parabolic Galerkin methods. The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos. Univ. Maryland Baltimore Md. 1972), Academic Press, New York, 491–504 (1972)

  23. 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)

    MATH  Google Scholar 

  24. 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)

    MATH  Google Scholar 

  25. 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)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  27. Feng, X., Prohl, A.: Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem Part I: error analysis under minimum regularities. 2001, IMA Technical Report #1798 downloadable at http://www.ima.umn.edu/preprints/jul01/jul01.html

  28. Feng, X., Prohl, A.: Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 3, 35–65 (2003)

    Google Scholar 

  29. Feng, X., Prohl, A.: Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem Part II: error analysis and convergence of the interface. Interfaces and Free Boundaries (submitted), 2003, IMA Technical Report #1799 downloadable at http://www.ima.umn.edu/preprints/jul01/jul01.html

  30. Feng, X., Prohl, A.: Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comp. 73, 541–567 (2004)

    Article  MATH  Google Scholar 

  31. French, D.A., Jensen, S.: Long-time behaviour of arbitrary order continuous time G, alerkin schemes for some one-dimensional phase transition problems. IMA J. Numer. Anal. 14(3), 421–442 (1994)

    MATH  Google Scholar 

  32. Friedman, A.: Variational principles and free-boundary problems. Robert E. Krieger Publishing Co. Inc. Malabar FL, second edition, 1988

  33. Mullins, W.W., Sekerka, J.: Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Math. 34, 322–329 (1963)

    Google Scholar 

  34. Novick-Cohen, A.: The Cahn-Hilliard equation: mathematical and modeling perspectives. Adv. Math. Sci. Appl. 8(2), 965–985 (1998)

    MATH  Google Scholar 

  35. Novick-Cohen, A.: Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system. Phys. D 137(1–2), 1–24 (2000)

    Google Scholar 

  36. Pego, R.L.: Front migration in the nonlinear Cahn-Hilliard equation. Proc. Roy. Soc. London Ser. A 422(1863), 261–278 (1989)

    MATH  Google Scholar 

  37. Prohl, A.: Projection and Quasi-Compressibility Methods for Solving the incompressible Navier-Stokes equations. Advances in Numerical Mathematics, B.G. Teubner-Verlag, Stuttgart, 1997

  38. Scholz R.: A mixed method for 4th order problems using linear finite elements. RAIRO Anal. Numér. 12(1), 85–90 (1978)

    Google Scholar 

  39. Stoth, B.: Convergence of the Cahn-Hilliard equation to the Mullins-Sekerka problem in spherical symmetry. J. Diff. Eqs. 125(1), 154–183 (1996)

    MATH  Google Scholar 

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Authors and Affiliations

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

    Xiaobing Feng

  2. Department of Mathematics, ETH Zurich, 8092, Zurich, Switzerland

    Andreas Prohl

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  1. Xiaobing Feng
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  2. Andreas Prohl
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Correspondence to Andreas Prohl.

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Mathematics Subject Classification (1991): 65M60, 65M12, 65M15, 35B25, 35K57, 35Q99, 53A10

Acknowledgments. The first author would like to thank Nicholas Alikakos for explaining all the fascinating properties of the Allen-Cahn and Cahn-Hilliard equations to him. He would also like to thank Nicholas Alikakos and Xinfu Chen for answering his questions regarding the spectrum estimate in Proposition 1. The second author gratefully acknowledges financial support by the DFG.

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Feng, X., Prohl, A. Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99, 47–84 (2004). https://doi.org/10.1007/s00211-004-0546-5

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