The stabilized semi-implicit finite element method for the surface Allen-Cahn equation

Xufeng Xiao; Xinlong Feng; Jinyun Yuan

doi:10.3934/dcdsb.2017154

Article Contents

2017, Volume 22, Issue 7: 2857-2877. Doi: 10.3934/dcdsb.2017154

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The stabilized semi-implicit finite element method for the surface Allen-Cahn equation

1.
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China
2.
Institute of Mathematics and Physics, Xinjiang University, Urumqi 830046, China
3.
Departamento de Matemática, Universidade Federal do Paraná, Centro Politácnico, Curitiba 81531-980, PR, Brazil

* Corresponding author: Institute of Mathematics and Physics, Xinjiang University, Urumqi 830046, P.R. China
* Corresponding author: Institute of Mathematics and Physics, Xinjiang University, Urumqi 830046, P.R. China

Received: May 31, 2016

Revised: February 28, 2017

Published: October 2017

The first author is supported by the Excellent Doctor Innovation Program of Xinjiang University (No. XJUBSCX-2016006) and the Graduate Student Research Innovation Program of Xinjiang (No. XJGRI2015009). The second author is supported by the NSF of Xinjiang Province (No.2016D01C058), NCET-13-0988, and the NSF of China (No. 11671345,11271313). The third author is supported by CAPES (No. 88881.068004/2014.01) and CNPq (No. 300326/2012-2,470934/2013-1, INCT-Matemática) of Brazil.

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Abstract

Two semi-implicit numerical methods are proposed for solving the surface Allen-Cahn equation which is a general mathematical model to describe phase separation on general surfaces. The spatial discretization is based on surface finite element method while the temporal discretization methods are first-and second-order stabilized semi-implicit schemes to guarantee the energy decay. The stability analysis and error estimate are provided for the stabilized semi-implicit schemes. Furthermore, the first-and second-order operator splitting methods are presented to compare with stabilized semi-implicit schemes. Some numerical experiments including phase separation and mean curvature flow on surfaces are performed to illustrate stability and accuracy of these methods.

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Mathematics Subject Classification: Primary:65M60, 76T99;Secondary:65M12.

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Figure 1. Simulation of phase separation on sphere by SSI1 with $\delta t=5\times10^{-4}$.

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Figure 2. Simulation of phase separation on sphere by SSI1 with $\delta t=10^{-4}$.

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Figure 3. Simulation of phase separation on sphere by SSI2 with $\delta t=5\times10^{-4}$.

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Figure 4. Simulation of phase separation on sphere by SSI2 with $\delta t=10^{-4}$.

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Figure 5. Non-dimensional discrete total energy line of SSI1 (a) and SSI2 (b) with $\delta t=5\times10^{-4}$(blue) and $\delta t=10^{-4}$(red) on sphere.

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Figure 6. Simulation of phase separation on torus by OS1 with $\delta t=5\times10^{-4}$.

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Figure 7. Simulation of phase separation on torus by OS1 with $\delta t=10^{-4}$.

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Figure 8. Non-dimensional discrete total energy curves of OS1 with different with $\delta t=5\times10^{-4}$(blue) and $\delta t=10^{-4}$(red) (a). And the side view of the solution of OS1 at t=0.05 with $\delta t=5\times10^{-4}$ on torus (b).

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Figure 9. Solutions of phase separation on torus by OS2 with different $\delta t$.

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Figure 10. Comparison between the SSI1, SSI2, OS1 and OS2 for the simulation of mean curvature flow on sphere with $\delta t=5\times10^{-4}$ (a) and $\delta t=10^{-4}$ (b).

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Figure 11. Simulation of motion of a circle on sphere by SSI2 with $\delta t=10^{-4}$.

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Figure 12. Comparison between the SSI1, SSI2, OS1 and OS2 for the simulation of mean curvature flow on hyperboloid with $\delta t=5\times10^{-4}$ (a) and $\delta t=10^{-4}$ (b).

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Figure 13. Simulation of motion of a circle on hyperboloid by OS1 with $\delta t=10^{-4}$.

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