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

Juan Wen; Yaling He; Yinnian He; Kun Wang

doi:10.3934/cpaa.2021074

Article Contents

2022, Volume 21, Issue 6: 1873-1894. 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
^* Corresponding author

Received: August 31, 2020

Revised: February 28, 2021

Published: June 2022

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q30, 65N30, 76D07.

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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).

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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).

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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).

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Figure 4. The dynamic of the scheme for Allen-Cahn equation at the $t = 5.5$ (left) and $t = 6.0$(right).

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Figure 5. The dynamic of the scheme for Cahn-Hilliard equation at the $t = 0.0001$ (left) and $t = 0.001$(right).

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Figure 6. The dynamic of the scheme for Cahn-Hilliard equation at the $t = 0.01$ (left) and $t = 0.1$(right).

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Figure 7. The dynamic of the scheme for Cahn-Hilliard equation at the $t = 0.5$ (left) and $t = 1.0$(right).

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Figure 8. The dynamic of the scheme for Cahn-Hilliard equation at the $t = 1.2$ (left) and $t = 1.5$(right).

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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

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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

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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

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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

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