Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. (English) Zbl 1115.76049

Feng, Xiaobing; Prohl, Andreas

Math. Comput. 73, No. 246, 541-567 (2004).

Summary: We propose and analyze a fully discrete finite element scheme for the phase field model describing the solidification process in materials science. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical method, in particular, by focusing on the dependence of the error bounds on the parameter \(\varepsilon\), known as the measure of the interface thickness. Optimal order error bounds are shown for the fully discrete scheme under some reasonable constraints on the mesh size \(h\) and the time step size \(k\). In particular, it is shown that all error bounds depend on \(\frac{1}{\varepsilon}\) only in some lower polynomial order for small \(\varepsilon\). The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Chen, and to establish a discrete counterpart of it for a linearized phase field operator to handle the nonlinear effect. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence of the solution of the fully discrete scheme to solutions of the sharp interface limits of the phase field model under different scaling in its coefficients. The sharp interface limits include the classical Stefan problem, the generalized Stefan problems with surface tension and surface kinetics, the motion by mean curvature flow, and the Hele-Shaw model.

MSC:

76M10	Finite element methods applied to problems in fluid mechanics
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M15	Error bounds for initial value and initial-boundary value problems involving PDEs
35B25	Singular perturbations in context of PDEs
35K57	Reaction-diffusion equations
35Q99	Partial differential equations of mathematical physics and other areas of application

References:

