Numerical analysis for stochastic time-space fractional diffusion equation driven by fractional Gaussion noise

TL;DR

This paper analyzes the strong convergence of a stochastic time-space fractional diffusion equation driven by fractional Gaussian noise, providing regularity estimates, numerical schemes, and error analysis.

Contribution

It introduces a finite element and backward Euler scheme for the equation and derives precise error estimates using inverse Laplace transform and fractional Ritz projection.

Findings

Established strong convergence of the numerical scheme.

Derived sharp regularity estimates for the mild solution.

Validated theoretical results with numerical experiments.

Abstract

In this paper, we consider the strong convergence of the time-space fractional diffusion equation driven by fractional Gaussion noise with Hurst index $H\in(\frac{1}{2},1)$. A sharp regularity estimate of the mild solution and the numerical scheme constructed by finite element method for integral fractional Laplacian and backward Euler convolution quadrature for Riemann-Liouville time fractional derivative are proposed. With the help of inverse Laplace transform and fractional Ritz projection, we obtain the accurate error estimates in time and space. Finally, our theoretical results are accompanied by numerical experiments.