Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits

Mathematics of Computation

Volumn 73, Issue 246, 2004, Pages 541-567

  Feng, Xiaobing ^a   Prohl, Andreas ^b  

^a UNIVERSITY OF TENNESSEE   (United States)

^b ETH ZURICH   (Switzerland)

Author keywords

Allen Cahn equation; Cahn Hilliard equation; Fully discrete finite element method; Hele Shaw model; Motion by mean curvature; Phase field model; Stefan problem

Indexed keywords

EID: 1642633953     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-03-01588-6     Document Type: Article

References (38)

1. R. A. Adams. Sobolev Spaces. Academic Press, New York, 1975. MR 56:9247
2. N. D. Alikakos and P. W. Bates. On the singular limit in a phase field model of phase transitions. Ann. Inst. H. Poincaré Anal., Non Linéaire, 5(2):141-178, 1988. MR 89h:35109
3. N. D. Alikakos, P. W. Bates, and X. Chen. Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Rational Mech. Anal., 128(2):165-205, 1994. MR 97b:35174
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(1-2):175-194, 2000. MR 2000j:80004
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(I):1-31, 1997. MR 97m:80011
7. J. F. Blowey and C. M. Elliott. A phase-field model with a double obstacle potential. In G. Buttazzo and A. Visintin, editors, Motion by mean curvature and related topics (Trento, 1992), pages 1-22. de Gruyter, Berlin, 1994. MR 95d:35093
8. S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods. Springer-Verlag, New York, 1994. MR 95f:65001
9. G. Caginalp. An analysis of a phase field model of a free boundary. Arch, Rational Mech. Anal., 92(3):205-245, 1986. MR 87c:80011
10. G. Caginalp. Stefan and Hele-Shaw type models as asymptotic limits of the phase-field equations. Phys. Rev. A (3), 39(11):5887-5896, 1989. MR 90c:80004
11. G. Caginalp and X. Chen. Convergence of the phase field model to its sharp interface limits. European J. Appl. Math., 9(4):417-445, 1998. MR 991:80011
12. G. Caginalp and J.-T. Lin. A numerical analysis of an anisotropic phase field model. IMA J. Appl. Math., 39(1):51-66, 1987. MR 90e:80005
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(1):106-126, 1994. MR 94k:65122
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. X. Chen. Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differential Equations, 19(7-8):1371-1395, 1994. MR 95f:35023
16. X. Chen, J. Hong, and F. Yi. Existence, uniqueness, and regularity of classical solutions of the Mullins-Sekerka problem. Comm. Partial Differential Equations, 21(11-12):1705-1727, 1996. MR 97h:35238
17. Z. M. Chen and K.-H. Hoffmann. An error estimate for a finite-element scheme for a phase field model. IMA J. Numer. Anal., 14(2):243-255, 1994. MR 95c:65152
18. P. G. Ciarlet. The finite element method for elliptic problems. North-Holland Publishing Co., Amsterdam, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 58:25001
19. J. B. Collins and H. Levine. Diffuse interface model of diffusion-limited crystal growth. Phys. Rev. B, 31:6119-6122, 1985.
20. P. de Mottoni and M. Schatzman. Geometrical evolution of developed interfaces. Trans. Amer. Math. Soc., 347(5):1533-1589, 1995. MR 2000a:35022
21. C. M. Elliott and S. M. Zheng. Global existence and stability of solutions to the phase field equations. In Free Boundary Value Problems (Oberwolfach, 1989), pages 46-58, Birkhäuser, Basel, 1990. MR 92g:35214
22. L. C. Evans, H. M. Soner, and P. E. Souganidis. Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math., 45(9):1097-1123, 1992. MR 93g:35064
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.
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.
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.
27. P. Fife. Models for phase separation and their mathematics. Electronic J. of Diff. Eqns, 2000(48):1-26, 2000. MR 2001k:35147
28. G. Fix. Phase field method for free boundary problems. In A. Fasano and M. Primicerio, editors, Free Boundary Problems, pages 580-589. Pitman, London, 1983. MR 84h:00003
29. G. J. Fix and J. T. Lin. Numerical simulations of nonlinear phase transitions. I. The Isotropic case. Nonlinear Anal., 12(8):811-823, 1988. MR 89h:80009
30. J. S. Langer. Models of patten formation in first-order phase transitions. In Directions in Condensed Matter Physics, pages 164-186. World Science Publishers, 1986. MR 88a:82023
31. J. T. Lin. The numerical analysis of a phase field model in moving boundary problems. SIAM J. Numer. Anal., 25(5):1015-1031, 1988. MR 89h:65161
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.
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.
35. N. Provatas, N. Goldenfeld, and J. Dantzig. Adaptive mesh refinement computation of solidification microstructures using dynamic data structures. J. Comput. Phys., 148(1):265-290, 1999. MR 99h:80012
36. H. M. Soner. Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling. Arch. Rational Mech. Anal., 131(2):139-197, 1995. MR 97d:80007
37. B. E. E. Stoth. A sharp interface limit of the phase field equations: one-dimensional and axisymmetric. European J. Appl. Math., 7(6):603-633, 1996. MR 981:80008
38. X. Y. Yue. Finite element analysis of the phase field model with nonsmooth initial data. Acta Math. Appl. Sinica, 19(1):15-24, 1996. MR 97d:65045