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September 2017, 22(7): 2857-2877. doi: 10.3934/dcdsb.2017154

The stabilized semi-implicit finite element method for the surface Allen-Cahn equation

1. 

College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China

2. 

Institute of Mathematics and Physics, Xinjiang University, Urumqi 830046, China

3. 

Departamento de Matemática, Universidade Federal do Paraná, Centro Politácnico, Curitiba 81531-980, PR, Brazil

* Corresponding author: Institute of Mathematics and Physics, Xinjiang University, Urumqi 830046, P.R. China

Received  June 2016 Revised  March 2017 Published  May 2017

Fund Project: The first author is supported by the Excellent Doctor Innovation Program of Xinjiang University (No. XJUBSCX-2016006) and the Graduate Student Research Innovation Program of Xinjiang (No. XJGRI2015009). The second author is supported by the NSF of Xinjiang Province (No.2016D01C058), NCET-13-0988, and the NSF of China (No. 11671345,11271313). The third author is supported by CAPES (No. 88881.068004/2014.01) and CNPq (No. 300326/2012-2,470934/2013-1, INCT-Matemática) of Brazil

Two semi-implicit numerical methods are proposed for solving the surface Allen-Cahn equation which is a general mathematical model to describe phase separation on general surfaces. The spatial discretization is based on surface finite element method while the temporal discretization methods are first-and second-order stabilized semi-implicit schemes to guarantee the energy decay. The stability analysis and error estimate are provided for the stabilized semi-implicit schemes. Furthermore, the first-and second-order operator splitting methods are presented to compare with stabilized semi-implicit schemes. Some numerical experiments including phase separation and mean curvature flow on surfaces are performed to illustrate stability and accuracy of these methods.

Citation: Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2857-2877. doi: 10.3934/dcdsb.2017154
References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall, 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2.

[2]

D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.

[3]

L. Chen, Phase-field models for microstructure evolution, Ann. Rev. Mater. Res., 32 (2002), 113-140. doi: 10.1146/annurev.matsci.32.112001.132041.

[4]

M. Cheng and J. A. Warren, An efficient algorithm for solving the phase field crystal model, J. Comput. Phys., 227 (2008), 6241-6248. doi: 10.1016/j.jcp.2008.03.012.

[5]

Y. ChoiD. JeongS. LeeM. Yoo and J. Kim, Motion by mean curvature of curves on surfaces using the Allen-Cahn equation, Int. J. Eng. Sci., 97 (2015), 126-132. doi: 10.1016/j.ijengsci.2015.10.002.

[6]

K. DeckelnickG. DziukC. M. Elliott and C. J. Heine, An $h$-narrow band finite-element method for elliptic equations on implicit surfaces, IMA. J. Numer. Anal., 30 (2010), 351-376. doi: 10.1093/imanum/drn049.

[7]

Q. DuL. Ju and L. Tian, Finite element approximation of the Cahn-Hilliard equation on surfaces, Comput. Methods Appl. Mech. Engrg., 200 (2011), 2458-2470. doi: 10.1016/j.cma.2011.04.018.

[8]

G. Dziuk and C. M. Elliott, Finite element methods for surface PDEs, Acta Numer., 22 (2013), 289-396. doi: 10.1017/S0962492913000056.

[9]

G. Dziuk and C. M. Elliott, Surface finite elements for parabolic equations, J. Comput. Math., 25 (2007), 385-407.

[10]

G. Dziuk, Finite Elements for the Beltrami operator on arbitrary surfaces, Partial Differential Equations and Calculus of Variations, Lecture Notes in Mathematics, Springer, 1357 (2006), 142-155. doi: 10.1007/BFb0082865.

[11]

C. M. Elliott and T. Ranner, Evolving surface finite element method for the Cahn-Hilliard equation, Numer. Math., 129 (2015), 483-534. doi: 10.1007/s00211-014-0644-y.

[12]

L. C. EvansH. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. doi: 10.1002/cpa.3160450903.

[13]

D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, 1998.

[14]

X. FengT. Tang and J. Yang, Long time numerical simulations for phase-field problems using p-adaptive spectral deferred correction methods, SIAM J. Sci. Comput., 37 (2015), 271-294. doi: 10.1137/130928662.

[15]

X. FengH. SongT. Tang and J. Yang, Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation, Inverse Prob. Imaging, 7 (2013), 679-695. doi: 10.3934/ipi.2013.7.679.

[16]

X. FengT. Tang and J. Yang, Stabilized Crank-Nicolson /Adams-Bashforth schemes for phase field models, East Asian J. Appl. Math., 3 (2013), 59-80. doi: 10.4208/eajam.200113.220213a.

[17]

B. Fornberg and N. Flyer, Solving PDEs with radial basis functions, Acta Numer., 24 (2015), 215-258. doi: 10.1017/S0962492914000130.

[18]

Z. GuanJ. S. LowengrubC. Wang and S. M. Wise, Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277 (2014), 48-71. doi: 10.1016/j.jcp.2014.08.001.

[19]

H. Holden, Splitting Method for Partial Differential Equations with Rough Solutions: Analysis and MATLAB Programs European Mathematical Society, Zurich, 2010. doi: 10.4171/078.

