The stabilized semi-implicit finite element method for the surface Allen-Cahn equation
Xufeng Xiao Xinlong Feng Jinyun Yuan
Discrete & Continuous Dynamical Systems - B 2017, 22(7): 2857-2877 doi: 10.3934/dcdsb.2017154

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.

keywords: Surface Allen-Cahn equation surface finite element method stabilized semi-implicit scheme operator splitting method error estimate

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