论文标题
关于光谱空间裂缝扩散问题的数值方法的调查
A Survey on Numerical Methods for Spectral Space-Fractional Diffusion Problems
论文作者
论文摘要
该调查专门用于分数方程的数值解决方案$ a^αu= f $，$ 0 <α<1 $，其中$ a $是对应于$ \ \ Mathbb r^d $中有限的椭圆边界值$ω$中的二阶椭圆边界值问题的对称的正定运算符。操作员分数功率是非本地运算符，并通过频谱定义。在过去十年中，由于对物理和工程的应用对物理和工程的应用的兴趣和需求不断增长，因此已经提出，研究和测试了几种数值方法。我们考虑使用$ n $二维有限元$ v_h $的椭圆运算符$ a $的离散化，或者在带有$ n $ grid点的统一网格上存在有限差异。 该方程的数值解决方案基于解决方案的以下三个等效表示：（1）Dunford-Taylor积分公式（或其等效的Balakrishnan公式），（2）在$ω\ times（0，\ iffty）\ subset \ subbb r^linational proper $ω\ times a二阶椭圆形问题扩展气缸$（x，t）\ inω\ times（0,1）$，（3）频谱表示和最佳统一有理近似（Bura）$ z^α$ on $ [0,1] $的最佳统一近似（BURA）。尽管原点及其分析基本不同，但这些方法可以解释为$ a^{ - α} $的某种合理近似。在本文中，我们介绍了这些方法的主要思想和相应的算法，讨论它们的准确性，计算复杂性并比较它们的效率和鲁棒性。
The survey is devoted to numerical solution of the fractional equation $A^αu=f$, $0 < α<1$, where $A$ is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain $Ω$ in $\mathbb R^d$. The operator fractional power is a non-local operator and is defined through the spectrum. Due to growing interest and demand in applications of sub-diffusion models to physics and engineering, in the last decade, several numerical approaches have been proposed, studied, and tested. We consider discretizations of the elliptic operator $A$ by using an $N$-dimensional finite element space $V_h$ or finite differences over a uniform mesh with $N$ grid points. The numerical solution of this equation is based on the following three equivalent representations of the solution: (1) Dunford-Taylor integral formula (or its equivalent Balakrishnan formula), (2) extension of the a second order elliptic problem in $Ω\times (0,\infty)\subset \mathbb R^{d+1}$ (with a local operator) or as a pseudo-parabolic equation in the cylinder $(x,t) \in Ω\times (0,1) $, (3) spectral representation and the best uniform rational approximation (BURA) of $z^α$ on $[0,1]$. Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of $A^{-α}$. In this paper we present the main ideas of these methods and the corresponding algorithms, discuss their accuracy, computational complexity and compare their efficiency and robustness.
