A posteriori estimates for the Cahn-Hilliard equation with obstacle free energy
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 5, pp. 1003-1026.

We derive a posteriori estimates for a discretization in space of the standard Cahn-Hilliard equation with a double obstacle free energy. The derived estimates are robust and efficient, and in practice are combined with a heuristic time step adaptation. We present numerical experiments in two and three space dimensions and compare our method with an existing heuristic spatial mesh adaptation algorithm.

DOI : 10.1051/m2an/2009015
Classification : 65M60, 65M15, 65M50, 35K55
Mots-clés : Cahn-Hilliard equation, obstacle free energy, linear finite elements, a posteriori estimates, adaptive numerical methods 
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     title = {A posteriori estimates for the {Cahn-Hilliard} equation with obstacle free energy},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
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Baňas, Ľubomír; Nürnberg, Robert. A posteriori estimates for the Cahn-Hilliard equation with obstacle free energy. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 5, pp. 1003-1026. doi : 10.1051/m2an/2009015. http://www.numdam.org/articles/10.1051/m2an/2009015/

[1] N.D. Alikakos, P.W. Bates and X.F. Chen, The convergence of solutions of the Cahn-Hilliard equation to the solution of the Hele-Shaw model. Arch. Rational Mech. Anal. 128 (1994) 165-205. | MR | Zbl

[2] Ľ. Baňas and R. Nürnberg, Adaptive finite element methods for Cahn-Hilliard equations. J. Comput. Appl. Math. 218 (2008) 2-11. | MR | Zbl

[3] Ľ. Baňas and R. Nürnberg, Finite element approximation of a three dimensional phase field model for void electromigration. J. Sci. Comp. 37 (2008) 202-232. | MR

[4] Ľ. Baňas and R. Nürnberg, Phase field computations for surface diffusion and void electromigration in ℝ 3 . Comput. Vis. Sci. (2008), doi: 10.1007/s00791-008-0114-0.

[5] J.W. Barrett and J.F. Blowey, 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. | MR | Zbl

[6] J.W. Barrett, J.F. Blowey and H. Garcke, Finite element approximation of the Cahn-Hilliard equation with degenerate mobility. SIAM J. Numer. Anal. 37 (1999) 286-318. | MR | Zbl

[7] J.W. Barrett, R. Nürnberg and V. Styles, Finite element approximation of a phase field model for void electromigration. SIAM J. Numer. Anal. 42 (2004) 738-772. | MR | Zbl

[8] J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part I: Mathematical analysis. European J. Appl. Math. 2 (1991) 233-279. | MR | Zbl

[9] J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part II: Numerical analysis. European J. Appl. Math. 3 (1992) 147-179. | MR | Zbl

[10] D. Braess, A posteriori error estimators for obstacle problems - another look. Numer. Math. 101 (2005) 415-421. | MR | Zbl

[11] J.W. Cahn, On spinodal decomposition. Acta Metall. 9 (1961) 795-801.

[12] J.W. Cahn and J.E. Hilliard, Free energy of a non-uniform system. I. Interfacial free energy. J. Chem. Phys. 28 (1958) 258-267.

[13] X. Chen, Spectrum for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interfaces. Comm. Partial Differ. Equ. 19 (1994) 1371-1395. | MR | Zbl

[14] Z. Chen and R.H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84 (2000) 527-548. | MR | Zbl

[15] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77-84. | Numdam | Zbl

[16] C.M. Elliott and Z. Songmu, On the Cahn-Hilliard equation. Arch. Rational Mech. Anal. 96 (1986) 339-357. | MR | Zbl

[17] C.M. Elliott, D.A. French and F.A. Milner, A second order splitting method for the Cahn-Hilliard equation. Numer. Math. 54 (1989) 575-590. | MR | Zbl

[18] X. Feng and A. Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation. Numer. Math. 99 (2004) 47-84. | MR | Zbl

[19] X. Feng and H. Wu, A posteriori error estimates for finite element approximations of the Cahn-Hilliard equation and the Hele-Shaw flow. J. Comput. Math. 26 (2008) 767-796. | MR | Zbl

[20] M. Hintermüller and R.H.W. Hoppe, Goal-oriented adaptivity in control constrained optimal control of partial differential equations. SIAM J. Control Optim. 47 (2008) 1721-1743. | MR | Zbl

[21] M. Hintermüller, R.H.W. Hoppe, Y. Iliash and M. Kieweg, An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM: COCV 14 (2008) 540-560. | Numdam | MR | Zbl

[22] J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys. 193 (2004) 511-543. | MR | Zbl

[23] L. Modica, Gradient theory of phase transitions with boundary contact energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987) 487-512. | Numdam | MR | Zbl

[24] K.-S. Moon, R.H. Nochetto, T. Von Petersdorff and C.-S. Zhang, A posteriori error analysis for parabolic variational inequalities. ESAIM: M2AN 41 (2007) 485-511. | Numdam | MR | Zbl

[25] R.H. Nochetto and L.B. Wahlbin, Positivity preserving finite element approximation. Math. Comp. 71 (2002) 1405-1419. | MR | Zbl

[26] R.L. Pego, Front migration in the nonlinear Cahn-Hilliard equation. Proc. Roy. Soc. London Ser. A 422 (1989) 261-278. | MR | Zbl

[27] A. Veeser, Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39 (2001) 146-167. | MR | Zbl

[28] A. Veeser, On a posteriori error estimation for constant obstacle problems, in Numerical methods for viscosity solutions and applications (Heraklion, 1999), M. Falcone and C. Makridakis Eds., Ser. Adv. Math. Appl. Sci. 59, World Sci. Publ., River Edge, USA (2001) 221-234. | MR | Zbl

[29] R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley, New York (1996). | Zbl

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