  The scientific literature contains a number of numerical approximation
results for stochastic partial differential equations (SPDEs) with
superlinearly growing nonlinearities but, to the best of our knowledge, none of
them prove strong or weak convergence rates for full-discrete numerical
approximations of space-time white noise driven SPDEs with superlinearly
growing nonlinearities. In particular, in the scientific literature there
exists neither a result which proves strong convergence rates nor a result
which proves weak convergence rates for full-discrete numerical approximations
of stochastic Allen-Cahn equations. In this article we bridge this gap and
establish strong convergence rates for full-discrete numerical approximations
of space-time white noise driven SPDEs with superlinearly growing
nonlinearities such as stochastic Allen-Cahn equations. Moreover, we also
establish lower bounds for strong temporal and spatial approximation errors
which demonstrate that our strong convergence rates are essentially sharp and
can, in general, not be improved.
