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doi: 10.3934/cpaa.2021074
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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
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
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