Numerical Approximation of Stochastic Time-Fractional Diffusion
For researchers in numerical analysis of stochastic fractional PDEs, this provides rigorous convergence guarantees for a finite element/Grünwald-Letnikov scheme.
The paper develops and analyzes a numerical method for stochastic time-fractional diffusion with additive fractionally integrated Gaussian noise, establishing sharp strong and weak convergence rates supported by numerical results.
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 $α\in(0,1)$, and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order $γ\in[0,1]$ 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 Grünwald-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.