March  2013, 8(1): 37-64. doi: 10.3934/nhm.2013.8.37

The degenerate and non-degenerate deep quench obstacle problem: A numerical comparison

1. 

Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom

2. 

Department of Mathematics, Technion-IIT, Haifa 32000

3. 

Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ

Received  March 2012 Revised  March 2013 Published  April 2013

The deep quench obstacle problem $$ {\rm{\bf{(DQ)}}} \begin{equation}\left\{ \begin{array}{l} \frac{\partial u}{\partial t}=\nabla \cdot M(u) \nabla w, \\ w + \epsilon^2 \triangle u + u \in \partial \Gamma(u), \end{array} \right. \end{equation}$$ for $(x,t) \in \Omega \times (0,T)$, models phase separation at low temperatures. In (DQ), $\epsilon>0,$ $\partial \Gamma(\cdot)$ is the sub-differential of the indicator function $I_{[-1,1]}(\cdot),$ and $u(x,t)$ should satisfy $\nu \cdot \nabla u=0$ on the ``free boundary'' where $u=\pm 1$. We shall assume that $u$ is sufficiently smooth to make these notions well-defined. The problem (DQ) corresponds to the zero temperature ``deep quench'' limit of the Cahn--Hilliard equation. We focus here on a degenerate variant of (DQ) in which $M(u)=1-u^2,$ as well as on a constant mobility non-degenerate variant in which $M(u)=1.$ Although historically more emphasis has been placed on models with non-degenerate mobilities, degenerate mobilities capture some of the underlying physics more accurately. In the present paper, a careful numerical study is undertaken, utilizing a variety of benchmarks as well as new upper bounds for coarsening, in order to clarify evolutionary properties and to explore the differences in the two variant models.
Citation: L’ubomír Baňas, Amy Novick-Cohen, Robert Nürnberg. The degenerate and non-degenerate deep quench obstacle problem: A numerical comparison. Networks & Heterogeneous Media, 2013, 8 (1) : 37-64. doi: 10.3934/nhm.2013.8.37
References:
[1]

N. D. Alikakos, P. W. Bates and X. F. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205. doi: 10.1007/BF00375025.  Google Scholar

[2]

N. D. Alikakos, G. Fusco and G. Karali, Motion of bubbles towards the boundary for the Cahn-Hilliard equation, European J. Appl. Math., 15 (2004), 103-124. doi: 10.1017/S0956792503005242.  Google Scholar

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C. H. Arns, M. A. Knackstedt, W. V. Pinczewski and K. R. Mecke, Euler-Poincaré characteristics of classes of disordered media, Phys. Rev. E, 63 (2001), 031112. doi: 10.1103/PhysRevE.63.031112.  Google Scholar

[4]

L'. Baňas and R. Nürnberg, Finite element approximation of a three dimensional phase field model for void electromigration, J. Sci. Comp., 37 (2008), 202-232. doi: 10.1007/s10915-008-9203-y.  Google Scholar

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L'. Baňas and R. Nürnberg, A multigrid method for the Cahn-Hilliard equation with obstacle potential, Appl. Math. Comput., 213 (2009), 290-303. doi: 10.1016/j.amc.2009.03.036.  Google Scholar

[6]

L'. Baňas and R. Nürnberg, Phase field computations for surface diffusion and void electromigration in $\mathbbR^3$, Comput. Vis. Sci., 12 (2009), 319-327. doi: 10.1007/s00791-008-0114-0.  Google Scholar

[7]

J. W. Barrett and J. F. Blowey, An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy, Numer. Math., 72 (1995), 1-20. doi: 10.1007/s002110050157.  Google Scholar

[8]

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. doi: 10.1137/S0036142997331669.  Google Scholar

[9]

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. doi: 10.1137/S0036142902413421.  Google Scholar

[10]

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), 233-280. doi: 10.1017/S095679250000053X.  Google Scholar

[11]

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. doi: 10.1017/S0956792500000759.  Google Scholar

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J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1.  Google Scholar

[13]

J. W. Cahn, C. M. Elliott and A. Novick-Cohen, The Cahn-Hilliard equation with a concentration dependent mobility: Motion by minus the Laplacian of the mean curvature, European J. Appl. Math., 7 (1996), 287-301. Google Scholar

[14]

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. Google Scholar

[15]

X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, J. Differential Geom., 44 (1996), 262-311.  Google Scholar

