A unified convergence analysis for the fractional diffusion equation driven by fractional Gaussion noise with Hurst index $H\in(0,1)$

TL;DR

This paper develops a unified numerical analysis framework for stochastic nonlinear fractional diffusion equations driven by fractional Gaussian noise, introducing new estimates and demonstrating the effectiveness of spectral Galerkin and backward Euler methods.

Contribution

It provides a novel second moment estimate for stochastic integrals with fractional Brownian motion, enabling comprehensive regularity and error analysis for the numerical scheme.

Findings

Established sharp error estimates for the numerical scheme.

Verified theoretical results through extensive numerical experiments.

Provided a unified approach applicable to fractional Gaussian noise with any Hurst index in (0,1).

Abstract

Here, we provide a unified framework for numerical analysis of stochastic nonlinear fractional diffusion equation driven by fractional Gaussian noise with Hurst index $H\in(0,1)$. A novel estimate of the second moment of the stochastic integral with respect to fractional Brownian motion is constructed, which greatly contributes to the regularity analyses of the solution in time and space for $H\in(0,1)$. Then we use spectral Galerkin method and backward Euler convolution quadrature to discretize the fractional Laplacian and Riemann-Liouville fractional derivative, respectively. The sharp error estimates of the built numerical scheme are also obtained. Finally, the extensive numerical experiments verify the theoretical results.