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 $[0,T]$.

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Cited by in corpus (8)

• Optimal error estimates of Galerkin finite element methods for stochastic Allen-Cahn equation with additive noise
• Strong convergence rates for explicit space-time discrete numerical approximations of stochastic Allen-Cahn equations
• Strong Approximation of Monotone Stochastic Partial Differential Equations Driven by Multiplicative Noise
• Numerical approximation of the Stochastic Cahn-Hilliard Equation near the Sharp Interface Limit
• On the Itô-Alekseev-Gröbner formula for stochastic differential equations
• Numerical Ergodicity and Uniform Estimate of Monotone SPDEs Driven by Multiplicative Noise
• Strong error estimates for a fully discrete SAV scheme for the stochastic Allen--Cahn equation with multiplicative noise
• A convergent stochastic scalar auxiliary variable method