Error estimates with low-order polynomial dependence for the fully-discrete finite element invariant energy quadratization scheme of the Allen–Cahn equation

Abstract

In this paper, for the Allen–Cahn equation, we obtain the error estimate of the temporal semi-discrete scheme, and the fully-discrete finite element numerical scheme, both of which are based on the invariant energy quadratization (IEQ) time-marching strategy. We establish the relationship between the [Formula: see text]-error bound and the [Formula: see text]-stabilities of the numerical solution. Then, by converting the numerical schemes to a form compatible with the original format of the Allen–Cahn equation, using mathematical induction, the superconvergence property of nonlinear terms, and the spectrum argument, the optimal error estimates that only depends on the low-order polynomial degree of [Formula: see text] instead of [Formula: see text] for both of the semi and fully-discrete schemes are derived. Numerical experiment also validates our theoretical convergence analysis.