June  2022, 21(6): 1873-1894. doi: 10.3934/cpaa.2021074

Stabilized finite element methods based on multiscale enrichment for Allen-Cahn and Cahn-Hilliard equations

1. 

School of Sciences, Xi'an University of Technology, Xi'an, Shaanxi 710048, China

2. 

Key Laboratory of Thermo-Fluid Science and Engineering of Ministry of Education, School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China

3. 

FirstCenter for Computational Geosciences, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, 710049, China

4. 

College of Mathematics and Statistics, Chongqing University, Chongqing, China

* Corresponding author

Received  September 2020 Revised  March 2021 Published  June 2022 Early access  April 2021

Fund Project: The first author is supported by the National Key R & D Program of China(Grant No.2018YFB1501001), the NSF of China (Grant Nos.11771348 and 11971379)

In this paper, we investigate fully discrete schemes for the Allen-Cahn and Cahn-Hilliard equations respectively, which consist of the stabilized finite element method based on multiscale enrichment for the spatial discretization and the semi-implicit scheme for the temporal discretization. With reasonable stability conditions, it is shown that the proposed schemes are energy stable. Furthermore, by defining a new projection operator, we deduce the optimal $ L^2 $ error estimates. Some numerical experiments are presented to confirm the theoretical predictions and the efficiency of the proposed schemes.

Citation: 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 and Applied Analysis, 2022, 21 (6) : 1873-1894. doi: 10.3934/cpaa.2021074
References:
[1] R. A. Adams, Sobolev Spaces, Acadamic Press, New York, 1975. 
[2]

S. M. Allen and J. W. Cahn, A mocroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall, 27 (1979), 1085-1095. 

[3]

R. ArayaG. R. Barrenechea and F. Valentin, Stabilized finite element methods based on multiscale enrichment for the Stokes problem, SIAM J. Numer. Anal., 44 (2006), 322-348.  doi: 10.1137/050623176.

[4]

Uri M. AscherJ. Ruuth and R. J. Spiteri, Implicit-explicit Runge-Kutta method for time dependent partial differential equations, Appl. Numer. Math., 25 (1997), 151-167.  doi: 10.1016/S0168-9274(97)00056-1.

[5]

A. L. BertozziS. Esedoglu and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for image inpainting, Multi. Model. Simul., 6 (2007), 913-936.  doi: 10.1137/060660631.

[6]

A. L. BertozziS. Esedoglu and A. Gillette, Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Proc., 16 (2007), 285-291.  doi: 10.1109/TIP.2006.887728.

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system, I: interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. 

[8]

L. Q. Chen and J. Shen, Application of semi-implict Fourier-spectral method to phase-filed equations, Comput. Phys. Comm., 108 (1998), 147-158. 

[9]

L. Q. Chen, Phase-filed models for microstructure evolution, Ann. Rev. Material Research, 32 (2002), 113-140. 

[10]

Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model pf phase transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322.  doi: 10.1137/0728069.

[11]

A. Ern and J. L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.

[12]

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

[13]

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

[14]

F. Guillén-González and G. Tierra, Second order schemes and time-step adaptivity for Allen-Cahn and Cahn-Hilliard models, Comput. Math. Appl., 68 (2014), 821-846.  doi: 10.1016/j.camwa.2014.07.014.

[15]

F. Guillén-González and G. Tierra, On linear schemes for a Cahn-Hilliard diffuse interface model, J. Comput. Phys., 234 (2013), 140-171.  doi: 10.1016/j.jcp.2012.09.020.

[16]

R. HeZ. Chen and X. Feng, Error estimate of fully discrete finite element solutions for the 2D Cahn-Hilliard equation with infinite time horizon, Numer. Meth. Partial Differ. Equ., 33 (2016), 742-762.  doi: 10.1002/num.22121.

[17]

Y. He, The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Math. Comp., 77 (2008), 2097-2124.  doi: 10.1090/S0025-5718-08-02127-3.

[18]

Y. HeY. Liu and T. Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math., 57 (2007), 616-628.  doi: 10.1016/j.apnum.2006.07.026.

