Weak error estimates of fully-discrete schemes for the stochastic Cahn-Hilliard equation

TL;DR

This paper establishes weak error estimates for fully-discrete numerical schemes solving stochastic Cahn-Hilliard equations, revealing that weak convergence order is twice the strong order, with new regularity results for the associated Kolmogorov equation.

Contribution

It provides the first weak error analysis for fully-discrete schemes of stochastic Cahn-Hilliard equations, including new regularity estimates for the Kolmogorov equation.

Findings

Weak convergence order is twice the strong order.

Regularity estimates for the Kolmogorov equation are established.

Results apply to space-time white noise and trace-class noise in various dimensions.

Abstract

We study a class of fully-discrete schemes for the numerical approximation of solutions of stochastic Cahn--Hilliard equations with cubic nonlinearity and driven by additive noise. The spatial (resp. temporal) discretization is performed with a spectral Galerkin method (resp. a tamed exponential Euler method). We consider two situations: space-time white noise in dimension $d=1$ and trace-class noise in dimensions $d=1,2,3$. In both situations, we prove weak error estimates, where the weak order of convergence is twice the strong order of convergence with respect to the spatial and temporal discretization parameters. To prove these results, we show appropriate regularity estimates for solutions of the Kolmogorov equation associated with the stochastic Cahn--Hilliard equation, which have not been established previously and may be of interest in other contexts.