Finite Element Approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise

TL;DR

This paper develops finite element methods to approximate solutions of a linear Cahn-Hilliard-Cook equation driven by space derivative of space-time white noise, providing error estimates for the regularized problem.

Contribution

It introduces a regularization and discretization scheme combining spline-based noise approximation with finite element and backward Euler methods, along with error analysis.

Findings

Derivation of strong a priori estimates for modeling and numerical errors

Development of fully-discrete finite element approximation scheme

Analysis of error bounds for the regularized stochastic problem

Abstract

We consider an initial- and Dirichlet boundary- value problem for a linear Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we propose an approximate regularized stochastic parabolic problem discretizing the noise using linear splines. Then fully-discrete approximations to the solution of the regularized problem are constructed using, for the discretization in space, a Galerkin finite element method based on H2-piecewise polynomials, and, for time-stepping, the Backward Euler method. Finally, we derive strong a priori estimates for the modeling error and for the numerical approximation error to the solution of the regularized problem.