[19]

C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by Fourier-spectral method, Physica D, 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7.

[20]

F. Liu and J. Shen, Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations, Math. Method Appl. Sci., 38 (2015), 4564-4575.  doi: 10.1002/mma.2869.

[21]

Q. LiuY. HouZ. Wang and J. Zhao, Two-level methods for the Cahn-Hilliard equation, Math. Comput. Simulat., 126 (2016), 89-103.  doi: 10.1016/j.matcom.2016.03.004.

[22]

X. Liu and Z. Chen, A virtual element method for the Cahn-Hilliard problem in mixed form, Appl. Math. Lett., 87 (2019), 115-124.  doi: 10.1016/j.aml.2018.07.031.

[23]

J. Lowengrub and L. Truskinovsky, Quasi-incrompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond. A., 454 (1998), 2617-2654.  doi: 10.1098/rspa.1998.0273.

[24]

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702.  doi: 10.3934/cpaa.2010.9.685.

[25]

J. Shen, Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 22 (2011), 147-195.  doi: 10.1142/9789814360906_0003.

[26]

J. Shen and X. F. Yang, Numerical approxomation of Allen-Cahn and Cahn-Hilliard equations, Disc. Conti. Dyn. Sys., 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.

[27]

J. ShenJ. Xu and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353 (2018), 407-416.  doi: 10.1016/j.jcp.2017.10.021.

[28]

H. Song, Energy stable and large time-stepping methods for the Cahn-Hilliard equation, Int. J. Comput. Math., 92 (2014), 2091-2108.  doi: 10.1080/00207160.2014.964694.

[29]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, 3$^rd$ edition, North-Holland, Amsterdam, 1984.

[30]

Y. YanW. ChenC. Wang and S.M. Wise, A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation, Commun. Comput. Phys., 23 (2018), 572-602.  doi: 10.4208/cicp.oa-2016-0197.

[31]

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

[32]

X. Yang and J. Zhao, Efficient linear schemes for the nonlocal Cahn-Hilliard equation of phase field models, Comp. Phys. Commun., 235 (2019), 234-245.  doi: 10.1016/j.cpc.2018.08.012.

[33]

J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, SIAM J. Sci. Comput., 31 (2009), 3042-3063.  doi: 10.1137/080738398.

[34]

S. Zhang and M. Wang, A nonconforming finite element method for the Cahn-Hilliard equation, J. Comput. Phys., 229 (2010), 7361-7372.  doi: 10.1016/j.jcp.2010.06.020.

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Acadamic Press, New York, 1975. 
[2]

S. M. Allen and J. W. Cahn, A mocroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall, 27 (1979), 1085-1095. 

[3]

R. ArayaG. R. Barrenechea and F. Valentin, Stabilized finite element methods based on multiscale enrichment for the Stokes problem, SIAM J. Numer. Anal., 44 (2006), 322-348.  doi: 10.1137/050623176.

[4]

Uri M. AscherJ. Ruuth and R. J. Spiteri, Implicit-explicit Runge-Kutta method for time dependent partial differential equations, Appl. Numer. Math., 25 (1997), 151-167.  doi: 10.1016/S0168-9274(97)00056-1.

[5]

A. L. BertozziS. Esedoglu and A. Gillette, Analysis of a two-scale Cahn-Hilliard model for image inpainting, Multi. Model. Simul., 6 (2007), 913-936.  doi: 10.1137/060660631.

[6]

A. L. BertozziS. Esedoglu and A. Gillette, Inpainting of binary images using the Cahn-Hilliard equation, IEEE Trans. Image Proc., 16 (2007), 285-291.  doi: 10.1109/TIP.2006.887728.

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system, I: interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. 

[8]

L. Q. Chen and J. Shen, Application of semi-implict Fourier-spectral method to phase-filed equations, Comput. Phys. Comm., 108 (1998), 147-158. 

[9]

L. Q. Chen, Phase-filed models for microstructure evolution, Ann. Rev. Material Research, 32 (2002), 113-140. 

[10]

Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model pf phase transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322.  doi: 10.1137/0728069.

