Uniform-in-time Strong Error Estimates of Tamed-FEM to Superlinear SPDEs driven by Multiplicative Noise

Abstract

We establish sharp, uniform-in-time strong error estimates for a nonlinearity-explicit tamed finite element method (FEM) applied to a class of superlinear stochastic partial differential equations (SPDEs) driven by multiplicative noise, including the stochastic Allen--Cahn equation with a moderately thick interface. This tamed-FEM was first introduced in [Z. Liu and J. Shen, arXiv:2502.19117] to ensure long-time unconditional stability and to preserve the Lyapunov structure of this class of SPDEs. We further prove that the scheme is exponentially ergodic and derive the convergence rate between the exact invariant measure and its numerical counterpart in the Wasserstein-2 distance. Finally, we present numerical experiments that verify the ergodicity as well as the sharpness and time-independence of the strong convergence rates for this tamed-FEM.