[Paper Review] Finite Element Methods for the Stochastic Allen-Cahn Equation with Gradient-type Multiplicative Noises

Xiaobing Feng, Yukun Li+1 more|arXiv (Cornell University)|May 14, 2015

Solidification and crystal growth phenomena3 references3 citations

TL;DR

This paper proposes two fully discrete finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noise, using different time-stepping strategies for the nonlinear term. It establishes strong convergence with sharp error rates by proving Hölder continuity in time for the solution's $L^2$ and $H^1$ norms, along with bounded high moments, and validates the methods through numerical experiments showing convergence to stochastic mean curvature flow.

ABSTRACT

This paper studies finite element approximations of the stochastic Allen-Cahn equation with gradient-type multiplicative noises that are white in time and correlated in space. The sharp interface limit as the parameter $ε ightarrow 0$ of the stochastic equation formally approximates a stochastic mean curvature flow which is described by a stochastically perturbed geometric law of the deterministic mean curvature flow. Both the stochastic Allen-Cahn equation and the stochastic mean curvature flow arise from materials science, fluid mechanics and cell biology applications. Two fully discrete finite element methods which are based on different time-stepping strategies for the nonlinear term are proposed. Strong convergence with sharp rates for both fully discrete finite element methods is proved. This is done with a crucial help of the Hölder continuity in time with respect to the spatial $L^2$-norm and $H^1$-seminorm for the strong solution of the stochastic Allen-Cahn equation, which are key technical lemmas proved in paper. It also relies on the fact that high moments of the strong solution are bounded in various spatial and temporal norms. Numerical experiments are provided to gauge the performance of the proposed fully discrete finite element methods and to study the interplay of the geometric evolution and gradient-type noises.

Motivation & Objective

• To develop robust numerical schemes for the stochastic Allen-Cahn equation with gradient-type multiplicative noise, which models phase transitions under stochastic perturbations.
• To establish strong convergence with optimal error rates for fully discrete finite element methods, accounting for the nonlinear and stochastic nature of the equation.
• To rigorously analyze the temporal regularity of the solution via Hölder continuity in time for $L^2$ and $H^1$ norms, a key technical contribution.
• To validate the theoretical findings through numerical experiments that demonstrate convergence to the stochastic mean curvature flow as $\epsilon \to 0$.
• To study the interplay between geometric evolution (mean curvature flow) and gradient-type noise in phase-field models.

Proposed method

• Two fully discrete finite element methods are proposed, differing in their time discretization strategy for the nonlinear term $f(u) = u^3 - u$.
• The methods use implicit time-stepping with Crank-Nicolson or backward Euler schemes, combined with conforming $P_1$ finite elements in space.
• The analysis relies on proving Hölder continuity in time of the strong solution with respect to $L^2$ and $H^1$ norms, which is essential for error estimation.
• High-order moment bounds of the solution in various norms are established to control the stochastic terms and ensure convergence.
• The Itô form of the SPDE is used, with the noise term $\delta \nabla u \cdot X \circ dW(t)$ transformed into Itô form via the correction term $\frac{\delta^2}{2} \nabla(\nabla u \cdot X) \cdot X$.
• A priori estimates and discrete energy methods are applied to derive optimal convergence rates in probability and expectation.

Experimental results

Research questions

• RQ1 What is the optimal convergence rate of fully discrete finite element methods for the stochastic Allen-Cahn equation with gradient-type multiplicative noise?
• RQ2 How does the Hölder continuity in time of the solution’s $L^2$ and $H^1$ norms affect the error analysis and convergence rates?
• RQ3 Can the numerical methods accurately capture the transition from the stochastic Allen-Cahn equation to the stochastic mean curvature flow as $\epsilon \to 0$?
• RQ4 How do the noise intensity $\delta$ and the interface thickness $\epsilon$ influence the evolution of the zero-level set in the numerical solution?
• RQ5 What is the role of bounded high moments of the solution in ensuring the stability and convergence of the finite element schemes?

Key findings

• The paper proves strong convergence with sharp rates for both fully discrete finite element methods, with convergence order $O(h^2 + \tau)$ in the $L^2$ norm and $O(h + \tau^{1/2})$ in the $H^1$ seminorm.
• Hölder continuity in time of the solution with exponent $1/2$ is established for both $L^2$ and $H^1$ norms, a crucial technical lemma for the error analysis.
• High moments of the solution are uniformly bounded in time and space, enabling control of the stochastic terms in the error estimates.
• Numerical experiments confirm the theoretical convergence rates and show that the zero-level set of the expected solution evolves toward the stochastic mean curvature flow as $\epsilon \to 0$.
• Larger noise intensity $\delta$ accelerates interface evolution and induces more pronounced shape changes, consistent with the stochastic geometric model.
• For fixed $\delta$, the interface evolves faster with larger $\epsilon$, and the zero-level set converges to the stochastic mean curvature flow as $\epsilon$ decreases.