[11]

A. Ern and J. L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.

[12]

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

[13]

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

[14]

F. Guillén-González and G. Tierra, Second order schemes and time-step adaptivity for Allen-Cahn and Cahn-Hilliard models, Comput. Math. Appl., 68 (2014), 821-846.  doi: 10.1016/j.camwa.2014.07.014.

[15]

F. Guillén-González and G. Tierra, On linear schemes for a Cahn-Hilliard diffuse interface model, J. Comput. Phys., 234 (2013), 140-171.  doi: 10.1016/j.jcp.2012.09.020.

[16]

R. HeZ. Chen and X. Feng, Error estimate of fully discrete finite element solutions for the 2D Cahn-Hilliard equation with infinite time horizon, Numer. Meth. Partial Differ. Equ., 33 (2016), 742-762.  doi: 10.1002/num.22121.

[17]

Y. He, The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Math. Comp., 77 (2008), 2097-2124.  doi: 10.1090/S0025-5718-08-02127-3.

[18]

Y. HeY. Liu and T. Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math., 57 (2007), 616-628.  doi: 10.1016/j.apnum.2006.07.026.

[19]

C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by Fourier-spectral method, Physica D, 179 (2003), 211-228.  doi: 10.1016/S0167-2789(03)00030-7.

[20]

F. Liu and J. Shen, Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations, Math. Method Appl. Sci., 38 (2015), 4564-4575.  doi: 10.1002/mma.2869.

[21]

Q. LiuY. HouZ. Wang and J. Zhao, Two-level methods for the Cahn-Hilliard equation, Math. Comput. Simulat., 126 (2016), 89-103.  doi: 10.1016/j.matcom.2016.03.004.

[22]

X. Liu and Z. Chen, A virtual element method for the Cahn-Hilliard problem in mixed form, Appl. Math. Lett., 87 (2019), 115-124.  doi: 10.1016/j.aml.2018.07.031.

[23]

J. Lowengrub and L. Truskinovsky, Quasi-incrompressible Cahn-Hilliard fluids and topological transitions, Proc. R. Soc. Lond. A., 454 (1998), 2617-2654.  doi: 10.1098/rspa.1998.0273.

[24]

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Commun. Pure Appl. Anal., 9 (2010), 685-702.  doi: 10.3934/cpaa.2010.9.685.

[25]

J. Shen, Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 22 (2011), 147-195.  doi: 10.1142/9789814360906_0003.

[26]

J. Shen and X. F. Yang, Numerical approxomation of Allen-Cahn and Cahn-Hilliard equations, Disc. Conti. Dyn. Sys., 28 (2010), 1669-1691.  doi: 10.3934/dcds.2010.28.1669.

[27]

J. ShenJ. Xu and J. Yang, The scalar auxiliary variable (SAV) approach for gradient flows, J. Comput. Phys., 353 (2018), 407-416.  doi: 10.1016/j.jcp.2017.10.021.

[28]

H. Song, Energy stable and large time-stepping methods for the Cahn-Hilliard equation, Int. J. Comput. Math., 92 (2014), 2091-2108.  doi: 10.1080/00207160.2014.964694.

[29]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, 3$^rd$ edition, North-Holland, Amsterdam, 1984.

[30]

Y. YanW. ChenC. Wang and S.M. Wise, A second-order energy stable BDF numerical scheme for the Cahn-Hilliard equation, Commun. Comput. Phys., 23 (2018), 572-602.  doi: 10.4208/cicp.oa-2016-0197.

[31]

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

[32]

X. Yang and J. Zhao, Efficient linear schemes for the nonlocal Cahn-Hilliard equation of phase field models, Comp. Phys. Commun., 235 (2019), 234-245.  doi: 10.1016/j.cpc.2018.08.012.

[33]

J. Zhang and Q. Du, Numerical studies of discrete approximations to the Allen-Cahn equation in the sharp interface limit, SIAM J. Sci. Comput., 31 (2009), 3042-3063.  doi: 10.1137/080738398.

