Space-time Finite Element Adaptive AMG For Multi-term Time Fractional Advection Diffusion Problems

Fractional advection diffusion equations are known as one of the foremost mathematical models in depictions for the transport process in complex systems governed by the abnormal diffusion and nonexponential relaxation patterns.In the thesis,we construct a space-time finite element discrete scheme and develop an efficient adaptive algebraic multigrid(AMG)algorithm for a class of multi-term time fractional advection diffusion equationsFirstly,a fully discrete scheme is obtained by using the linear finite element method in both temporal and spatial directions,whose coefficient matrix can be written and many characterizations are established:(1)Ah? is an M-matrix,whose each row-sum has a positive lower bound when h<1/7;(2)Ah? is also an M-matrix when ? is not less than some positive number,whose each row-sum has a positive lower bound when h:<1/7;(3)Ah?n is also an M-matrix in some cases.Secondly,we give out the condition number estimation?(Ah? n)(?)1+?n?0h-2?.If ?n=O(h(?))and(?)?0?2?,then ?(Ah?n)=O(1);The uniform convergence of the two-level classical AMG method is proved,and in some cases,we design an AMG method with optimal algorithmic complexity;Furthermore,we propose an adaptive variant with higher efficiency via the condition number estimation.Finally,through numerous numerical experiments,we verify that the space-time finite element solution possesses the saturation error order in the L2(?)norm sense,the correct-ness of our condition number estimation,and the well robustness and high efficiency of the proposed solver over the standard conjugate gradient and classical AMG methods.