Strong convergence rates for full-discrete approximations of the stochastic Allen-Cahn equations on 2D torus

In this paper we construct space-time full discretizations of stochastic Allen-Cahn equations driven by space-time white noise on 2D torus. The approximations are implemented by tamed exponential Euler discretization in time and spectral Galerkin method in space. We finally obtain the convergence rates with the spatial order of $\alpha -\delta$ and the temporal order of $\alpha \mathrm{/}6-\delta$ in ${\mathcal{C}}^{-\alpha }$ for $\alpha \in (0,1\mathrm{/}3)$ and $\delta >0$ arbitrarily small.