Figure 1.
Simulation of phase separation on sphere by SSI1 with $\delta t=5\times10^{-4}$ .
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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 |
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.
Keywords:
Surface Allen-Cahn equation,
surface finite element method,
stabilized semi-implicit scheme,
operator splitting method,
error estimate.
Mathematics Subject Classification:
Primary:65M60, 76T99;Secondary:65M12.
Citation:
Xufeng Xiao, Xinlong Feng, Jinyun Yuan. The stabilized semi-implicit finite element method for the surface Allen-Cahn equation. Discrete and Continuous Dynamical Systems - B,
2017, 22
(7)
: 2857-2877.
doi: 10.3934/dcdsb.2017154
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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. 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. |
| [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. |
| [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. |
| [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. 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. |
| [13] |
D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, 1998. |
| [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. |
| [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. |
| [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. |
| [17] |
B. Fornberg and N. Flyer,
Solving PDEs with radial basis functions, Acta Numer., 24 (2015), 215-258.
doi: 10.1017/S0962492914000130. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. |
| [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. |
| [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. 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. |
| [13] |
D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, Unpublished article, 1998. |
| [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. |
| [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. |
| [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. |
| [17] |
B. Fornberg and N. Flyer,
Solving PDEs with radial basis functions, Acta Numer., 24 (2015), 215-258.
doi: 10.1017/S0962492914000130. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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 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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