  This paper aims to investigate numerical approximation of a general second
order non-autonomous semilinear parabolic stochastic partial differential
equation (SPDE) driven by multiplicative noise. Numerical approximations of
autonomous SPDEs are thoroughly investigated in the literature, while the
non-autonomous case is not yet understood. We discretize the non-autonomous
SPDE driven by multiplicative noise by the finite element method in space and
the Magnus-type integrator in time. We provide a strong convergence proof of
the fully discrete scheme toward the mild solution in the root-mean-square
$L^2$ norm. The result reveals how the convergence orders in both space and
time depend on the regularity of the noise and the initial data. In particular,
for multiplicative trace class noise we achieve convergence order
$\mathcal{O}\left(h^2\left(1+\max(0,\ln\left(t_m/h^2\right)\right)+\Delta
t^{1/2}\right)$.
  Numerical simulations to illustrate our theoretical finding are provided.
