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 â (0,1), and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order γ â [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.
Original language | English |
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Pages (from-to) | 1245-1268 |
Number of pages | 24 |
Journal | ESAIM: Mathematical Modelling and Numerical Analysis |
Volume | 53 |
Issue number | 4 |
DOIs | |
Publication status | Published - 9 Jul 2019 |
Keywords
- Galerkin finite element method
- Grünwald-Letnikov method
- Stochastic time-fractional diffusion
- Strong convergence
- Weak convergence
ASJC Scopus subject areas
- Analysis
- Numerical Analysis
- Modelling and Simulation
- Computational Mathematics
- Applied Mathematics