  Strong approximation errors of both finite element semi-discretization and
spatio-temporal full discretization are analyzed for the stochastic Allen-Cahn
equation driven by additive noise in space dimension $d \leq 3$. The full
discretization is realized by combining the standard finite element method with
the backward Euler time-stepping scheme. Distinct from the globally Lipschitz
setting, the error analysis becomes rather challenging and demanding, due to
the presence of the cubic nonlinearity in the underlying model. By introducing
two auxiliary approximation processes, we propose an appropriate decomposition
of the considered error terms and introduce a novel approach of error analysis,
to successfully recover the convergence rates of the numerical schemes. The
approach is original and does not rely on high-order spatial regularity
properties of the approximation processes. It is shown that the fully discrete
scheme possesses convergence rates of order $ O(h^{\gamma} ) $ in space and
order $ O( \tau^{ \frac{\gamma}{2} } )$ in time, subject to the spatial
correlation of the noise process, characterized by $
\|A^{\frac{\gamma-1}2}Q^{\frac12}\|_{\mathcal{L}_2}<\infty, \, \gamma \in[\frac
d3,2] $, $ d\in\{1,2,3\}$. In particular, a classical convergence rate of order
$O(h^2 +\tau)$ is reachable, even in multiple space dimensions, when the
aforementioned condition is fulfilled with $ \gamma = 2 $. Numerical examples
confirm the previous findings.
