Fully Finite Element Adaptive AMG Method for Time-Space Caputo-Riesz Fractional Diffusion Equations (Advances in Applied Mathematics and Mechanics, 2018)
The paper aims to establish a fully discrete finite element (FE) scheme andprovide cost-effective solutions for one-dimensional time-space Caputo-Riesz fractionaldiffusion equations on a bounded domain $Ω$. Firstly, we construct a fully discretescheme of the linear FE method in both temporal and spatial directions, derivemany characterizations on the coefficient matrix and numerically verify that the fullydiscrete FE approximation possesses the saturation error order under $L^2(Ω)$ norm.Secondly, we theoretically prove the estimation $1+\mathcal{O}(τ^αh^{−2β})$ on the condition numberof the coefficient matrix, in which $τ$ and $h$ respectively denote time and spacestep sizes. Finally, on the grounds of the estimation and fast Fourier transform, wedevelop and analyze an adaptive algebraic multigrid (AMG) method with low algorithmiccomplexity, reveal a reference formula to measure the strength-of-connectiontolerance which severely affect the robustness of AMG methods in handling fractionaldiffusion equations, and illustrate the well robustness and high efficiency of the proposedalgorithm compared with the classical AMG, conjugate gradient and Jacobi iterativemethods.