  This paper deals with the backward Euler method applied to semilinear
parabolic stochastic partial differential equations (SPDEs) driven by additive
noise. The SPDE is discretized in space by the finite element method and in
time by the backward Euler. We consider a larger class of nonlinear drift
functions, which are of Nemytskii type and polynomial of any odd degree with
negative leading term, instead of only dealing with the special case of
stochastic Allen-Chan equation as in the up to date literature.Moreover our
linear operator is of second order and not necessarily self-adjoint, therefore
makes estimates more challenging than in the case of self-adjoint operator. We
prove the strong convergence of our fully discrete schemes toward the mild
solution and results indicate how the convergence rates depend on the
regularities of the initial data and the noise. In particular, for trace class
noise, we achieve convergence order $\mathcal{O} (h^{2}+\Delta
t^{1-\epsilon})$, where $\epsilon>0$ is positive number, small enough. We also
provide numerical experiments to illustrate our theoretical results.
