  In this study we construct a time-space finite element (FE) scheme and
furnish cost-efficient approximations for one-dimensional multi-term time
fractional advection diffusion equations on a bounded domain $\Omega$. Firstly,
a fully discrete scheme is obtained by the linear FE method in both temporal
and spatial directions, and many characterizations on the resulting matrix are
established. Secondly, the condition number estimation is proved, an adaptive
algebraic multigrid (AMG) method is further developed to lessen computational
cost and analyzed in the classical framework. Finally, some numerical
experiments are implemented to reach the saturation error order in the
$L^2(\Omega)$ norm sense, and present theoretical confirmations and predictable
behaviors of the proposed algorithm.
