Weak convergence of the backward Euler method for stochastic Cahn--Hilliard equation with additive noise

TL;DR

This paper establishes the weak convergence rates of a fully discrete numerical scheme combining spectral Galerkin and backward Euler methods for the stochastic Cahn--Hilliard equation with additive noise, using Malliavin calculus techniques.

Contribution

It introduces a novel approach to analyze weak convergence rates for the stochastic Cahn--Hilliard equation without relying on Kolmogorov equations.

Findings

First weak convergence rates for stochastic Cahn--Hilliard equation established.

The approach avoids Kolmogorov equations, using Malliavin calculus instead.

Provides a foundation for numerical analysis of complex SPDEs.

Abstract

We prove a weak rate of convergence of a fully discrete scheme for stochastic Cahn--Hilliard equation with additive noise, where the spectral Galerkin method is used in space and the backward Euler method is used in time. Compared with the Allen--Cahn type stochastic partial differential equation, the error analysis here is much more sophisticated due to the presence of the unbounded operator in front of the nonlinear term. To address such issues, a novel and direct approach has been exploited which does not rely on a Kolmogorov equation but on the integration by parts formula from Malliavin calculus. To the best of our knowledge, the rates of weak convergence are revealed in the stochastic Cahn--Hilliard equation setting for the first time.