Fast algorithm based on TT-M FE method for modified Cahn–Hilliard equation

Abstract

We consider numerical methods for solving the modified Cahn–Hilliard equation involving strong nonlinearities. A fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme is proposed to overcome the time-consuming computation of the nonlinear terms. The TT-M FE algorithm includes three main steps: Firstly, a nonlinear FE scheme is solved on a coarse time-mesh ${\tau }_c$. Here, the FE method is used for spatial discretization and the implicit second-order $𝜃$ scheme (containing both implicit Crank–Nicolson scheme and second-order backward difference method) is used for temporal discretization. Secondly, the Lagrange’s interpolation is used to obtain the interpolation result on the fine time-mesh. Finally, a linearized FE system is solved on a fine time-mesh $\tau$ $\left(\tau <{\tau }_c\right)$. The stability analysis and priori error estimates are provided in detail. Numerical examples are given to demonstrate the validity of the proposed scheme. The TT-M FE method is compared with the traditional Galerkin FE method and it is evident that the TT-M FE method can save the calculation time.