American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Asymptotic behaviour of an age and infection age structured model for the propagation of fungal diseases in plants
  • DCDS-B Home
  • This Issue
  • Next Article
    New convergence analysis for assumed stress hybrid quadrilateral finite element method
September  2017, 22(7): 2857-2877. doi: 10.3934/dcdsb.2017154

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

Xufeng Xiao 1, , Xinlong Feng 2,, and  Jinyun Yuan 3,

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.

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 & 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

Figure 1.  Simulation of phase separation on sphere by SSI1 with $\delta t=5\times10^{-4}$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Simulation of phase separation on sphere by SSI1 with $\delta t=10^{-4}$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Simulation of phase separation on sphere by SSI2 with $\delta t=5\times10^{-4}$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Simulation of phase separation on sphere by SSI2 with $\delta t=10^{-4}$.
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Simulation of phase separation on torus by OS1 with $\delta t=5\times10^{-4}$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Simulation of phase separation on torus by OS1 with $\delta t=10^{-4}$.
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Solutions of phase separation on torus by OS2 with different $\delta t$.
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 11.  Simulation of motion of a circle on sphere by SSI2 with $\delta t=10^{-4}$.
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 13.  Simulation of motion of a circle on hyperboloid by OS1 with $\delta t=10^{-4}$.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Xiaofeng Yang. Error analysis of stabilized semi-implicit method of Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1057-1070. doi: 10.3934/dcdsb.2009.11.1057
[2]

Juan Wen, Yaling He, Yinnian He, Kun Wang. Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021074
[3]

Hector D. Ceniceros. A semi-implicit moving mesh method for the focusing nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2002, 1 (1) : 1-18. doi: 10.3934/cpaa.2002.1.1
[4]

Xinlong Feng, Huailing Song, Tao Tang, Jiang Yang. Nonlinear stability of the implicit-explicit methods for the Allen-Cahn equation. Inverse Problems & Imaging, 2013, 7 (3) : 679-695. doi: 10.3934/ipi.2013.7.679
[5]

Charles-Edouard Bréhier, Ludovic Goudenège. Analysis of some splitting schemes for the stochastic Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4169-4190. doi: 10.3934/dcdsb.2019077
[6]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025
[7]

Tina Hartley, Thomas Wanner. A semi-implicit spectral method for stochastic nonlocal phase-field models. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 399-429. doi: 10.3934/dcds.2009.25.399
[8]

Yinnian He, Yanping Lin, Weiwei Sun. Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 41-68. doi: 10.3934/dcdsb.2006.6.41
[9]

Xiaoxiao He, Fei Song, Weibing Deng. A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021163
[10]

Gianni Gilardi. On an Allen-Cahn type integrodifferential equation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 703-709. doi: 10.3934/dcdss.2013.6.703
[11]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[12]

Hassan Belhadj, Mohamed Fihri, Samir Khallouq, Nabila Nagid. Optimal number of Schur subdomains: Application to semi-implicit finite volume discretization of semilinear reaction diffusion problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 21-34. doi: 10.3934/dcdss.2018002
[13]

Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663
[14]

Hongmei Cheng, Rong Yuan. Multidimensional stability of disturbed pyramidal traveling fronts in the Allen-Cahn equation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015
[15]

Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Representation formulas of solutions and bifurcation sheets to a nonlocal Allen-Cahn equation. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4907-4925. doi: 10.3934/dcds.2020205
[16]

Christos Sourdis. On the growth of the energy of entire solutions to the vector Allen-Cahn equation. Communications on Pure & Applied Analysis, 2015, 14 (2) : 577-584. doi: 10.3934/cpaa.2015.14.577
[17]

Paul H. Rabinowitz, Ed Stredulinsky. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 319-332. doi: 10.3934/dcds.2008.21.319
[18]

Ciprian G. Gal, Maurizio Grasselli. The non-isothermal Allen-Cahn equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 1009-1040. doi: 10.3934/dcds.2008.22.1009
[19]

Eleonora Cinti. Saddle-shaped solutions for the fractional Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 441-463. doi: 10.3934/dcdss.2018024
[20]

Zhuoran Du, Baishun Lai. Transition layers for an inhomogeneous Allen-Cahn equation in Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1407-1429. doi: 10.3934/dcds.2013.33.1407

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (162)
  • HTML views (132)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences