  This paper aims to investigate the numerical approximation of a general
second order parabolic stochastic partial differential equation(SPDE) driven by
multiplicative and additive noise under more relaxed conditions. The SPDE is
discretized in space by the finite element method and in time by the linear
implicit Euler method. This extends the current results in the literature to
not necessary self-adjoint operator with more general boundary conditions. As a
consequence key part of the proof does not rely on the spectral decomposition
of the linear operator. We achieve optimal convergence orders which depend on
the regularity of the noise and the initial data. In particular, for
multiplicative noise we achieve optimal order $\mathcal{O}(h^2+\Delta t^{1/2})$
and for additive noise, we achieve optimal order $\mathcal{O}(h^2+\Delta t)$.
In contrast to current work in the literature, where the optimal convergence
orders are achieved for additive noise by incorporating further regularity
assumptions on the nonlinear drift function, our optimal convergence orders are
obtained under only the standard Lipschitz condition of the nonlinear drift
term. Numerical experiments to sustain our theoretical results are provided.
