Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem

Interfaces and Free Boundaries

Volumn 7, Issue 1, 2005, Pages 1-28

  Feng, Xiaobing ^a   Prohl, Andreas ^b  

^a UNIVERSITY OF TENNESSEE   (United States)

^b ETH ZURICH   (Switzerland)

Author keywords

Biharmonic problem; Cahn Hilliard equation; Ciarlet Raviart element; Fully discrete mixed finite element method; Hele Shaw (Mullins Sekerka) problem; Phase transition

Indexed keywords

EID: 15544366700     PISSN: 14639963     EISSN: None     Source Type: Journal    
DOI: 10.4171/IFB/111     Document Type: Article

References (34)

1. ADAMS, R. A. Sobolev Spaces. Academic Press, New York, 1975. Zbl 0314.46030 MR 0450957
2. ALIKAKOS, N. D., BATES, P. W., & CHEN, X. Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Rat. Mech. Anal. 128 (1994), 165-205. Zbl 0828.35105 MR 1308851
3. ALIKAKOS, N. D. & FUSCO, G. The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions. Indiana Univ. Math. J. 42 (1993), 637-674. Zbl 0798.35123 MR 1237062
4. ALLEN, S. & CAHN, J.W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979), 1084-1095.
5. 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 (1996), 257-287. Zbl 0849.65069 MR 1382718
6. 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 (1997), 1-34. Zbl 0882.65129 MR 1464653
7. 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 (2001), 713-748. Zbl 0987.35071 MR 1863277
8. BATES, P. W. & FIFE, P. C. The dynamics of nucleation for the Cahn-Hilliard equation. SIAM J. Appl. Math. 53 (1993), 990-1008. Zbl 0788.35061 MR 1232163
9. BELLETTINI, G. & PAOLINI, M. Quasi-optimal error estimates for the mean curvature flow with a forcing term. Differential Integral Equations 8 (1995), 735-752. Zbl 0820.49019 MR 1306590
10. BLUM, H. & RANNACHER, R. On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2 (1980), 556-581. Zbl 0445.35023 MR 0595625
11. BRENNER, S. C. & SCOTT, L. R. The Mathematical Theory of Finite Element Methods. Springer, New York (1994). Zbl 0804.65101 MR 1278258
12. CAHN, J. W. & HILLIARD, J. E. Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 28 (1958), 258-267.
13. 2-projection onto finite element spaces. Math. Comp. 71 (2002), 157-163. Zbl 0989.65123 MR 1862993
14. CHEN, X. Generation and propagation of interfaces for reaction-diffusion equations. J. Differential Equations 96 (1992), 116-141. Zbl 0765.35024 MR 1153311
15. CHEN, X. Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differential Equations 19 (1994), 1371-1395. Zbl 0811.35098 MR 1284813
16. CHEN, X. Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differential Geom. 44 (1996), 262-311. Zbl 0874.35045 MR 1425577
17. CIARLET, P. G. The Finite Element Method for Elliptic Problems. Stud. Math. Appl. 4, North-Holland, Amsterdam (1978). Zbl 0383.65058 MR 0520174
18. COPETTI, M. I. M. & ELLIOTT, C. M. Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy. Numer. Math. 63 (1992), 39-65. Zbl 0762.65074 MR 1182511
19. DU, Q. & NICOLAIDES, R. A. Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal. 28 (1991), 1310-1322. Zbl 0744.65089 MR 1119272
20. 2 error estimates for parabolic Galerkin methods. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Baltimore, MD 1972), Academic Press, New York (1972), 491-504. Zbl 0279.65086 MR 0403255
21. ELLIOTT, C. M. & FRENCH, D. A. Numerical studies of the Cahn-Hilliard equation for phase separation. IMA J. Appl. Math. 38 (1987), 97-128. Zbl 0632.65113 MR 0983721
22. ELLIOTT, C. M. & FRENCH, D. A. A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 26 (1989), 884-903. Zbl 0686.65086 MR 1005515
23. ELLIOTT, C. M., FRENCH, D. A., & MILNER, F. A. A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54 (1989), 575-590. Zbl 0668.65097 MR 0978609
24. ELLIOTT, C. M. & SONOMU, Z. On the Cahn-Hilliard equation. Arch. Rat. Mech. Anal. 96 (1986), 339-357. Zbl 0624.35048 MR 0855754
25. EVANS, L. C., SONER, H. M., & SOUGANIDIS, P. E. Phase transitions and generalized motion by mean curvature. Comm. Pure Appl. Math. 45 (1992), 1097-1123. Zbl 0801.35045 MR 1177477
26. 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. IMA Technical Report #1799, Univ. Minnesota (2001).
27. 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 (2003), 541-567. Zbl pre02041046 MR 2028419
28. FENG, X. & PROHL, A. Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows. Numer. Math. 94 (2003), 33-65. Zbl 1029.65093 MR 1971212
29. FENG, X. & PROHL, A. Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99 (2004), 47-84. MR 2101784
30. FRENCH, D. A. & JENSEN, S. Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems. IMA J. Numer. Anal. 14 (1994), 421-442. Zbl 0806.65132 MR 1283945
31. NOCHETTO, R. H., PAOLINI, M., & VERDI, C. Optimal interface error estimates for the mean curvature flow. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), 193-212. Zbl 0886.35079 MR 1288364
32. NOCHETTO, R. H. & VERDI, C. Convergence past singularities for a fully discrete approximation of curvature-driven interfaces. SIAM J. Numer. Anal. 34 (1997), 490-512. Zbl 0876.35053 MR 1442924
33. SCHOLZ, R. A mixed method for 4th order problems using linear finite elements. RAIRO Anal. Numér. 12(1978), 85-90. Zbl 0382.65059 MR 0483557
34. 2 error estimates for Galerkin approximations to parabolic partial differential equations. SIAM J. Numer. Anal. 10 (1973), 723-759. Zbl 0258.35041 MR 0351124