[20]

D. Marenduzzo and E. Orlandini, Phase separation dynamics on curved surfaces, Soft Matter, 9 (2013), 1178-1187. doi: 10.1039/C2SM27081A.

[21]

C. Piret, The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces, J. Comput. Phys., 231 (2012), 4662-4675. doi: 10.1016/j.jcp.2012.03.007.

[22]

S. J. Ruuth and B. Merrimanb, A simple embedding method for solving partial differential equations on surfaces, J. Comput. Phys., 227 (2008), 1943-1961. doi: 10.1016/j.jcp.2007.10.009.

[23]

O. Schönborn and R. C. Desai, Kinetics of phase ordering on curved surfaces, Phys. A, 239 (1997), 412-419.

[24]

J. ShenT. Tang and J. Yang, On the maximum principle preserving schemes for the generalized Allen-Cahn equation, Comm. Math. Sci., 14 (2016), 1517-1534. doi: 10.4310/CMS.2016.v14.n6.a3.

[25]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst. Ser. A, 28 (2010), 1669-1691. doi: 10.3934/dcds.2010.28.1669.

[26]

J. ShenC. WangX. Wang and S. M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105-125. doi: 10.1137/110822839.

[27]

P. Tang, F. Qiu, H. Zhang, et al. , Phase separation patterns for diblock copolymers on spherical surfaces: A finite volume method Phys. Rev. E. 72 (2005), 016710. doi: 10.1103/PhysRevE. 72. 016710.

[28]

V. Thomee, Galerkin Finite Element Methods for Parabolic Problems Springer-Verlag, Berlin, 2006.

[29]

C. WangX. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst. Ser. A, 28 (2010), 405-423. doi: 10.3934/dcds.2010.28.405.

[30]

X. XiaoD. Gui and X. Feng, A highly efficient operator-splitting finite element method for 2D/3D nonlinear Allen-Cahn equation, Inter. J. Numer. Methods Heat Fluid Flow, 27 (2017), 530-542. doi: 10.1108/HFF-12-2015-0521.

[31]

X. Yang, Error analysis of stabilized semi-implicit method of Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 1057-1070. doi: 10.3934/dcdsb.2009.11.1057.

show all references

References:
[1]

S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall, 27 (1979), 1085-1095. doi: 10.1016/0001-6160(79)90196-2.

[2]

D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.

[3]

L. Chen, Phase-field models for microstructure evolution, Ann. Rev. Mater. Res., 32 (2002), 113-140. doi: 10.1146/annurev.matsci.32.112001.132041.

[4]

M. Cheng and J. A. Warren, An efficient algorithm for solving the phase field crystal model, J. Comput. Phys., 227 (2008), 6241-6248. doi: 10.1016/j.jcp.2008.03.012.

[5]

Y. ChoiD. JeongS. LeeM. Yoo and J. Kim, Motion by mean curvature of curves on surfaces using the Allen-Cahn equation, Int. J. Eng. Sci., 97 (2015), 126-132. doi: 10.1016/j.ijengsci.2015.10.002.

[6]

K. DeckelnickG. DziukC. M. Elliott and C. J. Heine, An $h$-narrow band finite-element method for elliptic equations on implicit surfaces, IMA. J. Numer. Anal., 30 (2010), 351-376. doi: 10.1093/imanum/drn049.

[7]

Q. DuL. Ju and L. Tian, Finite element approximation of the Cahn-Hilliard equation on surfaces, Comput. Methods Appl. Mech. Engrg., 200 (2011), 2458-2470. doi: 10.1016/j.cma.2011.04.018.

[8]

G. Dziuk and C. M. Elliott, Finite element methods for surface PDEs, Acta Numer., 22 (2013), 289-396. doi: 10.1017/S0962492913000056.

[9]

G. Dziuk and C. M. Elliott, Surface finite elements for parabolic equations, J. Comput. Math., 25 (2007), 385-407.

[10]

G. Dziuk, Finite Elements for the Beltrami operator on arbitrary surfaces, Partial Differential Equations and Calculus of Variations, Lecture Notes in Mathematics, Springer, 1357 (2006), 142-155. doi: 10.1007/BFb0082865.

[11]

C. M. Elliott and T. Ranner, Evolving surface finite element method for the Cahn-Hilliard equation, Numer. Math., 129 (2015), 483-534. doi: 10.1007/s00211-014-0644-y.

[12]

L. C. EvansH. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., 45 (1992), 1097-1123. doi: 10.1002/cpa.3160450903.

[13]

D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, 1998.

[14]

X. FengT. Tang and J. Yang, Long time numerical simulations for phase-field problems using p-adaptive spectral deferred correction methods, SIAM J. Sci. Comput., 37 (2015), 271-294. doi: 10.1137/130928662.

[15]

X. FengH. SongT. Tang and J. Yang, Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation, Inverse Prob. Imaging, 7 (2013), 679-695. doi: 10.3934/ipi.2013.7.679.

[16]

X. FengT. Tang and J. Yang, Stabilized Crank-Nicolson /Adams-Bashforth schemes for phase field models, East Asian J. Appl. Math., 3 (2013), 59-80. doi: 10.4208/eajam.200113.220213a.

