# American Institute of Mathematical Sciences

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. Araya, G. 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. Ascher, J. 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. Bertozzi, S. 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. Bertozzi, S. 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. Feng, H. Song, T. 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. Feng, T. 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. He, Z. 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. He, Y. 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. Liu, Y. Hou, Z. 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. Shen, J. 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. Yan, W. Chen, C. 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. Araya, G. 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. Ascher, J. 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. Bertozzi, S. 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. Bertozzi, S. 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. Feng, H. Song, T. 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. Feng, T. 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. He, Z. 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. He, Y. 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. Liu, Y. Hou, Z. 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. Shen, J. 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. Yan, W. Chen, C. 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.
The dynamic of the scheme for Allen-Cahn equation at the $t = 0.0001$(left), $t = 0.001$(middle)and $t = 0.01$(right).
The dynamic of the scheme for Allen-Cahn equation at the $t = 0.1$ (left), $t = 1.0$(middle)and $t = 2.0$(right).
The dynamic of the scheme for Allen-Cahn equation at the $t = 3.0$ (left), $t = 4.0$(middle)and $t = 5.0$(right).
The dynamic of the scheme for Allen-Cahn equation at the $t = 5.5$ (left) and $t = 6.0$(right).
The dynamic of the scheme for Cahn-Hilliard equation at the $t = 0.0001$ (left) and $t = 0.001$(right).
The dynamic of the scheme for Cahn-Hilliard equation at the $t = 0.01$ (left) and $t = 0.1$(right).
The dynamic of the scheme for Cahn-Hilliard equation at the $t = 0.5$ (left) and $t = 1.0$(right).
The dynamic of the scheme for Cahn-Hilliard equation at the $t = 1.2$ (left) and $t = 1.5$(right).
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
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
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
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
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