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

Numerische Mathematik

Volumn 99, Issue 1, 2004, Pages 47-84

  Feng, Xiaobing ^a   Prohl, Andreas ^b  

^a UNIVERSITY OF TENNESSEE   (United States)

^b ETH ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 11144340252     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-004-0546-5     Document Type: Article

References (39)

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)
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)
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)
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)
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)
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)
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)
8. Blowey, J.F., Elliott, C.M.: The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy, II. Numer. Anal. Fur. J. Appl. Math. 3(2), 147-179 (1992)
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)
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)
12. Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system I, Interfacial free energy. J. Chem. Phys. 28, 258-267 (1958)
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. 2-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)
16. Chen, X.: Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Diff. Geom. 44(2), 262-311 (1996)
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)
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)
20. de Mottoni, P, Schatzman, M.: Geometrical evolution of developed interfaces. Trans. Am. Math. Soc. 347(5), 1533-1589 (1995)
21. Du, Q., Nicolaides, R.A.: Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal. 28(5) 1310-1322 (1991)
22. 2 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)
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)
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)
26. Elliott, C.M., Songmu Z.: On the Cahn-Hilriard equation. Arch. Rational Mech. Anal. 96(4), 339-357 (1986)
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)
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)
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)
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)
34. Novick-Cohen, A.: The Cahn-Hilliard equation: mathematical and modeling perspectives. Adv. Math. Sci. Appl. 8(2), 965-985 (1998)
35. Novick-Cohen, A.: Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system. Phys. D 137(1-2), 1-24 (2000)
36. Pego, R.L.: Front migration in the nonlinear Cahn-Hilliard equation. Proc. Roy. Soc. London Ser. A 422(1863), 261-278 (1989)
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)
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)