[34]

S. Zhang and M. Wang, A nonconforming finite element method for the Cahn-Hilliard equation, J. Comput. Phys., 229 (2010), 7361-7372.  doi: 10.1016/j.jcp.2010.06.020.

Figure 1.  The dynamic of the scheme for Allen-Cahn equation at the $t = 0.0001$(left), $t = 0.001$(middle)and $t = 0.01$(right).
Figure 2.  The dynamic of the scheme for Allen-Cahn equation at the $t = 0.1$ (left), $t = 1.0$(middle)and $t = 2.0$(right).
Figure 3.  The dynamic of the scheme for Allen-Cahn equation at the $t = 3.0$ (left), $t = 4.0$(middle)and $t = 5.0$(right).
Figure 4.  The dynamic of the scheme for Allen-Cahn equation at the $t = 5.5$ (left) and $t = 6.0$(right).
Figure 5.  The dynamic of the scheme for Cahn-Hilliard equation at the $t = 0.0001$ (left) and $t = 0.001$(right).
Figure 6.  The dynamic of the scheme for Cahn-Hilliard equation at the $t = 0.01$ (left) and $t = 0.1$(right).
Figure 7.  The dynamic of the scheme for Cahn-Hilliard equation at the $t = 0.5$ (left) and $t = 1.0$(right).
Figure 8.  The dynamic of the scheme for Cahn-Hilliard equation at the $t = 1.2$ (left) and $t = 1.5$(right).
Table 1.  Convergence performance of the time discretization for the 2D Allen-Cahn equation
Time step $ dt = 10^{-2} $ $ dt/2 $ $ dt/3 $ $ dt/4 $ $ dt/5 $ $ dt/6 $ $ dt/7 $ $ dt/8 $ $ dt/9 $
$\frac{{{{\left\| {u - {u_h}} \right\|}_0}}}{{{{\left\| u \right\|}_0}}}$ 0.257853 0.165833 0.121715 0.095751 0.078637 0.066504 0.057452 0.050440 0.044848
$ u_{L^{2}rate} $ \ 0.636817 0.762828 0.834011 0.882443 0.919165 0.949093 0.974767 0.997637
Time step $ dt = 10^{-2} $ $ dt/2 $ $ dt/3 $ $ dt/4 $ $ dt/5 $ $ dt/6 $ $ dt/7 $ $ dt/8 $ $ dt/9 $
$\frac{{{{\left\| {u - {u_h}} \right\|}_0}}}{{{{\left\| u \right\|}_0}}}$ 0.257853 0.165833 0.121715 0.095751 0.078637 0.066504 0.057452 0.050440 0.044848
$ u_{L^{2}rate} $ \ 0.636817 0.762828 0.834011 0.882443 0.919165 0.949093 0.974767 0.997637
Table 2.  Convergence performance of the spatial discretization for the 2D Allen-Cahn equation
Mesh $ h = 2pi/8 $ $ h = 2pi/16 $ $ h = 2pi/32 $ $ h = 2pi/64 $ $ h = 2pi/128 $
$\frac{|u-u_{h}|_{0}}{|u|_{0}} $ 0.122619 0.031633 0.007850 0.001841 0.000349
$ u_{L^{2}rate} $ \ 1.954660 2.010630 2.092350 2.401100
Mesh $ h = 2pi/8 $ $ h = 2pi/16 $ $ h = 2pi/32 $ $ h = 2pi/64 $ $ h = 2pi/128 $
$\frac{|u-u_{h}|_{0}}{|u|_{0}} $ 0.122619 0.031633 0.007850 0.001841 0.000349
$ u_{L^{2}rate} $ \ 1.954660 2.010630 2.092350 2.401100
Table 3.  Convergence performance of the time discretization for the 2D Cahn-Hilliard equation
Time step $ dt = 10^{-4} $ $ dt/2 $ $ dt/3 $ $ dt/4 $ $ dt/5 $ $ dt/6 $ $ dt/7 $ $ dt/8 $ $ dt/9 $
$ \frac{\|u-u_{h}\|_{0}}{\|u\|_{0}} $ 1.940e-04 9.65e-05 6.38e-05 4.74e-05 3.76e-05 3.11e-05 2.65e-05 2.30e-05 2.04e-05
$ u_{L^{2}rate} $ \ 1.00706 1.02108 1.03078 1.03783 1.04254 1.04493 1.04497 1.04261
$ \frac{\|w-w_{h}\|_{0}}{\|w\|_{0}} $ 0.019526 0.009760 0.006461 0.004804 0.003807 0.003141 0.002665 0.002308 0.002030
$ w_{L^{2}rate} $ \ 1.00045 1.01715 1.03026 1.04246 1.05439 1.06629 1.07827 1.09041
