Strong convergence for efficient full discretization of the stochastic Allen–Cahn equation with multiplicative noise

In this paper, we study the strong convergence of the full discretization based on a semi-implicit tamed approach in time and the finite element method with truncated noise in space for the stochastic Allen–Cahn equation driven by multiplicative noise. The proposed fully discrete scheme is efficient thanks to its low computational complexity and mean-square unconditional stability. The low regularity of the solution due to the multiplicative infinite-dimensional driving noise and the non-global Lipschitz difficulty introduced by the cubic nonlinear drift term make the strong convergence analysis of the fully discrete solution considerably complicated. By constructing an appropriate auxiliary procedure, the full discretization error can be cleverly decomposed, and the spatio-temporal strong convergence order is successfully derived under certain weak assumptions. Numerical experiments are finally reported to validate the theoretical result. • The time discretization scheme is mean-square unconditionally stable and only requires solving a second-order linear elliptic equation at each time step, which means that the computational cost required is lower than the popular backward Euler method (Qi et al., 2023; Majee and Prohl, 2018). • The existence of cubic nonlinear drift term in the underlying model causes intrinsic difficulties to the error estimation, and the strong convergence analysis becomes much more complicated than that of SPDEs satisfying globally Lipschitz setting. The main theoretical result is the proof of the strong convergence rate $\mathcal{O}(\Delta t̂{{1}/{2}}+ĥ{2-\varepsilon_0})$ of the spatio-temporal full discretization under certain weak assumptions, where $\varepsilon_0$ is an infinitesimal positive number, $h$ and $\Delta t$ are respectively the spatial and temporal mesh sizes.