Numerical approximation of stochastic time-fractional diffusion

¹ Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK
² Department of Mathematics, University of Chester, Chester CHI 4BJ, UK
³ Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

^* Corresponding author: bangti.jin@gmail.com, b.jin@ucl.ac.uk

Received: 24 September 2018
Accepted: 16 March 2019

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 α ∈ (0,1), and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order γ ∈ [0,1] in the front). The numerical scheme approximates the model in space by the standard Galerkin method with continuous piecewise linear finite elements and in time by the classical Grünwald-Letnikov method (for both Caputo fractional derivative and Riemann-Liouville fractional integral), and the noise by the L²-projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the discrete solution operators for the deterministic inhomogeneous problem. One- and two-dimensional numerical results are presented to support the theoretical findings.