Optimal error estimates for the semi-discrete and fully-discrete finite element schemes of the Allen–Cahn equation

Abstract

In this paper, we study the numerical approximation of the Allen–Cahn(AC) equation. Firstly, based on the Lagrange multiplier strategy, we present an equivalent form of the AC equation. Secondly, we provide corresponding semi-discrete and fully-discrete finite element schemes which satisfy the energy law of dissipation. Thirdly, we derive optimal error estimates of the semi-discrete and fully-discrete schemes. That is, by eliminating the Lagrange multiplier of the above discrete schemes, we obtain equivalent equations with only the phase field variable, then we establish the $L^{\infty }$ bounds and $H^2$ bounds of the numerical solution using mathematical induction. Due to the spatial discretization of the fully-discrete scheme, we derive more necessary conclusions to make preparation for the optimal error estimation. With the use of the spectrum argument, the superconvergence property of nonlinear terms and the mathematical induction, we obtain error bounds that solely depend on low-order polynomial of ${\varepsilon }^{-1}$ rather than the exponential factor $e^{\frac{T}{{\varepsilon }^2}}$. Finally, we present some numerical experiments to validate our theoretical convergence analysis.