  In this study, a Galerkin finite element method is presented for
time-fractional stochastic heat equation driven by multiplicative noise, which
arises from the consideration of heat transport in porous media with thermal
memory with random effects. The spatial and temporal regularity properties of
mild solution to the given problem under certain sufficient conditions are
obtained. Numerical techniques are developed by the standard Galerkin finite
element method in spatial direction, and Gorenflo-Mainardi-Moretti-Paradisi
scheme is applied in temporal direction. The convergence error estimates for
both semi-discrete and fully discrete schemes are established. Finally,
numerical example is provided to verify the theoretical results.
