Strong convergence of a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation

TL;DR

This paper proves that a fully discrete finite element method converges strongly to the solution of the stochastic Cahn-Hilliard equation with additive noise in a convex polygonal domain, providing optimal error estimates.

Contribution

It establishes strong convergence and optimal error bounds for a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation in dimensions up to three.

Findings

Proved strong convergence of the numerical scheme.

Derived optimal error estimates with high probability.

Established uniform-in-time moment bounds for the solution.

Abstract

We consider the stochastic Cahn-Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension $d\le 3$. We discretize the equation using a standard finite element method in space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.