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This paper deals with the backward Euler method applied to semilinearparabolic stochastic partial differential equations (SPDEs) driven by additivenoise. The SPDE is discretized in space by the finite element method and intime by the backward Euler. We consider a larger class of nonlinear driftfunctions, which are of Nemytskii type and polynomial of any odd degree withnegative leading term, instead of only dealing with the special case ofstochastic Allen-Chan equation as in the up to date literature.Moreover ourlinear operator is of second order and not necessarily self-adjoint, thereforemakes estimates more challenging than in the case of self-adjoint operator. Weprove the strong convergence of our fully discrete schemes toward the mildsolution and results indicate how the convergence rates depend on theregularities of the initial data and the noise. In particular, for trace classnoise, we achieve convergence order $\mathcal{O} (h^{2}+\Deltat^{1-\epsilon})$, where $\epsilon>0$ is positive number, small enough. We alsoprovide numerical experiments to illustrate our theoretical results.

Strong convergence of the backward Euler approximation for the finite  element discretization of semilinear parabolic SPDEs with non-global  Lipschitz drift driven by additive noise