Numerical Approximation of Stochastic Time-Fractional Diffusion

TL;DR

This paper introduces a numerical scheme for solving stochastic time-fractional diffusion equations involving nonlocal derivatives and noise, providing convergence analysis and numerical validation.

Contribution

The paper develops a novel numerical method combining Galerkin, Gr"unwald-Letnikov, and $L^2$-projection techniques for stochastic time-fractional diffusion with convergence proofs.

Findings

Established sharp strong and weak convergence rates.

Validated theoretical results with numerical experiments.

Handled nonlocal fractional derivatives and noise in the scheme.

Abstract

We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order $\alpha\in(0,1)$, and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order $\gamma \in[0,1]$ in the front). The numerical scheme approximates the model in space by the Galerkin method with continuous piecewise linear finite elements and in time by the classical Gr\"unwald-Letnikov method, and the noise by the $L^2$-projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the deterministic counterpart. Numerical results are presented to support the theoretical findings.