Volume 53, Number 4, July-August 2019
|Page(s)||1245 - 1268|
|Published online||09 July 2019|
Numerical approximation of stochastic time-fractional diffusion
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
Accepted: 16 March 2019
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.
Mathematics Subject Classification: 60H15 / 60H35 / 65M12
Key words: stochastic time-fractional diffusion / Galerkin finite element method / Grünwald-Letnikov method / strong convergence / weak convergence
© EDP Sciences, SMAI 2019
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.