An efficient time adaptivity based on chemical potential for surface Cahn–Hilliard equation using finite element approximation

摘要

We present numerical simulations for the surface Cahn–Hilliard equation which describes phase separation phenomenon occurred 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 which guarantee the free energy decay. An efficient and parameter-free adaptive time-stepping strategy is proposed based on the numerical energy stability generated by stabilized semi-implicit scheme. The main idea is to use discrete chemical potential, the byproduct of stabilized semi-implicit scheme, to estimate the variation of numerical energy that is used as a indicator to update the time step, the operation avoid calculating numerical energy with Gauss integral in each time step and reduce the calculated cost. In addition, optimal error estimate of first-order stabilized semi-implicit scheme in the case of curved surface are provided. Finally, numerical experiments are presented to demonstrate the stability, accuracy and efficiency of the proposed algorithms.