  For the numerical solution of the cubic nonlinear Schr\"{o}dinger equation
with periodic boundary conditions, a pseudospectral method in space combined
with a filtered Lie splitting scheme in time is considered. This scheme is
shown to converge even for initial data with very low regularity. In
particular, for data in $H^s(\mathbb T^2)$, where $s>0$, convergence of order
$\mathcal O(\tau^{s/2}+N^{-s})$ is proved in $L^2$. Here $\tau$ denotes the
time step size and $N$ the number of Fourier modes considered. The proof of
this result is carried out in an abstract framework of discrete Bourgain
spaces, the final convergence result, however, is given in $L^2$. The stated
convergence behavior is illustrated by several numerical examples.
