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 $\mathcal{A}(x)=\Delta x-(|x{|}^2-1)x$. We use the fact that $\mathcal{A}(x)=-{\mathcal{J}}^{\prime }(x)$ 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 $\underset{1\le j\le J}{\sup }\mathbb{E}[\Vert X_{t_j}-Y^j{\Vert }_{{\mathbb{L}}^2}^2]\le C_{\delta }(k^{1-\delta }+h^2)$ for all small $\delta >0$, where $X$ is the strong variational solution of the stochastic Allen-Cahn equation, while $\{Y^j:0\le j\le J\}$ solves a structure preserving finite element based space-time discretization of the problem on a temporal mesh $\{t_j;1\le j\le J\}$ of size $k>0$ which covers $[0,T]$.

References in corpus (1)

• Rate of Convergence of Space Time Approximations for stochastic evolution equations