[16]

S. Conti, B. Niethammer and F. Otto, Coarsening rates in off-critical mixtures, SIAM J. Math. Anal., 37 (2006), 1732-1741. doi: 10.1137/040620059.  Google Scholar

[17]

M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65. doi: 10.1007/BF01385847.  Google Scholar

[18]

R. Dal Passo, L. Giacomelli and A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility, Interfaces Free Bound., 1 (1999), 199-226. doi: 10.4171/IFB/9.  Google Scholar

[19]

C. M. Elliott and D. A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38 (1987), 97-128. doi: 10.1093/imamat/38.2.97.  Google Scholar

[20]

C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423. doi: 10.1137/S0036141094267662.  Google Scholar

[21]

C. M. Elliott and S. Luckhaus, A generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy (1991),, IMA, ().   Google Scholar

[22]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.  Google Scholar

[23]

P. Fratzl and J. L. Lebowitz, Universality of scaled structure functions in quenched systems undergoing phase separation, Acta Metall., 37 (1989), 3245-3248. doi: 10.1016/0001-6160(89)90196-X.  Google Scholar

[24]

P. Fratzl, J. L. Lebowitz, O. Penrose and J. Amar, Scaling functions, self-similarity, and the morphology of phase-separating systems, Phys. Rev. B, 44 (1991), 4794-4811. doi: 10.1103/PhysRevB.44.4794.  Google Scholar

[25]

M. Gameiro, K. Mischaikow and T. Wanner, Evolution of pattern complexity in the Cahn-Hilliard theory of phase separation, Acta Mater., 53 (2005), 693-704. doi: 10.1016/j.actamat.2004.10.022.  Google Scholar

[26]

H. Garcke, B. Niethammer, M. Rumpf and U. Weikard, Transient coarsening behaviour in the Cahn-Hilliard model, Acta Mater., 51 (2003), 2823-2830. doi: 10.1016/S1359-6454(03)00087-9.  Google Scholar

[27]

J. D. Gunton, M. San Miguel and P. S. Sahni, The dynamics of first-order phase transitions, in "Phase Transitions and Critical Phenomena," 8, Academic Press, London (1983), 267-482.  Google Scholar

[28]

R. Hilfer, Review on scale dependent characterization of the microstructure of porous media, Transp. Porous Media, 46 (2002), 373-390. doi: 10.1023/A:1015014302642.  Google Scholar

[29]

J. E. Hilliard, Spinodal decomposition,, in, (): 497.   Google Scholar

[30]

D. J. Horntrop, Concentration effects in mesoscopic simulation of coarsening, Math. Comput. Simulation, 80 (2010), 1082-1088. doi: 10.1016/j.matcom.2009.10.002.  Google Scholar

[31]

J. M. Hyde, M. K. Miller, M. G. Hetherington, A. Cerezo, G. D. W. Smith and C. M. Elliott, Spinodal decomposition in Fe-Cr alloys: Experimental study at the atomic level and comparison with computer models-II. Development of domain size and composition amplitude, Acta Metall. Mater., 43 (1995), 3403-3413. doi: 10.1016/0956-7151(95)00041-S.  Google Scholar

[32]

J. M. Hyde, M. K. Miller, M. G. Hetherington, A. Cerezo, G. D. W. Smith and C. M. Elliott, Spinodal decomposition in Fe-Cr alloys: Experimental study at the atomic level and comparison with computer models-III. Development of morphology, Acta Metall. Mater., 43 (1995), 3415-3426. doi: 10.1016/0956-7151(95)00042-T.  Google Scholar

[33]

T. Izumitani, M. Takenaka and T. Hashimoto, Slow spinodal decomposition in binary liquid mixtures of polymers. III. Scaling analyses of later-stage unmixing, J. Chem. Phys., 92 (1990), 3213-3221. doi: 10.1063/1.457871.  Google Scholar

[34]

T. Kaczynski, K. Mischaikow and M. Mrozek, "Computational Homology," 157 of Applied Mathematical Sciences, Springer-Verlag, New York, 2004.  Google Scholar

[35]

W. Kalies and P. Pilarczyk, CHomP software,, Available from , ().   Google Scholar

[36]

R. V. Kohn and F. Otto, Upper bounds on coarsening rates, Comm. Math. Phys., 229 (2002), 375-395. doi: 10.1007/s00220-002-0693-4.  Google Scholar

