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This paper aims to investigate the numerical approximation of a generalsecond order parabolic stochastic partial differential equation(SPDE) driven bymultiplicative and additive noise under more relaxed conditions. The SPDE isdiscretized in space by the finite element method and in time by the linearimplicit Euler method. This extends the current results in the literature tonot necessary self-adjoint operator with more general boundary conditions. As aconsequence key part of the proof does not rely on the spectral decompositionof the linear operator. We achieve optimal convergence orders which depend onthe regularity of the noise and the initial data. In particular, formultiplicative 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 convergenceorders are achieved for additive noise by incorporating further regularityassumptions on the nonlinear drift function, our optimal convergence orders areobtained under only the standard Lipschitz condition of the nonlinear driftterm. Numerical experiments to sustain our theoretical results are provided.

Strong Convergence of the Linear Implicit Euler Method for the Finite  Element Discretization of Semilinear SPDEs Driven by Multiplicative and  Additive Noise