# Numerical approximation of the Stochastic Cahn-Hilliard Equation near the Sharp Interface Limit

## TL;DR

This paper analyzes the numerical approximation of the stochastic Cahn-Hilliard equation near the sharp interface limit, establishing error estimates, convergence results, and computational evidence on how noise influences geometric evolution.

## Contribution

It provides strong error estimates and convergence analysis for a discretization of the stochastic Cahn-Hilliard equation, including the effects of noise strength on the sharp interface limit.

## Key findings

- Convergence to deterministic Hele-Shaw/Mullins-Sekerka problem for large noise exponent b3.
- Numerical evidence that noise strength b3 influences the limit behavior.
- Validation of theoretical results through computational simulations.

## Abstract

We consider the stochastic Cahn-Hilliard equation with additive noise term $\varepsilon^\gamma g\, \dot{W}$ ($\gamma >0$) that scales with the interfacial width parameter $\varepsilon$. We verify strong error estimates for a gradient flow structure-inheriting time-implicit discretization, where $\varepsilon^{-1}$ only enters polynomially; the proof is based on higher-moment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For $\gamma$ sufficiently large, convergence in probability of iterates towards the deterministic Hele-Shaw/Mullins-Sekerka problem in the sharp-interface limit $\varepsilon \rightarrow 0$ is shown. These convergence results are partly generalized to a fully discrete finite element based discretization. We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its 'strength' $\gamma$) on the geometric evolution in the sharp-interface limit. For this purpose we compare the simulations with those from a fully discrete finite element numerical scheme for the (stochastic) Mullins-Sekerka problem. The computational results indicate that the limit for $\gamma\geq 1$ is the deterministic problem, and for $\gamma=0$ we obtain agreement with a (new) stochastic version of the Mullins-Sekerka problem.