A posteriori estimates for the Cahn-Hilliard equation with obstacle free energy
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 5, p. 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 : https://doi.org/10.1051/m2an/2009015
Classification:  65M60,  65M15,  65M50,  35K55
Keywords: Cahn-Hilliard equation, obstacle free energy, linear finite elements, a posteriori estimates, adaptive numerical methods
@article{M2AN_2009__43_5_1003_0,
     author = {Ba\v nas, \v Lubom\'\i r and N\"urnberg, Robert},
     title = {A posteriori estimates for the Cahn-Hilliard equation with obstacle free energy},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {5},
     year = {2009},
     pages = {1003-1026},
     doi = {10.1051/m2an/2009015},
     zbl = {pre05608360},
     mrnumber = {2559742},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2009__43_5_1003_0}
}
Baňas, Ľubomír; Nürnberg, Robert. A posteriori estimates for the Cahn-Hilliard equation with obstacle free energy. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 5, pp. 1003-1026. doi : 10.1051/m2an/2009015. http://www.numdam.org/item/M2AN_2009__43_5_1003_0/

[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 1308851 | Zbl 0828.35105

[2] Ľ. Baňas and R. Nürnberg, Adaptive finite element methods for Cahn-Hilliard equations. J. Comput. Appl. Math. 218 (2008) 2-11. | MR 2431593 | Zbl 1143.65076

[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 2453219

[4] Ľ. Baňas and R. Nürnberg, Phase field computations for surface diffusion and void electromigration in 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 1464653 | Zbl 0882.65129

[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 1742748 | Zbl 0947.65109

[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 2084234 | Zbl 1076.78012

[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 1123143 | Zbl 0797.35172

[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 1166255 | Zbl 0810.35158

[10] D. Braess, A posteriori error estimators for obstacle problems - another look. Numer. Math. 101 (2005) 415-421. | MR 2194822 | Zbl 1118.65068

[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 1284813 | Zbl 0811.35098

[14] Z. Chen and R.H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84 (2000) 527-548. | MR 1742264 | Zbl 0943.65075

[15] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77-84. | Numdam | Zbl 0368.65008

[16] C.M. Elliott and Z. Songmu, On the Cahn-Hilliard equation. Arch. Rational Mech. Anal. 96 (1986) 339-357. | MR 855754 | Zbl 0624.35048

[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 978609 | Zbl 0668.65097

[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 2101784 | Zbl 1071.65128

[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 2464735 | Zbl 1174.65035

[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 2421327 | Zbl 1167.49029

[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 2434065 | Zbl 1157.65039

[22] J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys. 193 (2004) 511-543. | MR 2030475 | Zbl 1109.76348

[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 921549 | Zbl 0642.49009

[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 2355709 | Zbl 1142.65053

[25] R.H. Nochetto and L.B. Wahlbin, Positivity preserving finite element approximation. Math. Comp. 71 (2002) 1405-1419. | MR 1933037 | Zbl 1001.41011

[26] R.L. Pego, Front migration in the nonlinear Cahn-Hilliard equation. Proc. Roy. Soc. London Ser. A 422 (1989) 261-278. | MR 997638 | Zbl 0701.35159

[27] A. Veeser, Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39 (2001) 146-167. | MR 1860720 | Zbl 0992.65073

[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 1886715 | Zbl 0988.65047

[29] R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Teubner-Wiley, New York (1996). | Zbl 0853.65108