LAUSR

In this study, we construct a space‐time 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 associated condition number estimation 1+τnα0h−2γ is derived, where τn and h, respectively, are the current time and space step sizes. Next, we propose a lossless in robustness and adaptive algebraic multigrid (AMG) with O(MlogM) computational cost and O(M) matrix‐free storage, in contrast to classical AMG with O(M2) , where M is the number of spatial segments. Meanwhile, uniform convergence analyses on the two‐level V(0,1)‐cycle of classical and adaptive AMGs are provided. Finally, we demonstrate the optimal error order of space‐time FE approximations in the L2(Ω) norm sense, present theoretical confirmations and predictable behaviors of the proposed algorithm in numerical experiments.