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

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.