TY - JOUR
T1 - Strong convergence rates for the approximation of a stochastic time-fractional Allen–Cahn equation
AU - Al-Maskari, Mariam
AU - Karaa, Samir
N1 - Publisher Copyright:
© 2023 Elsevier B.V.
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PY - 2023/5/1
Y1 - 2023/5/1
N2 - The paper is concerned with the strong approximation of a stochastic time-fractional Allen-Cahn equation driven by an additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time; namely, a Caputo fractional derivative of order α∈(0,1), and a Riemann–Liouville fractional integral operator of order γ∈[0,1] applied to a Gaussian noise. We approximate the model by a standard piecewise linear finite element method (FEM) in space and the classical Grünwald–Letnikov method in time (for both time-fractional operators), and the noise by the L2-projection. Spatially semidiscrete and fully discrete schemes are analyzed and strong convergence rates are obtained by exploiting the temporal Hölder continuity property of the solution. Numerical experiments are presented to illustrate the theoretical results.
AB - The paper is concerned with the strong approximation of a stochastic time-fractional Allen-Cahn equation driven by an additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time; namely, a Caputo fractional derivative of order α∈(0,1), and a Riemann–Liouville fractional integral operator of order γ∈[0,1] applied to a Gaussian noise. We approximate the model by a standard piecewise linear finite element method (FEM) in space and the classical Grünwald–Letnikov method in time (for both time-fractional operators), and the noise by the L2-projection. Spatially semidiscrete and fully discrete schemes are analyzed and strong convergence rates are obtained by exploiting the temporal Hölder continuity property of the solution. Numerical experiments are presented to illustrate the theoretical results.
KW - Error estimates
KW - Finite element method
KW - Fractional derivative
KW - Stochastic Allen–Cahn equation
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U2 - 10.1016/j.cnsns.2023.107099
DO - 10.1016/j.cnsns.2023.107099
M3 - Article
AN - SCOPUS:85145978265
SN - 1007-5704
VL - 119
SP - 107099
JO - Communications in Nonlinear Science and Numerical Simulation
JF - Communications in Nonlinear Science and Numerical Simulation
M1 - 107099
ER -