Time-space Finite Element Adaptive AMG for Multi-term Time Fractional Advection Diffusion Equations

TL;DR

This paper develops a finite element and adaptive AMG method for efficiently solving multi-term time fractional advection diffusion equations, with theoretical analysis and numerical validation of accuracy and computational cost.

Contribution

It introduces a cost-efficient adaptive AMG algorithm tailored for multi-term time fractional advection diffusion equations with rigorous matrix analysis and condition number estimation.

Findings

The scheme achieves saturation error order in $L^2( abla)$ norm.

The adaptive AMG reduces computational cost effectively.

Numerical experiments confirm theoretical predictions.

Abstract

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.