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

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 L-2-error bound and the L-infinity/H-2-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 epsilon(-1) instead of e(T)/epsilon(2) for both of the semi and fully-discrete schemes are derived. Numerical experiment also validates our theoretical convergence analysis.