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Weak Convergence Analysis for the Finite Element Approximation to Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise

Minxing Zhang, Yongkui Zou, Ran Zhang, Yanzhao Cao

TL;DR

The paper studies weak convergence of finite element discretizations for the stochastic Allen–Cahn equation driven by multiplicative noise. It develops a two-step framework: (i) a temporal splitting with an auxiliary SPDE that yields a strong in-time rate of $O(\tau)$ between the exact and auxiliary solutions, and (ii) a Kolmogorov–Malliavin calculus-based weak error analysis to obtain a near-$O(h)$ convergence for the spatial FE discretization. The authors establish uniform regularity estimates for the Kolmogorov solution and its derivatives, derive Malliavin-regularity for the FE scheme, and prove that the weak error decays as $O(h^{1-\varepsilon})$ for any $\varepsilon\in(0,1)$ when $\tau\to0$, with numerical experiments corroborating the theory. This work extends weak-error analysis to multiplicative-noise SPDEs with one-sided Lipschitz nonlinearities and provides a practical pathway for accurate distributional approximations in stochastic phase-field models.

Abstract

In this paper, we aim to study the optimal weak convergence order for the finite element approximation to a stochastic Allen-Cahn equation driven by multiplicative white noise. We first construct an auxiliary equation based on the splitting-up technique and derive prior estimates for the corresponding Kolmogorov equation and obtain the strong convergence order of 1 in time between the auxiliary and exact solutions. Then, we prove the optimal weak convergence order of the finite element approximation to the stochastic Allen-Cahn equation by deriving the weak convergence order between the finite element approximation and the auxiliary solution via the theory of Kolmogorov equation and Malliavin calculus. Finally, we present a numerical experiment to illustrate the theoretical analysis.

Weak Convergence Analysis for the Finite Element Approximation to Stochastic Allen-Cahn Equation Driven by Multiplicative White Noise

TL;DR

The paper studies weak convergence of finite element discretizations for the stochastic Allen–Cahn equation driven by multiplicative noise. It develops a two-step framework: (i) a temporal splitting with an auxiliary SPDE that yields a strong in-time rate of between the exact and auxiliary solutions, and (ii) a Kolmogorov–Malliavin calculus-based weak error analysis to obtain a near- convergence for the spatial FE discretization. The authors establish uniform regularity estimates for the Kolmogorov solution and its derivatives, derive Malliavin-regularity for the FE scheme, and prove that the weak error decays as for any when , with numerical experiments corroborating the theory. This work extends weak-error analysis to multiplicative-noise SPDEs with one-sided Lipschitz nonlinearities and provides a practical pathway for accurate distributional approximations in stochastic phase-field models.

Abstract

In this paper, we aim to study the optimal weak convergence order for the finite element approximation to a stochastic Allen-Cahn equation driven by multiplicative white noise. We first construct an auxiliary equation based on the splitting-up technique and derive prior estimates for the corresponding Kolmogorov equation and obtain the strong convergence order of 1 in time between the auxiliary and exact solutions. Then, we prove the optimal weak convergence order of the finite element approximation to the stochastic Allen-Cahn equation by deriving the weak convergence order between the finite element approximation and the auxiliary solution via the theory of Kolmogorov equation and Malliavin calculus. Finally, we present a numerical experiment to illustrate the theoretical analysis.

Paper Structure

This paper contains 14 sections, 12 theorems, 137 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some spaces and properties
  4. Stochastic Allen-Cahn equation and assumptions
  5. Malliavin calculus
  6. Auxiliary equation and regularity estimates
  7. Construction of auxiliary equation and strong convergence rate
  8. Kolmogorov equation and a priori estimates
  9. Finite element approximation and regularity estimates
  10. Semi-disretized finite element approximation
  11. Regularity estimate
  12. Weak convergence analysis for spatial semi-discretization
  13. Numerical experiments
  14. Conclusion and Discussion

Key Result

Lemma 1

For any $u\in \dot{H}^{-\frac{1}{2}-\delta}$ and $v,w \in H$, there exists a constant $C>0$, such that Moreover, for any $u \in \dot{H}^1$, there exists a constant $C>0$, such that

Figures (1)

  • Figure 1: Convergence order in $\log-\log$ scale

Theorems & Definitions (21)

  • Remark 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • ...and 11 more