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Robust a posteriori error analysis of the stochastic Cahn-Hilliard equation with rough noise

Lubomir Banas, Jean Daniel Mukam

TL;DR

This work derives robust, fully computable a posteriori error estimates for a fully discrete adaptive finite element approximation of the stochastic Cahn–Hilliard equation with rough noise by regularizing space-time white noise and splitting the solution into a linear SPDE part and a random PDE part. A refined linear transformation and subspace-based probabilistic analysis yield error bounds that are robust with respect to the interfacial width parameter and noise regularization. The authors develop an adaptive algorithm, prove pathwise and probabilistic error estimates, and validate the theory through numerical experiments that reveal interface dynamics and eigenvalue behavior under stochastic forcing. The framework enables reliable, mesh-adaptive simulations of stochastic CH phenomena in low-regularity settings, with practical implications for uncertainty quantification in phase-field models.

Abstract

We derive a posteriori error estimate for a fully discrete adaptive finite element approximation of the stochastic Cahn-Hilliard equation with rough noise. The considered model is derived from the stochastic Cahn-Hilliard equation with additive space-time white noise through suitable spatial regularization of the white noise. The a posteriori estimate is robust with respect to the interfacial width parameter as well as the noise regularization parameter. We propose a practical adaptive algorithm for the considered problem and perform numerical simulations to illustrate the theoretical findings.

Robust a posteriori error analysis of the stochastic Cahn-Hilliard equation with rough noise

TL;DR

This work derives robust, fully computable a posteriori error estimates for a fully discrete adaptive finite element approximation of the stochastic Cahn–Hilliard equation with rough noise by regularizing space-time white noise and splitting the solution into a linear SPDE part and a random PDE part. A refined linear transformation and subspace-based probabilistic analysis yield error bounds that are robust with respect to the interfacial width parameter and noise regularization. The authors develop an adaptive algorithm, prove pathwise and probabilistic error estimates, and validate the theory through numerical experiments that reveal interface dynamics and eigenvalue behavior under stochastic forcing. The framework enables reliable, mesh-adaptive simulations of stochastic CH phenomena in low-regularity settings, with practical implications for uncertainty quantification in phase-field models.

Abstract

We derive a posteriori error estimate for a fully discrete adaptive finite element approximation of the stochastic Cahn-Hilliard equation with rough noise. The considered model is derived from the stochastic Cahn-Hilliard equation with additive space-time white noise through suitable spatial regularization of the white noise. The a posteriori estimate is robust with respect to the interfacial width parameter as well as the noise regularization parameter. We propose a practical adaptive algorithm for the considered problem and perform numerical simulations to illustrate the theoretical findings.

Paper Structure

This paper contains 10 sections, 25 theorems, 232 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The regularized stochastic Cahn-Hilliard equation
  4. Fully discrete numerical approximation
  5. Error estimate for the linear SPDE
  6. Error estimate for the random PDE
  7. Error estimate for the approximation of the stochastic Cahn-Hilliard equation
  8. Numerical experiments
  9. Regularity estimates of the solution to the stochastic Cahn-Hilliard equation and some useful inequalities
  10. Rate of convergence of the backward Euler method for linear stochastic Cahn-Hilliard equation with rough noise

Key Result

Lemma 5.1

Suppose that $\mathbb{V}_{\tilde{h}}\subset\mathbb{V}^n_h$ for all $n=1,\cdots,N$. Then it holds that:

Figures (5)

  • Figure 1: Numerical solution at time $t=0,0.0065,0.009,0.0095,0.0097,0.012$.
  • Figure 2: Finite element mesh at time $t=0,0.0065,0.009,0.0095,0.0097,0.012$.
  • Figure 3: Evolution of the discrete principal eigenvalue for different realizations of the noise for $\tilde{h}=1/16$ (left) and for $\tilde{h}=1/32$ (right)
  • Figure 4: Histogram of the peak-times of the principal eigenvalue for $\tilde{h}=1/16$, $\tilde{h}=1/32$ and the evolution of the (scaled) principal eigenvalue of the deterministic problem.
  • Figure 5: Evolution of the expected value of the discrete energy (left) and of the principal eigenvalue (right) for $\tilde{h}=1/16$, $\tilde{h}=1/32$ and for the deterministic problem.

Theorems & Definitions (47)

  • Remark 3.1
  • Remark 3.2
  • Lemma 5.1
  • Lemma 5.2
  • proof
  • Lemma 5.3
  • Remark 5.1
  • Lemma 5.4
  • proof
  • Proposition 5.1
  • ...and 37 more