  This paper aims to investigate the numerical approximation of semilinear
non-autonomous stochastic partial differential equations (SPDEs) driven by
multiplicative or additive noise. Such equations are more realistic than
autonomous SPDEs while modeling real world phenomena. Numerical approximations
for autonomous SPDEs are thoroughly investigated in the literature, while the
non-autonomous case is not yet well understood. The non-autonomous SPDE is
discretized in space by the finite element method and in time by the linear
implicit Euler method. We break the complexity in the analysis of the time
depending, not necessarily self-adjoint linear operators with the corresponding
semi group and provide the strong convergence result of the fully discrete
scheme toward the exact solution in the root-mean-square $L^2$ norm. The
results indicate how the converge order depends on the regularity of the
initial solution and the noise. In particular, for multiplicative trace class
noise we achieve convergence order $\mathcal{O}(h^{2-\epsilon}+\Delta t^{1/2})$
and for additive noise with trace class, we achieve convergence order
$\mathcal{O}(h^{2-\epsilon}+\Delta t^{1-\epsilon})$, for an arbitrarily small
$\epsilon>0$. Numerical experiments to sustain our theoretical results are
provided.
