Robust a posteriori estimates for the stochastic Cahn-Hilliard equation

TL;DR

This paper develops robust a posteriori error estimates for the fully discrete finite element approximation of the stochastic Cahn-Hilliard equation, enabling adaptive algorithms that handle stochastic noise and topological changes effectively.

Contribution

It introduces a novel splitting approach to derive error bounds that are robust to parameters, noise, and topological changes, with practical computability.

Findings

Error estimates are robust with respect to interfacial width and noise.

The estimates are computable using the discrete principal eigenvalue.

Numerical simulations validate the adaptive algorithm's effectiveness.

Abstract

We derive a posteriori error estimates for a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. The a posteriori bound is obtained by a splitting of the equation into a linear stochastic partial differential equation (SPDE) and a nonlinear random partial differential equation (RPDE). The resulting estimate is robust with respect to the interfacial width parameter and is computable since it involves the discrete principal eigenvalue of a linearized (stochastic) Cahn-Hilliard operator. Furthermore, the estimate is robust with respect to topological changes as well as the intensity of the stochastic noise. We provide numerical simulations to demonstrate the practicability of the proposed adaptive algorithm.