Strong convergence of the linear implicit Euler method for the finite element discretization of semilinear SPDEs driven by multiplicative or additive noise

This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative or additive noise. The linear operator is not necessary self-adjoint, so more useful in concrete applications. The SPDE is discretized in space by the finite element method and in time by the linear implicit Euler method. The corresponding scheme is more stable and efficient to solve stochastic advection-dominated reactive transport in porous media. This extends the current results in the literature to not necessary self-adjoint operator. As a challenge, key part of the proof does not rely anymore on the spectral decomposition of the linear operator. The results reveal how the convergence orders depend on the regularity of the noise and the initial data. In particular for multiplicative trace class noise, we achieve optimal convergence order O(h2+Δt1/2) and for additive trace class noise, we achieve optimal convergence order in space and sup-optimal convergence order in time of the form O(h2+Δt1−ϵ), for an arbitrarily small ϵ > 0. Numerical experiments to sustain our theoretical results are provided.