Time step $ dt = 10^{-4} $ $ dt/2 $ $ dt/3 $ $ dt/4 $ $ dt/5 $ $ dt/6 $ $ dt/7 $ $ dt/8 $ $ dt/9 $
$ \frac{\|u-u_{h}\|_{0}}{\|u\|_{0}} $ 1.940e-04 9.65e-05 6.38e-05 4.74e-05 3.76e-05 3.11e-05 2.65e-05 2.30e-05 2.04e-05
$ u_{L^{2}rate} $ \ 1.00706 1.02108 1.03078 1.03783 1.04254 1.04493 1.04497 1.04261
$ \frac{\|w-w_{h}\|_{0}}{\|w\|_{0}} $ 0.019526 0.009760 0.006461 0.004804 0.003807 0.003141 0.002665 0.002308 0.002030
$ w_{L^{2}rate} $ \ 1.00045 1.01715 1.03026 1.04246 1.05439 1.06629 1.07827 1.09041
Table 4.  Convergence performance of the spatial discretization for the 2D Cahn-Hilliard equation
Mesh $ h = 2\pi/8 $ $ h = 2\pi/16 $ $ h = 2\pi/32 $ $ h = 2\pi/64 $ $ h = 2\pi/128 $
$ \frac{\|u-u_{h}\|_{0}}{\|u\|_{0}} $ {0.116925} 0.030137 0.007479 0.001755 0.000337
$ u_{L^{2}rate} $ \ 1.95596 2.01063 2.09102 2.38675
$ \frac{\|w-w_{h}\|_{0}}{\|w\|_{0}} $ {0.120028} 0.030952 0.007681 0.001802 0.000343
$ w_{L^{2}rate} $ \ 1.95523 2.01063 2.09177 2.39460
Mesh $ h = 2\pi/8 $ $ h = 2\pi/16 $ $ h = 2\pi/32 $ $ h = 2\pi/64 $ $ h = 2\pi/128 $
$ \frac{\|u-u_{h}\|_{0}}{\|u\|_{0}} $ {0.116925} 0.030137 0.007479 0.001755 0.000337
$ u_{L^{2}rate} $ \ 1.95596 2.01063 2.09102 2.38675
$ \frac{\|w-w_{h}\|_{0}}{\|w\|_{0}} $ {0.120028} 0.030952 0.007681 0.001802 0.000343
$ w_{L^{2}rate} $ \ 1.95523 2.01063 2.09177 2.39460
[1]

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

[2]

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

[3]

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

[4]

Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2607-2620. doi: 10.3934/dcdsb.2020024

[5]

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

[6]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations and Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012

[7]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations and Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517

[8]

Christopher P. Grant. Grain sizes in the discrete Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 127-146. doi: 10.3934/dcds.2001.7.127

[9]

Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099

[10]

Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301

[11]

Jie Shen, Xiaofeng Yang. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1669-1691. doi: 10.3934/dcds.2010.28.1669

[12]

Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3633-3651. doi: 10.3934/dcdsb.2018308

[13]

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

[14]

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

[15]

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

[16]

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

[17]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511

[18]

Irena Pawłow. Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1169-1191. doi: 10.3934/dcds.2006.15.1169

[19]

Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027

[20]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (474)
  • HTML views (479)
  • Cited by (0)

Other articles
by authors

[Back to Top]