Title: Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen-Cahn Equation with multiplicative noise,
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 $[0,T]$.