# Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen-Cahn Equation with multiplicative noise

## TL;DR

This paper establishes optimal strong convergence rates for a space-time discretization of the stochastic Allen-Cahn equation with multiplicative noise, using weak monotonicity to derive uniform bounds and error estimates.

## Contribution

It introduces a novel approach leveraging weak monotonicity to obtain optimal convergence rates for discretizations of the stochastic Allen-Cahn equation.

## Key findings

- Achieves an error bound of order $k^{1- ext{delta}} + h^2$.
- Provides uniform strong norm bounds for solutions.
- Demonstrates structure-preserving finite element discretization.

## Abstract

The stochastic Allen-Cahn equation with multiplicative noise involves the nonlinear drift operator ${\mathscr A}(x) = \Delta x - \bigl(\vert x\vert^2 -1\bigr)x$. We use the fact that ${\mathscr A}(x) = -{\mathcal J}^{\prime}(x)$ satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate $ \underset{1 \leq j \leq J}\sup {\mathbb E}\bigl[ \Vert X_{t_j} - Y^j\Vert_{{\mathbb L}^2}^2\bigr] \leq C_{\delta}(k^{1-\delta} + h^2)$ for all small $\delta>0$, where $X$ is the strong variational solution of the stochastic Allen-Cahn equation, while $\big\{Y^j:0\le j\le J\big\}$ solves a structure preserving finite element based space-time discretization of the problem on a temporal mesh $\{ t_j;\, 1 \leq j \leq J\}$ of size $k>0$ which covers $[0,T]$.