# American Institute of Mathematical Sciences

• Previous Article
Analysis of COVID-19 epidemic transmission trend based on a time-delayed dynamic model
• CPAA Home
• This Issue
• Next Article
Local well-posedness in Sobolev spaces for first-order barotropic causal relativistic viscous hydrodynamics
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 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 & Applied Analysis, doi: 10.3934/cpaa.2021074
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
 [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] 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 [3] 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 [4] Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - B, 2015, 20 (4) : 1015-1029. doi: 10.3934/dcdsb.2015.20.1015 [16] Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511 [17] 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 [18] Irena Pawłow. Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids. Discrete & 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 & 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 & Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207

2020 Impact Factor: 1.916