An IMEX Finite Element Method for a linearized Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise

TL;DR

This paper develops an IMEX finite element method for a linearized Cahn-Hilliard-Cook equation driven by space-time white noise, providing regular approximations and error estimates for the solution.

Contribution

It introduces a novel finite element approach combining Galerkin methods and IMEX time-stepping for stochastic PDEs driven by white noise.

Findings

Derived strong a priori error estimates for the approximation.

Constructed regularized solutions depending on finite random variables.

Validated the effectiveness of the method through theoretical analysis.

Abstract

We consider a model initial- and Dirichlet boundary- value problem for a linearized Cahn-Hilliard-Cook equation, in one space dimension, forced by the space derivative of a space-time white noise. First, we introduce a canvas problem the solution to which is a regular approximation of the mild solution to the problem and depends on a finite number of random variables. Then, fully-discrete approximations of the solution to the canvas problem are constructed using, for discretization in space, a Galerkin finite element method based on $H^2$ piecewise polynomials, and, for time-stepping, an implicit/explicit method. Finally, we derive a strong a priori estimate of the error approximating the mild solution to the problem by the canvas problem solution, and of the numerical approximation error of the solution to the canvas problem.