[37]

T. Y. Kong and A. Rosenfeld, Digital topology: Introduction and survey, Comput. Vision Graph. Image Process., 48 (1989), 357-393. Google Scholar

[38]

P. Leßle, M. Dong and S. Schmauder, Self-consistent matricity model to simulate the mechanical behaviour of interpenetrating microstructures, Comput. Mater. Sci., 15 (1999), 455-465. Google Scholar

[39]

P. Leßle, M. Dong, E. Soppa and S. Schmauder, Simulation of interpenetrating microstructures by self consistent matricity models, Scripta Mater., 38 (1998), 1327-1332. Google Scholar

[40]

S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics, Arch. Ration. Mech. Anal., 151 (2000), 187-219. doi: 10.1007/s002050050196.  Google Scholar

[41]

A. Novick-Cohen, Upper bounds for coarsening for the deep quench obstacle problem, J. Stat. Phys., 141 (2010), 142-157. doi: 10.1007/s10955-010-0040-7.  Google Scholar

[42]

A. Novick-Cohen, "The Cahn-Hilliard Equation: From Backwards Diffusion to Surface Diffusion," Cambridge Univ. Press, Cambridge, 2013. (in preparation). Google Scholar

[43]

A. Novick-Cohen and A. Shishkov, Upper bounds for coarsening for the degenerate Cahn-Hilliard equation, Discrete Contin. Dyn. Syst., 25 (2009), 251-272. doi: 10.3934/dcds.2009.25.251.  Google Scholar

[44]

Y. Oono and S. Puri, Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling, Phys. Rev. A, 38 (1988), 434-453. doi: 10.1103/PhysRevA.38.434.  Google Scholar

[45]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278. doi: 10.1098/rspa.1989.0027.  Google Scholar

[46]

E. Sander and T. Wanner, Unexpectedly linear behavior for the Cahn-Hilliard equation, SIAM J. Appl. Math., 60 (2000), 2182-2202. doi: 10.1137/S0036139999352225.  Google Scholar

[47]

A. Schmidt and K. G. Siebert, "Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA," 42 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 2005.  Google Scholar

[48]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. doi: 10.1007/BF00253122.  Google Scholar

[49]

T. Sullivan and P. Palffy-Muhoray, The effects of pattern morphology on late time scaling in the Cahn-Hilliard model, Abstract, American Physical Society, APS March Meeting, (2007). Google Scholar

[50]

R. Toral, A. Chakrabarti and J. D. Gunton, Large scale simulations of the two-dimensional Cahn-Hilliard model, Physica A, 213 (1995), 41-49. doi: 10.1016/0378-4371(94)00146-K.  Google Scholar

[51]

T. Ujihara and K. Osamura, Kinetic analysis of spinodal decomposition process in Fe-Cr alloys by small angle neutron scattering, Acta Mater., 48 (2000), 1629-1637. doi: 10.1016/S1359-6454(99)00441-3.  Google Scholar

show all references

References:
[1]

N. D. Alikakos, P. W. Bates and X. F. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model, Arch. Rational Mech. Anal., 128 (1994), 165-205. doi: 10.1007/BF00375025.  Google Scholar

[2]

N. D. Alikakos, G. Fusco and G. Karali, Motion of bubbles towards the boundary for the Cahn-Hilliard equation, European J. Appl. Math., 15 (2004), 103-124. doi: 10.1017/S0956792503005242.  Google Scholar

[3]

C. H. Arns, M. A. Knackstedt, W. V. Pinczewski and K. R. Mecke, Euler-Poincaré characteristics of classes of disordered media, Phys. Rev. E, 63 (2001), 031112. doi: 10.1103/PhysRevE.63.031112.  Google Scholar

[4]

L'. Baňas and R. Nürnberg, Finite element approximation of a three dimensional phase field model for void electromigration, J. Sci. Comp., 37 (2008), 202-232. doi: 10.1007/s10915-008-9203-y.  Google Scholar

[5]

L'. Baňas and R. Nürnberg, A multigrid method for the Cahn-Hilliard equation with obstacle potential, Appl. Math. Comput., 213 (2009), 290-303. doi: 10.1016/j.amc.2009.03.036.  Google Scholar

[6]

L'. Baňas and R. Nürnberg, Phase field computations for surface diffusion and void electromigration in $\mathbbR^3$, Comput. Vis. Sci., 12 (2009), 319-327. doi: 10.1007/s00791-008-0114-0.  Google Scholar

