Strong convergence of an fractional exponential integrator scheme for the finite element discretization of time-fractional SPDE driven by standard and fractional Brownian motions

TL;DR

This paper establishes the first strong convergence results for a numerical scheme approximating a time-fractional stochastic PDE driven by both standard and fractional Brownian motions, with convergence rates depending on data regularity.

Contribution

It introduces a novel finite element and fractional exponential integrator scheme for a complex stochastic PDE with fractional derivatives and proves its strong convergence.

Findings

Convergence order depends on initial data regularity.

The scheme effectively handles both standard and fractional Brownian motions.

Provides existence and uniqueness results for the PDE.

Abstract

The aim of this work is to provide the first strong convergence result of numerical approximation of a general time-fractional second order stochastic partial differential equation involving a Caputo derivative in time of order $\alpha\in(\frac 12; 1)$ and driven simultaneously by a multiplicative standard Brownian motion and additive fBm with Hurst parameter $H\in(\frac 12, 1)$, more realistic to model the random effects on transport of particles in medium with thermal memory. We prove the existence and uniqueness results and perform the spatial discretization using the finite element and the temporal discretization using a fractional exponential integrator scheme. We provide the temporal and spatial convergence proofs for our fully discrete scheme and the result shows that the convergence orders depend on the regularity of the initial data, the power of the fractional derivative, and the Hurst parameter $H$.