Crank-Nicolson finite element approximations for a linear stochastic fourth order equation with additive space-time white noise

TL;DR

This paper develops a Crank-Nicolson finite element method for a linear stochastic fourth-order PDE with white noise, providing strong convergence error estimates comparable to backward Euler methods.

Contribution

It introduces a fully-discrete Crank-Nicolson finite element scheme for a stochastic fourth-order PDE and establishes its strong convergence properties.

Findings

The method achieves strong convergence with an order matching backward Euler methods.

Error estimates are derived for the fully-discrete scheme.

The approach effectively handles additive space-time white noise.

Abstract

We consider a model initial- and Dirichlet boundary- value problem for a fourth-order linear stochastic parabolic equation, in one space dimension, forced by an additive space-time white noise. First, we approximate its solution by the solution of an auxiliary fourth-order stochastic parabolic problem with additive, finite dimensional, spectral-type stochastic load. Then, fully-discrete approximations of the solution to the approximate problem are constructed by using, for the discretization in space, a standard Galerkin finite element method based on $H^2$-piecewise polynomials, and, for time-stepping, the Crank-Nicolson method. Analyzing the convergence of the proposed discretization approach, we derive strong error estimates which show that the order of strong convergence of the Crank-Nicolson finite element method is equal to that reported in [Kosioris and Zouraris MMAN 44 (2010)] for the Backward Euler finite element method.