[17]

B. Fornberg and N. Flyer, Solving PDEs with radial basis functions, Acta Numer., 24 (2015), 215-258. doi: 10.1017/S0962492914000130.

[18]

Z. GuanJ. S. LowengrubC. Wang and S. M. Wise, Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys., 277 (2014), 48-71. doi: 10.1016/j.jcp.2014.08.001.

[19]

H. Holden, Splitting Method for Partial Differential Equations with Rough Solutions: Analysis and MATLAB Programs European Mathematical Society, Zurich, 2010. doi: 10.4171/078.

[20]

D. Marenduzzo and E. Orlandini, Phase separation dynamics on curved surfaces, Soft Matter, 9 (2013), 1178-1187. doi: 10.1039/C2SM27081A.

[21]

C. Piret, The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces, J. Comput. Phys., 231 (2012), 4662-4675. doi: 10.1016/j.jcp.2012.03.007.

[22]

S. J. Ruuth and B. Merrimanb, A simple embedding method for solving partial differential equations on surfaces, J. Comput. Phys., 227 (2008), 1943-1961. doi: 10.1016/j.jcp.2007.10.009.

[23]

O. Schönborn and R. C. Desai, Kinetics of phase ordering on curved surfaces, Phys. A, 239 (1997), 412-419.

[24]

J. ShenT. Tang and J. Yang, On the maximum principle preserving schemes for the generalized Allen-Cahn equation, Comm. Math. Sci., 14 (2016), 1517-1534. doi: 10.4310/CMS.2016.v14.n6.a3.

[25]

J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete Contin. Dyn. Syst. Ser. A, 28 (2010), 1669-1691. doi: 10.3934/dcds.2010.28.1669.

[26]

J. ShenC. WangX. Wang and S. M. Wise, Second-order convex splitting schemes for gradient flows with Ehrlich-Schwoebel type energy: application to thin film epitaxy, SIAM J. Numer. Anal., 50 (2012), 105-125. doi: 10.1137/110822839.

[27]

P. Tang, F. Qiu, H. Zhang, et al. , Phase separation patterns for diblock copolymers on spherical surfaces: A finite volume method Phys. Rev. E. 72 (2005), 016710. doi: 10.1103/PhysRevE. 72. 016710.

[28]

V. Thomee, Galerkin Finite Element Methods for Parabolic Problems Springer-Verlag, Berlin, 2006.

[29]

C. WangX. Wang and S. M. Wise, Unconditionally stable schemes for equations of thin film epitaxy, Discrete Contin. Dyn. Syst. Ser. A, 28 (2010), 405-423. doi: 10.3934/dcds.2010.28.405.

[30]

X. XiaoD. Gui and X. Feng, A highly efficient operator-splitting finite element method for 2D/3D nonlinear Allen-Cahn equation, Inter. J. Numer. Methods Heat Fluid Flow, 27 (2017), 530-542. doi: 10.1108/HFF-12-2015-0521.

[31]

X. Yang, Error analysis of stabilized semi-implicit method of Allen-Cahn equation, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 1057-1070. doi: 10.3934/dcdsb.2009.11.1057.

Figure 1.  Simulation of phase separation on sphere by SSI1 with $\delta t=5\times10^{-4}$.
Figure 2.  Simulation of phase separation on sphere by SSI1 with $\delta t=10^{-4}$.
Figure 3.  Simulation of phase separation on sphere by SSI2 with $\delta t=5\times10^{-4}$.
Figure 4.  Simulation of phase separation on sphere by SSI2 with $\delta t=10^{-4}$.
Figure 5.  Non-dimensional discrete total energy line of SSI1 (a) and SSI2 (b) with $\delta t=5\times10^{-4}$(blue) and $\delta t=10^{-4}$(red) on sphere.
Figure 6.  Simulation of phase separation on torus by OS1 with $\delta t=5\times10^{-4}$.
Figure 7.  Simulation of phase separation on torus by OS1 with $\delta t=10^{-4}$.
Figure 8.  Non-dimensional discrete total energy curves of OS1 with different with $\delta t=5\times10^{-4}$(blue) and $\delta t=10^{-4}$(red) (a). And the side view of the solution of OS1 at t=0.05 with $\delta t=5\times10^{-4}$ on torus (b).
Figure 9.  Solutions of phase separation on torus by OS2 with different $\delta t$.
Figure 10.  Comparison between the SSI1, SSI2, OS1 and OS2 for the simulation of mean curvature flow on sphere with $\delta t=5\times10^{-4}$ (a) and $\delta t=10^{-4}$ (b).
Figure 11.  Simulation of motion of a circle on sphere by SSI2 with $\delta t=10^{-4}$.
Figure 12.  Comparison between the SSI1, SSI2, OS1 and OS2 for the simulation of mean curvature flow on hyperboloid with $\delta t=5\times10^{-4}$ (a) and $\delta t=10^{-4}$ (b).
Figure 13.  Simulation of motion of a circle on hyperboloid by OS1 with $\delta t=10^{-4}$.
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