# Fully Discrete Mixed Finite Element Methods for the Stochastic Cahn-Hilliard Equation with Gradient-type Multiplicative Noise

## TL;DR

This paper introduces and analyzes fully discrete mixed finite element methods for the stochastic Cahn-Hilliard equation with gradient-type multiplicative noise, proving strong convergence and optimal rates, supported by numerical experiments.

## Contribution

The paper develops new finite element methods for the stochastic Cahn-Hilliard equation and establishes their strong convergence with optimal rates, addressing low regularity challenges.

## Key findings

- Proved strong convergence with optimal rates.
- Established Hölder continuity in time for solutions.
- Validated results through numerical experiments.

## Abstract

This paper develops and analyzes some fully discrete mixed finite element methods for the stochastic Cahn-Hilliard equation with gradient-type multiplicative noise that is white in time and correlated in space. The stochastic Cahn-Hilliard equation is formally derived as a phase field formulation of the stochastically perturbed Hele-Shaw flow. The main result of this paper is to prove strong convergence with optimal rates for the proposed mixed finite element methods. To overcome the difficulty caused by the low regularity in time of the solution to the stochastic Cahn-Hilliard equation, the H\"{o}lder continuity in time with respect to various norms for the stochastic PDE solution is established, and it plays a crucial role in the error analysis. Numerical experiments are also provided to validate the theoretical results and to study the impact of noise on the Hele-Shaw flow as well as the interplay of the geometric evolution and gradient-type noise.