Fully Finite Element Adaptive Algebraic Multigrid Method for Time-Space Caputo-Riesz Fractional Diffusion Equations

TL;DR

This paper develops a fully discrete finite element scheme and an adaptive algebraic multigrid method for efficiently solving one-dimensional time-space Caputo-Riesz fractional diffusion equations, demonstrating high accuracy and robustness.

Contribution

It introduces a novel fully discrete FE scheme with proven error bounds and an adaptive AMG algorithm optimized for fractional diffusion equations, improving computational efficiency.

Findings

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

The condition number of the coefficient matrix is estimated as $1+ ext{O}( au^eta h^{-2eta})$.

The proposed AMG method shows superior robustness and efficiency compared to classical methods.

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 $\Omega$. 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 FE approximation possesses the saturation error order under $L^2(\Omega)$ norm. Secondly, we theoretically prove the estimation $1+\mathcal{O}(\tau^\alpha h^{-2\beta})$ on the condition number of the coefficient matrix, in which $\tau$ 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.