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 Ω. 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 (Ω) norm sense, and present theoretical confirmations and predictable behaviors of the proposed algorithm.Another important note is that no matter which fully discrete scheme is utilized, there always exists the computational challenge in nonlocality caused by fractional differential operators [27]. 15 Quite many scholars are working to identify algorithms most appropriate to overcome the challenge and utilize computer resources. Multigrid exploits a hierarchy of grids or multiscale representations, and reduces the error of the approximation at a number of frequencies (from global smooth to local oscillation) simultaneously [28-31], which makes multigrid as an extremely superior solver or preconditioner of particular interest. Pang and Sun presented an efficient V-cycle geometric multi-20 grid (GMG) with fast Fourier transform (FFT) to solve one-dimensional space-fractional diffusion equations (SFDEs) discretized by an implicit FD scheme [32]. Bu et al. extended and analyzed the V-cycle GMG for one-dimensional MTFADEs via the FD in temporal and FE in spatial directions [26]. Zhou and Wu discussed the FE F-cycle GMG to linear stationary FADEs in Riemann-Liouvlle

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