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This paper aims to investigate the numerical approximation of semilinearnon-autonomous stochastic partial differential equations (SPDEs) driven bymultiplicative or additive noise. Such equations are more realistic thanautonomous SPDEs while modeling real world phenomena. Numerical approximationsfor autonomous SPDEs are thoroughly investigated in the literature, while thenon-autonomous case is not yet well understood. The non-autonomous SPDE isdiscretized in space by the finite element method and in time by the linearimplicit Euler method. We break the complexity in the analysis of the timedepending, not necessarily self-adjoint linear operators with the correspondingsemi group and provide the strong convergence result of the fully discretescheme toward the exact solution in the root-mean-square $L^2$ norm. Theresults indicate how the converge order depends on the regularity of theinitial solution and the noise. In particular, for multiplicative trace classnoise 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 areprovided.

Strong Convergence of the Linear Implicit Euler Method for the Finite  Element Discretization of Semilinear non-Autonomous SPDEs Driven by  Multiplicative or Additive Noise