[7]

J. W. Barrett and J. F. Blowey, An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy, Numer. Math., 72 (1995), 1-20. doi: 10.1007/s002110050157.  Google Scholar

[8]

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. doi: 10.1137/S0036142997331669.  Google Scholar

[9]

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. doi: 10.1137/S0036142902413421.  Google Scholar

[10]

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), 233-280. doi: 10.1017/S095679250000053X.  Google Scholar

[11]

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. doi: 10.1017/S0956792500000759.  Google Scholar

[12]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801. doi: 10.1016/0001-6160(61)90182-1.  Google Scholar

[13]

J. W. Cahn, C. M. Elliott and A. Novick-Cohen, The Cahn-Hilliard equation with a concentration dependent mobility: Motion by minus the Laplacian of the mean curvature, European J. Appl. Math., 7 (1996), 287-301. Google Scholar

[14]

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. Google Scholar

[15]

X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, J. Differential Geom., 44 (1996), 262-311.  Google Scholar

[16]

S. Conti, B. Niethammer and F. Otto, Coarsening rates in off-critical mixtures, SIAM J. Math. Anal., 37 (2006), 1732-1741. doi: 10.1137/040620059.  Google Scholar

[17]

M. I. M. Copetti and C. M. Elliott, Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy, Numer. Math., 63 (1992), 39-65. doi: 10.1007/BF01385847.  Google Scholar

[18]

R. Dal Passo, L. Giacomelli and A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility, Interfaces Free Bound., 1 (1999), 199-226. doi: 10.4171/IFB/9.  Google Scholar

[19]

C. M. Elliott and D. A. French, Numerical studies of the Cahn-Hilliard equation for phase separation, IMA J. Appl. Math., 38 (1987), 97-128. doi: 10.1093/imamat/38.2.97.  Google Scholar

[20]

C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423. doi: 10.1137/S0036141094267662.  Google Scholar

[21]

C. M. Elliott and S. Luckhaus, A generalized diffusion equation for phase separation of a multi-component mixture with interfacial free energy (1991),, IMA, ().   Google Scholar

[22]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.  Google Scholar

[23]

P. Fratzl and J. L. Lebowitz, Universality of scaled structure functions in quenched systems undergoing phase separation, Acta Metall., 37 (1989), 3245-3248. doi: 10.1016/0001-6160(89)90196-X.  Google Scholar

[24]

P. Fratzl, J. L. Lebowitz, O. Penrose and J. Amar, Scaling functions, self-similarity, and the morphology of phase-separating systems, Phys. Rev. B, 44 (1991), 4794-4811. doi: 10.1103/PhysRevB.44.4794.  Google Scholar

[25]

M. Gameiro, K. Mischaikow and T. Wanner, Evolution of pattern complexity in the Cahn-Hilliard theory of phase separation, Acta Mater., 53 (2005), 693-704. doi: 10.1016/j.actamat.2004.10.022.  Google Scholar

[26]

H. Garcke, B. Niethammer, M. Rumpf and U. Weikard, Transient coarsening behaviour in the Cahn-Hilliard model, Acta Mater., 51 (2003), 2823-2830. doi: 10.1016/S1359-6454(03)00087-9.  Google Scholar

[27]

J. D. Gunton, M. San Miguel and P. S. Sahni, The dynamics of first-order phase transitions, in "Phase Transitions and Critical Phenomena," 8, Academic Press, London (1983), 267-482.  Google Scholar

[28]

R. Hilfer, Review on scale dependent characterization of the microstructure of porous media, Transp. Porous Media, 46 (2002), 373-390. doi: 10.1023/A:1015014302642.  Google Scholar

[29]

J. E. Hilliard, Spinodal decomposition,, in, (): 497.   Google Scholar

[30]

D. J. Horntrop, Concentration effects in mesoscopic simulation of coarsening, Math. Comput. Simulation, 80 (2010), 1082-1088. doi: 10.1016/j.matcom.2009.10.002.  Google Scholar

[31]

J. M. Hyde, M. K. Miller, M. G. Hetherington, A. Cerezo, G. D. W. Smith and C. M. Elliott, Spinodal decomposition in Fe-Cr alloys: Experimental study at the atomic level and comparison with computer models-II. Development of domain size and composition amplitude, Acta Metall. Mater., 43 (1995), 3403-3413. doi: 10.1016/0956-7151(95)00041-S.  Google Scholar

