Implicit scheme for semilinear SPDEs with additive noise under nonglobally Lipschitz and irregular drift

We consider the standard finite element method combined with the implicit Euler scheme to approximate second-order semilinear parabolic stochastic partial differential equations (SPDEs) with additive noise. The nonlinearity is of Nemytskii type, satisfies a one-sided Lipschitz condition, exhibits polynomial growth and includes irregular components. Such SPDEs serve as suitable models for various phenomena, including advection–reaction–diffusion processes in a convex polyhedral domain ⊂ Rᵈ (d∈ \1,2, 3\). We prove strong convergence of the fully discrete scheme towards the mild solution, achieving an almost first-order temporal rate. We obtain a spatial convergence rate that, in the case $d=3$, depends on the spatial dimension and the polynomial growth order of the nonlinearity. The analysis is challenging because of the irregularities of the nonlinear drift function and the absence of a global Lipschitz condition. Numerical experiments are provided to illustrate the theoretical results.