[1]	Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030
[2]	Nicholas D. Alikakos and Peter W. Bates, On the singular limit in a phase field model of phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1988), no. 2, 141 – 178 (English, with French summary). · Zbl 0696.35060
[3]	Nicholas D. Alikakos, Peter W. Bates, and Xinfu Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal. 128 (1994), no. 2, 165 – 205. · Zbl 0828.35105
[4]	S. Allen and J. W. Cahn. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall., 27:1084-1095, 1979.
[5]	D. M. Anderson, G. B. McFadden, and A. A. Wheeler, A phase-field model of solidification with convection, Phys. D 135 (2000), no. 1-2, 175 – 194. · Zbl 0951.35112
[6]	P. W. Bates, P. C. Fife, R. A. Gardner, and C. K. R. T. Jones, Phase field models for hypercooled solidification, Phys. D 104 (1997), no. 1, 1 – 31. · Zbl 0890.35161
[7]	J. F. Blowey and C. M. Elliott, A phase-field model with a double obstacle potential, Motion by mean curvature and related topics (Trento, 1992) de Gruyter, Berlin, 1994, pp. 1 – 22. · Zbl 0809.35168
[8]	Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. · Zbl 0804.65101
[9]	Gunduz Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal. 92 (1986), no. 3, 205 – 245. · Zbl 0608.35080
[10]	G. Caginalp, Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations, Phys. Rev. A (3) 39 (1989), no. 11, 5887 – 5896. · Zbl 1027.80505
[11]	Gunduz Caginalp and Xinfu Chen, Convergence of the phase field model to its sharp interface limits, European J. Appl. Math. 9 (1998), no. 4, 417 – 445. · Zbl 0930.35024
[12]	Gunduz Caginalp and Jian-Tong Lin, A numerical analysis of an anisotropic phase field model, IMA J. Appl. Math. 39 (1987), no. 1, 51 – 66. · Zbl 0663.35098
[13]	G. Caginalp and E. Socolovsky, Phase field computations of single-needle crystals, crystal growth, and motion by mean curvature, SIAM J. Sci. Comput. 15 (1994), no. 1, 106 – 126. · Zbl 0793.65099
[14]	J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys., 28:258-267, 1958.
[15]	Xinfu Chen, Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces, Comm. Partial Differential Equations 19 (1994), no. 7-8, 1371 – 1395. · Zbl 0811.35098
[16]	Xinfu Chen, Jiaxing Hong, and Fahuai Yi, Existence, uniqueness, and regularity of classical solutions of the Mullins-Sekerka problem, Comm. Partial Differential Equations 21 (1996), no. 11-12, 1705 – 1727. · Zbl 0884.35177
[17]	Zhi Ming Chen and K.-H. Hoffmann, An error estimate for a finite-element scheme for a phase field model, IMA J. Numer. Anal. 14 (1994), no. 2, 243 – 255. · Zbl 0801.65091
[18]	Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058
[19]	J. B. Collins and H. Levine. Diffuse interface model of diffusion-limited crystal growth. Phys. Rev. B, 31:6119-6122, 1985.
[20]	Piero de Mottoni and Michelle Schatzman, Geometrical evolution of developed interfaces, Trans. Amer. Math. Soc. 347 (1995), no. 5, 1533 – 1589 (English, with English and French summaries). · Zbl 0840.35010
[21]	C. M. Elliott and Song Mu Zheng, Global existence and stability of solutions to the phase field equations, Free boundary value problems (Oberwolfach, 1989) Internat. Ser. Numer. Math., vol. 95, Birkhäuser, Basel, 1990, pp. 46 – 58.
[22]	L. C. Evans, H. M. Soner, and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992), no. 9, 1097 – 1123. · Zbl 0801.35045
[23]	X. Feng and A. Prohl. Numerical analysis of the Allen-Cahn equation and approximation of the mean curvature flows. Numer. Math., 94(1):33-65, 2003. · Zbl 1029.65093
[24]	X. Feng and A. Prohl. Error Analysis of a Mixed Finite Element Method for the Cahn-Hilliard Equation, Numer. Math. (submitted), IMA-Preprint #1798, 2001. · Zbl 1071.65128
[25]	X. Feng and A. Prohl. Numerical Analysis of the Cahn-Hilliard Equation and Approximation for the Hele-Shaw problem, Interfaces and Free Boundaries (submitted), IMA-Preprint #1799, 2001.
[26]	X. Feng and A. Prohl. Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interphase limits. IMA-Preprint #1817, 2001. · Zbl 1115.76049
[27]	Paul C. Fife, Models for phase separation and their mathematics, Electron. J. Differential Equations (2000), No. 48, 26. · Zbl 0957.35062
[28]	Antonio Fasano and Mario Primicerio , Free boundary problems: theory and applications. Vol. I, II, Research Notes in Mathematics, vol. 78, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1983. · Zbl 0511.35088
[29]	G. J. Fix and J. T. Lin, Numerical simulations of nonlinear phase transitions. I. The isotropic case, Nonlinear Anal. 12 (1988), no. 8, 811 – 823. · Zbl 0659.65131
[30]	J. S. Langer, Models of pattern formation in first-order phase transitions, Directions in condensed matter physics, World Sci. Ser. Dir. Condensed Matter Phys., vol. 1, World Sci. Publishing, Singapore, 1986, pp. 165 – 186.
[31]	J. T. Lin, The numerical analysis of a phase field model in moving boundary problems, SIAM J. Numer. Anal. 25 (1988), no. 5, 1015 – 1031. · Zbl 0664.65114
[32]	G. B. McFadden, A. A. Wheeler, R. J. Braun, S. R. Coriell, and R. F. Sekerka. Phase-field models for anisotropic interfaces. Phys. Rev. E (3), 48(3):2016-2024, 1993. · Zbl 0791.35159
[33]	W. W. Mullins and J. Sekerka. Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Math., 34:322-329, 1963.
[34]	O. Penrose and P. C. Fife. On the relation between the standard phase-field model and a “thermodynamically consistent” phase-field model. Phys. D, 69(1-2):107-113, 1993. · Zbl 0799.76084
[35]	Nikolas Provatas, Nigel Goldenfeld, and Jonathan Dantzig, Adaptive mesh refinement computation of solidification microstructures using dynamic data structures, J. Comput. Phys. 148 (1999), no. 1, 265 – 290. · Zbl 0932.80003
[36]	H. Mete Soner, Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling, Arch. Rational Mech. Anal. 131 (1995), no. 2, 139 – 197. · Zbl 0829.73010
[37]	Barbara E. E. Stoth, A sharp interface limit of the phase field equations: one-dimensional and axisymmetric, European J. Appl. Math. 7 (1996), no. 6, 603 – 633. · Zbl 0876.35133
[38]	Xing Ye Yue, Finite element analysis of the phase field model with nonsmooth initial data, Acta Math. Appl. Sinica 19 (1996), no. 1, 15 – 24 (Chinese, with English and Chinese summaries).