[32]

J. M. Hyde, M. K. Miller, M. G. Hetherington, A. Cerezo, G. D. W. Smith and C. M. Elliott, Spinodal decomposition in Fe-Cr alloys: Experimental study at the atomic level and comparison with computer models-III. Development of morphology, Acta Metall. Mater., 43 (1995), 3415-3426. doi: 10.1016/0956-7151(95)00042-T.  Google Scholar

[33]

T. Izumitani, M. Takenaka and T. Hashimoto, Slow spinodal decomposition in binary liquid mixtures of polymers. III. Scaling analyses of later-stage unmixing, J. Chem. Phys., 92 (1990), 3213-3221. doi: 10.1063/1.457871.  Google Scholar

[34]

T. Kaczynski, K. Mischaikow and M. Mrozek, "Computational Homology," 157 of Applied Mathematical Sciences, Springer-Verlag, New York, 2004.  Google Scholar

[35]

W. Kalies and P. Pilarczyk, CHomP software,, Available from , ().   Google Scholar

[36]

R. V. Kohn and F. Otto, Upper bounds on coarsening rates, Comm. Math. Phys., 229 (2002), 375-395. doi: 10.1007/s00220-002-0693-4.  Google Scholar

[37]

T. Y. Kong and A. Rosenfeld, Digital topology: Introduction and survey, Comput. Vision Graph. Image Process., 48 (1989), 357-393. Google Scholar

[38]

P. Leßle, M. Dong and S. Schmauder, Self-consistent matricity model to simulate the mechanical behaviour of interpenetrating microstructures, Comput. Mater. Sci., 15 (1999), 455-465. Google Scholar

[39]

P. Leßle, M. Dong, E. Soppa and S. Schmauder, Simulation of interpenetrating microstructures by self consistent matricity models, Scripta Mater., 38 (1998), 1327-1332. Google Scholar

[40]

S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics, Arch. Ration. Mech. Anal., 151 (2000), 187-219. doi: 10.1007/s002050050196.  Google Scholar

[41]

A. Novick-Cohen, Upper bounds for coarsening for the deep quench obstacle problem, J. Stat. Phys., 141 (2010), 142-157. doi: 10.1007/s10955-010-0040-7.  Google Scholar

[42]

A. Novick-Cohen, "The Cahn-Hilliard Equation: From Backwards Diffusion to Surface Diffusion," Cambridge Univ. Press, Cambridge, 2013. (in preparation). Google Scholar

[43]

A. Novick-Cohen and A. Shishkov, Upper bounds for coarsening for the degenerate Cahn-Hilliard equation, Discrete Contin. Dyn. Syst., 25 (2009), 251-272. doi: 10.3934/dcds.2009.25.251.  Google Scholar

[44]

Y. Oono and S. Puri, Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling, Phys. Rev. A, 38 (1988), 434-453. doi: 10.1103/PhysRevA.38.434.  Google Scholar

[45]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278. doi: 10.1098/rspa.1989.0027.  Google Scholar

[46]

E. Sander and T. Wanner, Unexpectedly linear behavior for the Cahn-Hilliard equation, SIAM J. Appl. Math., 60 (2000), 2182-2202. doi: 10.1137/S0036139999352225.  Google Scholar

[47]

A. Schmidt and K. G. Siebert, "Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA," 42 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 2005.  Google Scholar

[48]

P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rational Mech. Anal., 101 (1988), 209-260. doi: 10.1007/BF00253122.  Google Scholar

[49]

T. Sullivan and P. Palffy-Muhoray, The effects of pattern morphology on late time scaling in the Cahn-Hilliard model, Abstract, American Physical Society, APS March Meeting, (2007). Google Scholar

[50]

R. Toral, A. Chakrabarti and J. D. Gunton, Large scale simulations of the two-dimensional Cahn-Hilliard model, Physica A, 213 (1995), 41-49. doi: 10.1016/0378-4371(94)00146-K.  Google Scholar

[51]

T. Ujihara and K. Osamura, Kinetic analysis of spinodal decomposition process in Fe-Cr alloys by small angle neutron scattering, Acta Mater., 48 (2000), 1629-1637. doi: 10.1016/S1359-6454(99)00441-3.  Google Scholar

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