Abstract
In this article, we consider numerical methods for solving Allen–Cahn 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 nonlinear terms. The TT-M FE algorithm includes the following 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 backward difference scheme is used for temporal discretization. Secondly, the Lagrange’s interpolation is used to obtain the interpolation result on the fine grid. Finally, a linearized FE system is solved on a fine time mesh $$\tau (\tau < \tau _{c})$$. 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.
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