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 {\it Z. Liu and J. Shen, J. Sci. Comput., 109:Paper No. 61, 26, 2026} 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 of the numerical invariant measure to the exact one in the Wasserstein-2 distance. Finally, we present numerical experiments that verify the ergodicity, sharpness, and time-independence of the strong convergence rates for this tamed-FEM.
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BibTeX
@misc{cai2026uniformintime,
title = {{Uniform-in-time Strong Error Estimates of Tamed-FEM to Superlinear SPDEs driven by Multiplicative Noise}},
author = {Jingjing Cai and Zhihui Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.09173},
eprint = {2606.09173},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.09173},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Uniform-in-time Strong Error Estimates of Tamed-FEM to Superlinear SPDEs driven by Multiplicative Noise
AU - Jingjing Cai
AU - Zhihui Liu
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2606.09173
UR - https://arxiv.org/abs/2606.09173
AB - 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 {\it Z. Liu and J. Shen, J. Sci. Comput., 109:Paper No. 61, 26, 2026} 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 of the numerical invariant measure to the exact one in the Wasserstein-2 distance. Finally, we present numerical experiments that verify the ergodicity, sharpness, and time-independence of the strong convergence rates for this tamed-FEM.
ER -