  This paper develops and analyzes a fully discrete finite element method for a
class of semilinear stochastic partial differential equations (SPDEs) with
multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is
assumed to be globally Lipschitz and the nonlinearity in the drift term is only
assumed to satisfy a one-side Lipschitz condition. The semilinear SPDEs
considered in this paper is a direct generalization of the SODEs considered in
[13]. There are several difficulties which need to be overcome for this
generalization. First, obviously the spatial discretization, which does not
appear in the SODE case, adds an extra layer of difficulty. It turns out a
special discretization must be designed to guarantee certain properties for the
numerical scheme and its stiffness matrix. In this paper we use a finite
element interpolation technique to discretize the nonlinear drift term. Second,
in order to prove the strong convergence of the proposed fully discrete finite
element method, stability estimates for higher order moments of the
$H^1$-seminorm of the numerical solution must be established, which are
difficult and delicate. A judicious combination of the properties of the drift
and diffusion terms and a nontrivial technique borrowed from [16] is used in
this paper to achieve the goal. Finally, stability estimates for the second and
higher order moments of the $L^2$-norm of the numerical solution is also
difficult to obtain due to the fact that the mass matrix may not be diagonally
dominant. This is done by utilizing the interpolation theory and the higher
moment estimates for the $H^1$-seminorm of the numerical solution. After
overcoming these difficulties, it is proved that the proposed fully discrete
finite element method is convergent in strong norms with nearly optimal rates
of convergence.
