A SCR-based error estimation and adaptive finite element method for the Allen–Cahn equation

In this paper, we consider the adaptive finite element method for the Allen–Cahn equation. The adaptive method is based on a second order accurate unconditionally energy stable finite element scheme and a recovery-type a posteriori error estimator. A SCR-based a posteriori error estimation is derived to control the mesh refinement and coarsening. A time–space adaptive algorithm is proposed for numerical approximation of the Allen–Cahn equation. Numerical experiments are presented to illustrate the reliability and efficiency of the proposed SCR-based error estimator and the corresponding adaptive algorithm. The extension of the proposed adaptive algorithm to the Cahn–Hilliard equation is also discussed.