Strong convergence rates of a fully discrete scheme for the Cahn-Hilliard-Cook equation

TL;DR

This paper establishes the existence, uniqueness, and regularity of solutions to the stochastic Cahn-Hilliard-Cook equation in low dimensions, and proves strong convergence rates for a fully discrete numerical scheme combining spectral Galerkin and backward Euler methods.

Contribution

It introduces a novel analysis of the regularity and convergence of a fully discrete scheme for the stochastic Cahn-Hilliard-Cook equation, extending previous work to include detailed convergence rates.

Findings

Proved existence, uniqueness, and regularity of solutions in dimensions up to 3.

Established strong convergence rates for the fully discrete spectral Galerkin and backward Euler scheme.

Demonstrated the scheme's effectiveness through rigorous mathematical analysis.

Abstract

The first aim of this paper is to examine existence, uniqueness and regularity for the Cahn-Hilliard-Cook (CHC) equation in space dimension $d\leq 3$. By applying a spectral Galerkin method to the infinite dimensional equation, we elaborate the well-posedness and regularity of the finite dimensional approximate problem. The key idea lies in transforming the stochastic problem {\color{black}{with additive noise}} into an equivalent random equation. The regularity of the solution to the equivalent random equation is obtained, in one dimension, with the aid of the Gagliardo-Nirenberg inequality and done in two and three dimensions, by the energy argument. Further, the approximate solution is shown to be strongly convergent to the unique mild solution of the original CHC equation, whose spatio-temporal regularity can be attained by similar arguments. In addition, a fully discrete approximation of such problem is investigated, performed by the spectral Galerkin method in space and the backward Euler method in time. The previously obtained regularity results of the problem help us to identify strong convergence rates of the fully discrete scheme.