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Structure Preserving Discretization of Allen–Cahn Type Problems Modeling the Motion of Phase Boundaries

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Abstract

We study the systematic numerical approximation of a class of Allen–Cahn type problems modeling the motion of phase interfaces. The common feature of these models is an underlying gradient flow structure which gives rise to a decay of an associated energy functional along solution trajectories. We first study the discretization in space by a conforming Galerkin approximation of a variational principle which characterizes smooth solutions of the problem. Well-posedness of the resulting semi-discretization is established and the energy decay along discrete solution trajectories is proven. A problem adapted implicit time-stepping scheme is then proposed and we establish its well-posed and decay of the free energy for the fully discrete scheme. Some details about the numerical realization by finite elements are discussed, in particular the iterative solution of the nonlinear problems arising in every time-step. The theoretical results are illustrated by numerical tests which also provide further evidence for asymptotic expansions of the interface velocities derived by Alber et al. and support the observation that their hybrid Allen–Cahn model avoids the problem of mesh-locking to a large extent.

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References

  1. Alber, H. -D.: Asymptotics and numerical efficiency of the Allen–Cahn model for phase interfaces with low energy in solids. arXiv:1505.05442 (2015)

  2. Alber, H.-D., Zhu, P.: Comparison of a rapidely converging phase field model for interfaces in solids with the Allen–Cahn model. J. Elast. 111, 153–221 (2013)

    Article  MathSciNet  Google Scholar 

  3. Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1085–1095 (1979)

    Article  Google Scholar 

  4. Amann, H.: Linear and Quasilinear Parabolic Problems. Vol. I. Monographs in Mathematics, vol. 89. Birkhäuser, Boston (1995)

    Book  Google Scholar 

  5. Bartels, S.: A posteriori error analysis for time-dependent Ginzburg-Landau type equations. Numer. Math. 99, 557–583 (2005)

    Article  MathSciNet  Google Scholar 

  6. Blowey, J.F., Elliott, C.M.: Curvature dependent phase boundary motion and parabolic double obstacle problems. In: Ni, W.-M., Peletier, L.A., Vazquez, J.L. (eds.) Degenerate Diffusions. The IMA Volumes in Mathematics and Its Applications, vol. 47, pp 19–60. Springer, New York (1993)

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

    MathSciNet  MATH  Google Scholar 

  8. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Publishing Co., Amsterdam–New York–Oxford (1978)

    MATH  Google Scholar 

  9. Clarenz, U., Haußer, F., Rumpf, M., Voigt, A., Weikard, U.: On level set formulations for anisotropic mean curvature flow and surface diffusion. In: Voigt, A. (ed.) Multiscale Modeling in Epitaxial Growth. ISNM International Series of Numerical Mathematics, vol. 149, pp 227–237. Birkhäuser, Basel (2005)

  10. De Giorgi, E.: New problems in Γ-convergence and G-convergence. In: Free Boundary Problems, Vol. II (Pavia, 1979), pp 183–194. Ist. Naz. Alta Mat. Francesco Severi, Rome (1980)

  11. Deckelnick, K., Dziuk, G.: Convergence of a finite element method for non-parametric mean curvature flow. Numer. Math. 72, 197–222 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin–Heidelberg (1985)

    Book  Google Scholar 

  14. Droske, M., Rumpf, M.: A level set formulation for Willmore flow. Interfaces Free Bound. 6, 361–378 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Egger, H., Leitão, A.: Nonlinear regularization methods for ill-posed problems with piecewise constant or strongly varying solutions. Inverse Probl. 25, 115014 (2009)

    Article  MathSciNet  Google Scholar 

  17. Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)

    Book  Google Scholar 

  18. Eshelby, J.D.: The elastic field outside an ellipsoidal inclusion. Proc. R. Soc. Ser. A 252, 561–569 (1959)

    MathSciNet  MATH  Google Scholar 

  19. Eshelby, J.D.: Elastic inclusions and inhomogeneities. In: Sneddon, I. N., Hill, R (eds.) Progress in Solid Mechanics, vol. II, pp 87–140. North-Holland, Amsterdam (1961)

  20. Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45, 1097–1123 (1992)

    Article  MathSciNet  Google Scholar 

  21. Feng, X., Prohl, A.: Numerical analysis of the Allen–Cahn equation and approximation for mean curvature flows. Numer. Math. 94, 33–65 (2003)

