Article Contents
Article Contents

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

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

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

 Citation:

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

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

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

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

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.

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

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

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).

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

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).

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

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).

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

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