Strong convergence rates for the approximation of a stochastic time-fractional Allen-Cahn equation

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 alpha E (0, 1), and a Riemann-Liouville fractional integral operator of order gamma E [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 Grunwald-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 Holder continuity property of the solution. Numerical experiments are presented to illustrate the theoretical results. (c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper focuses on the strong approximation problem of a stochastic time-fractional Allen-Cahn equation driven by an additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time: a Caputo fractional derivative of order alpha E (0, 1) and a Riemann-Liouville fractional integral operator of order gamma E [0, 1] applied to a Gaussian noise. The model is approximated using a standard piecewise linear finite element method (FEM) in space and the classical Grunwald-Letnikov method in time, with the noise handled through L2-projection. Spatially semidiscrete and fully discrete schemes are analyzed, and strong convergence rates are obtained by exploiting the temporal Holder continuity property of the solution. Numerical experiments are conducted to illustrate the theoretical results.