    Article  MathSciNet  Google Scholar 

  22. Feng, X., Song, H., Tang, T., Yang, J.: Nonlinear stability of the implicit-explicit methods for the Allen–Cahn equation. Inverse Probl. Imaging 7, 679–695 (2013)

    Article  MathSciNet  Google Scholar 

  23. Feng, X., Tang, T., Yang, J.: Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models. East Asian J. Appl. Math. 3, 59–80 (2013)

    Article  MathSciNet  Google Scholar 

  24. Feng, X., Wu, H.-j.: A posteriori error estimates and an adaptive finite element method for the Allen–Cahn equation and the mean curvature flow. J. Sci. Comput. 24, 121–146 (2005)

    Article  MathSciNet  Google Scholar 

  25. Finel, A., Le Bouar, Y., Dabas, B., Appolaire, B., Yamada, Y., Mohri, T.: Sharp phase field method. Phys. Rev. Lett. 121, 025501 (2018)

    Article  Google Scholar 

  26. Fried, E., Gurtin, M.E.: Dynamic solid-solid transitions with phase characterized by an order parameter. Phys. D 72, 287–308 (1994)

    Article  MathSciNet  Google Scholar 

  27. Garcke, H.: On Mathematical Models for Phase Separation in Elastically Stressed Solids. University Bonn, Habilitation thesis (2000)

  28. Garcke, H.: Curvature driven interface evolution. Jahresber. Dtsch. Math. Ver. 115, 63–100 (2013)

    Article  MathSciNet  Google Scholar 

  29. Grayson, M.A.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom. 26, 285–314 (1987)

    Article  MathSciNet  Google Scholar 

  30. Ilmanen, T.: Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38, 417–461 (1993)

    Article  MathSciNet  Google Scholar 

  31. Jian, H.Y.: A relation between Γ-convergence of functionals and their associated gradient flows. Sci. China Ser. A-Math. 42, 133–139 (1999)

    Article  MathSciNet  Google Scholar 

  32. Kessler, D., Nochetto, R.H., Schmidt, A.: A posteriori error control for the Allen–Cahn problem: circumventing Gronwall’s inequality. ESAIM Math. Model. Numer. Anal. 38, 129–142 (2004)

    Article  MathSciNet  Google Scholar 

  33. Ladyženskaja, O. A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type. Translations of Mathematical Monographs. Vol. 23 (Transl. from the Russian by S. Smith). American Mathematical Society, Providence (1968)

    Book  Google Scholar 

  34. Leray, J., Schauder, J.: Topologie et équations fonctionnelles. Ann. Sci. l’É.N.S. 51, 45–78 (1934)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  36. Osher, S., Paragios, N.: Geometric Level Set Methods in Imaging, Vision and Graphics. Springer, New York (2003)

    Book  Google Scholar 

  37. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Department of Mathematics, University of Maryland, College Park, Md. (1974)

  38. Rubinstein, J., Sternberg, P., Keller, J.B.: Fast reaction, slow diffusion, and curve shortening. SIAM J. Appl. Math. 49, 116–133 (1989)

    Article  MathSciNet  Google Scholar 

  39. Shen, J., Tang, T., Yang, J.: On the maximum principle preserving schemes for the generalized Allen–Cahn equation. Commun. Math. Sci. 14, 1517–1534 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  41. Visintin, A.: Models of Phase Transitions. Progress in Nonlinear Differential Equations and Their Applications, vol. 28. Birkhäuser, Boston (1996)

    Google Scholar 

  42. Walkington, N.J.: Algorithms for computing motion by mean curvature. SIAM J. Numer. Anal. 33, 2215–2238 (1996)

    Article  MathSciNet  Google Scholar 

  43. Zhang, J., Du, Q.: Numerical studies of discrete approximations to the Allen–Cahn equation in the sharp interface limit. SIAM J. Sci. Comput. 31, 3042–3063 (2009)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the German Research Foundation (DFG) via grants IRTG 1529, TRR 146 projection C3, TRR 154 project C4, and Eg-331/1-1 and by the German Excellence Initiative via grant GSC 233.

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  1. Department of Mathematics, TU Darmstadt, Darmstadt, Germany

    Anke Böttcher & Herbert Egger

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  1. Anke Böttcher
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  2. Herbert Egger
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Correspondence to Herbert Egger.

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Böttcher, A., Egger, H. Structure Preserving Discretization of Allen–Cahn Type Problems Modeling the Motion of Phase Boundaries. Vietnam J. Math. 48, 847–863 (2020). https://doi.org/10.1007/s10013-020-00428-w

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