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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. Google Scholar [2] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Google Scholar [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. Google Scholar [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. Google Scholar [5] Y. Choi, D. Jeong, S. Lee, M. 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. Google Scholar [6] K. Deckelnick, G. Dziuk, C. 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. Google Scholar [7] Q. Du, L. 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. Google Scholar [8] G. Dziuk and C. M. Elliott, Finite element methods for surface PDEs, Acta Numer., 22 (2013), 289-396. doi: 10.1017/S0962492913000056. Google Scholar [9] G. Dziuk and C. M. Elliott, Surface finite elements for parabolic equations, J. Comput. Math., 25 (2007), 385-407. Google Scholar [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. Google Scholar [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. Google Scholar [12] L. C. Evans, H. 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. Google Scholar [13] D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, 1998.Google Scholar [14] X. Feng, T. 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. Google Scholar [15] X. Feng, H. Song, T. 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. Google Scholar [16] X. Feng, T. 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. Google Scholar [17] B. Fornberg and N. Flyer, Solving PDEs with radial basis functions, Acta Numer., 24 (2015), 215-258. doi: 10.1017/S0962492914000130. Google Scholar [18] Z. Guan, J. S. Lowengrub, C. 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. Google Scholar [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. Google Scholar [20] D. Marenduzzo and E. Orlandini, Phase separation dynamics on curved surfaces, Soft Matter, 9 (2013), 1178-1187. doi: 10.1039/C2SM27081A. Google Scholar [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. Google Scholar [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. Google Scholar [23] O. Schönborn and R. C. Desai, Kinetics of phase ordering on curved surfaces, Phys. A, 239 (1997), 412-419. Google Scholar [24] J. Shen, T. 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. Google Scholar [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. Google Scholar [26] J. Shen, C. Wang, X. 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. Google Scholar [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. Google Scholar [28] V. Thomee, Galerkin Finite Element Methods for Parabolic Problems Springer-Verlag, Berlin, 2006. Google Scholar [29] C. Wang, X. 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. Google Scholar [30] X. Xiao, D. 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. Google Scholar [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. Google Scholar

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. Google Scholar [2] D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982. Google Scholar [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. Google Scholar [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. Google Scholar [5] Y. Choi, D. Jeong, S. Lee, M. 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. Google Scholar [6] K. Deckelnick, G. Dziuk, C. 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. Google Scholar [7] Q. Du, L. 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. Google Scholar [8] G. Dziuk and C. M. Elliott, Finite element methods for surface PDEs, Acta Numer., 22 (2013), 289-396. doi: 10.1017/S0962492913000056. Google Scholar [9] G. Dziuk and C. M. Elliott, Surface finite elements for parabolic equations, J. Comput. Math., 25 (2007), 385-407. Google Scholar [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. Google Scholar [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. Google Scholar [12] L. C. Evans, H. 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. Google Scholar [13] D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, 1998.Google Scholar [14] X. Feng, T. 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. Google Scholar [15] X. Feng, H. Song, T. 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. Google Scholar [16] X. Feng, T. 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. Google Scholar [17] B. Fornberg and N. Flyer, Solving PDEs with radial basis functions, Acta Numer., 24 (2015), 215-258. doi: 10.1017/S0962492914000130. Google Scholar [18] Z. Guan, J. S. Lowengrub, C. 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. Google Scholar [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. Google Scholar [20] D. Marenduzzo and E. Orlandini, Phase separation dynamics on curved surfaces, Soft Matter, 9 (2013), 1178-1187. doi: 10.1039/C2SM27081A. Google Scholar [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. Google Scholar [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. Google Scholar [23] O. Schönborn and R. C. Desai, Kinetics of phase ordering on curved surfaces, Phys. A, 239 (1997), 412-419. Google Scholar [24] J. Shen, T. 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. Google Scholar [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. Google Scholar [26] J. Shen, C. Wang, X. 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. Google Scholar [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. Google Scholar [28] V. Thomee, Galerkin Finite Element Methods for Parabolic Problems Springer-Verlag, Berlin, 2006. Google Scholar [29] C. Wang, X. 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. Google Scholar [30] X. Xiao, D. 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. Google Scholar [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. Google Scholar
Simulation of phase separation on sphere by SSI1 with $\delta t=5\times10^{-4}$.
Simulation of phase separation on sphere by SSI1 with $\delta t=10^{-4}$.
Simulation of phase separation on sphere by SSI2 with $\delta t=5\times10^{-4}$.
Simulation of phase separation on sphere by SSI2 with $\delta t=10^{-4}$.
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.
Simulation of phase separation on torus by OS1 with $\delta t=5\times10^{-4}$.
Simulation of phase separation on torus by OS1 with $\delta t=10^{-4}$.
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).
Solutions of phase separation on torus by OS2 with different $\delta t$.
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).
Simulation of motion of a circle on sphere by SSI2 with $\delta t=10^{-4}$.
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).
Simulation of motion of a circle on hyperboloid by OS1 with $\delta t=10^{-4}$.
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