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

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.