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
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