Volume 10, Issue 5
Fully Finite Element Adaptive AMG Method for Time-Space Caputo-Riesz Fractional Diffusion Equations

Adv. Appl. Math. Mech., 10 (2018), pp. 1103-1125.

Published online: 2018-07

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• Abstract

The paper aims to establish a fully discrete finite element (FE) scheme and provide cost-effective solutions for one-dimensional time-space Caputo-Riesz fractional diffusion equations on a bounded domain $Ω$. Firstly, we construct a fully discrete scheme of the linear FE method in both temporal and spatial directions, derive many characterizations on the coefficient matrix and numerically verify that the fully discrete 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 number of the coefficient matrix, in which $τ$ and $h$ respectively denote time and space step sizes. Finally, on the grounds of the estimation and fast Fourier transform, we develop and analyze an adaptive algebraic multigrid (AMG) method with low algorithmic complexity, reveal a reference formula to measure the strength-of-connection tolerance which severely affect the robustness of AMG methods in handling fractional diffusion equations, and illustrate the well robustness and high efficiency of the proposed algorithm compared with the classical AMG, conjugate gradient and Jacobi iterative methods.

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@Article{AAMM-10-1103, author = {X. Q. Yue , and W. P. Bu , and S. Shu , and M. H. Liu , and Wang , S.}, title = {Fully Finite Element Adaptive AMG Method for Time-Space Caputo-Riesz Fractional Diffusion Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {5}, pages = {1103--1125}, abstract = {

The paper aims to establish a fully discrete finite element (FE) scheme and provide cost-effective solutions for one-dimensional time-space Caputo-Riesz fractional diffusion equations on a bounded domain $Ω$. Firstly, we construct a fully discrete scheme of the linear FE method in both temporal and spatial directions, derive many characterizations on the coefficient matrix and numerically verify that the fully discrete 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 number of the coefficient matrix, in which $τ$ and $h$ respectively denote time and space step sizes. Finally, on the grounds of the estimation and fast Fourier transform, we develop and analyze an adaptive algebraic multigrid (AMG) method with low algorithmic complexity, reveal a reference formula to measure the strength-of-connection tolerance which severely affect the robustness of AMG methods in handling fractional diffusion equations, and illustrate the well robustness and high efficiency of the proposed algorithm compared with the classical AMG, conjugate gradient and Jacobi iterative methods.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0046}, url = {http://global-sci.org/intro/article_detail/aamm/12591.html} }
TY - JOUR T1 - Fully Finite Element Adaptive AMG Method for Time-Space Caputo-Riesz Fractional Diffusion Equations AU - X. Q. Yue , AU - W. P. Bu , AU - S. Shu , AU - M. H. Liu , AU - Wang , S. JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1103 EP - 1125 PY - 2018 DA - 2018/07 SN - 10 DO - http://doi.org/10.4208/aamm.OA-2018-0046 UR - https://global-sci.org/intro/article_detail/aamm/12591.html KW - AB -

The paper aims to establish a fully discrete finite element (FE) scheme and provide cost-effective solutions for one-dimensional time-space Caputo-Riesz fractional diffusion equations on a bounded domain $Ω$. Firstly, we construct a fully discrete scheme of the linear FE method in both temporal and spatial directions, derive many characterizations on the coefficient matrix and numerically verify that the fully discrete 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 number of the coefficient matrix, in which $τ$ and $h$ respectively denote time and space step sizes. Finally, on the grounds of the estimation and fast Fourier transform, we develop and analyze an adaptive algebraic multigrid (AMG) method with low algorithmic complexity, reveal a reference formula to measure the strength-of-connection tolerance which severely affect the robustness of AMG methods in handling fractional diffusion equations, and illustrate the well robustness and high efficiency of the proposed algorithm compared with the classical AMG, conjugate gradient and Jacobi iterative methods.

X. Q. Yue, W. P. Bu, S. Shu, M. H. Liu & S. Wang. (1970). Fully Finite Element Adaptive AMG Method for Time-Space Caputo-Riesz Fractional Diffusion Equations. Advances in Applied Mathematics and Mechanics. 10 (5). 1103-1125. doi:10.4208/aamm.OA-2018-0046
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