  This paper discusses the fractional diffusion equation forced by a tempered
fractional Gaussian noise. The fractional diffusion equation governs the
probability density function of the subordinated killed Brownian motion. The
tempered fractional Gaussian noise plays the role of fluctuating external
source with the property of localization. We first establish the regularity of
the infinite dimensional stochastic integration of the tempered fractional
Brownian motion and then build the regularity of the mild solution of the
fractional stochastic diffusion equation. The spectral Galerkin method is used
for space approximation; after that the system is transformed into an
equivalent form having better regularity than the original one in time. Then we
use the semi-implicit Euler scheme to discretize the time derivative. In terms
of the temporal-spatial error splitting technique, we obtain the error
estimates of the fully discrete scheme in the sense of mean-squared $L^2$-norm.
Extensive numerical experiments confirm the theoretical estimates.
