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

TL;DR

This paper develops a numerical scheme combining Crank-Nicolson and finite element methods to approximate solutions of a one-dimensional stochastic heat equation driven by space-time white noise, providing error estimates.

Contribution

It introduces a combined numerical approach for a stochastic heat equation with additive noise and derives error bounds for the approximation.

Findings

Strong a priori error estimates for the regularized solution

Error bounds for the Crank-Nicolson finite element approximation

Validation of the numerical scheme's effectiveness in stochastic PDEs

Abstract

We formulate an initial- and Dirichlet boundary- value problem for a linear stochastic heat equation, in one space dimension, forced by an additive space-time white noise. First, we approximate the mild solution to the problem by the solution of the regularized second-order linear stochastic parabolic problem with random forcing proposed by Allen, Novosel and Zhang (Stochastics Stochastics Rep., 64, 1998). Then, we construct numerical approximations of the solution to the regularized problem by combining the Crank-Nicolson method in time with a standard Galerkin finite element method in space. We derive strong a priori estimates of the modeling error made in approximating the mild solution to the problem by the solution to the regularized problem, and of the numerical approximation error of the Crank-Nicolson finite element method.