Abstract The survey is devoted to numerical solution of the equation Aαu=f $ {\mathcal A}^\alpha u=f $ , 0 < α<1, where A $ {\mathcal A} $ is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in ℝd. The fractional power Aα $ {\mathcal A}^\alpha $ is a non-local operator and is defined though the spectrum of A $ {\mathcal A} $ . 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 $ {\mathcal A} $ by using an N-dimensional finite element space Vh or finite differences over a uniform mesh with N points. In the case of finite element approximation we get a symmetric and positive definite operator Ah:Vh→Vh $ {\mathcal A}_h: V_h \to V_h $ , which results in an operator equation Ahαuh=fh $ {\mathcal A}_h^{\alpha} u_h = f_h $ for uh ∈ Vh. 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, (2.5), (2) extension of the a second order elliptic problem in Ω × (0, ∞)⊂ ℝd+1 [17,55] (with a local operator) or as a pseudo-parabolic equation in the cylinder (x, t) ∈ Ω × (0, 1), [70, 29], (3) spectral representation (2.6) and the best uniform rational approximation (BURA) of zα on [0, 1], [37,40]. Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of Ah−α $ {\mathcal A}_h^{-\alpha} $ . 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.

中文翻译：

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谱空间分数扩散问题的数值方法综述

摘要 该调查致力于方程 Aαu=f $ {\mathcal A}^\alpha u=f $ , 0 < α<1 的数值解，其中 A $ {\mathcal A} $ 是对称正定算子对应的到 ℝd 中的有界域 Ω 中的二阶椭圆边值问题。分数幂 Aα $ {\mathcal A}^\alpha $ 是一个非局部算子，通过 A $ {\mathcal A} $ 的谱定义。由于对子扩散模型在物理和工程中的应用的兴趣和需求不断增长，在过去十年中，已经提出、研究和测试了几种数值方法。我们通过使用 N 维有限元空间 Vh 或具有 N 点的均匀网格上的有限差分来考虑椭圆算子 A $ {\mathcal A} $ 的离散化。在有限元近似的情况下，我们得到一个对称的正定算子 Ah:Vh→Vh $ {\mathcal A}_h: V_h \to V_h $ ，这导致了一个算子方程 Ahαuh=fh $ {\mathcal A}_h ^{\alpha} u_h = f_h $ uh ∈ Vh。该方程的数值解基于解的以下三个等效表示： (1) Dunford-Taylor 积分公式（或其等效的 Balakrishnan 公式，（2.5），（2）Ω 中的二阶椭圆问题的扩展× (0, ∞)⊂ ℝd+1 [17,55]（使用局部算子）或作为圆柱体中的伪抛物线方程 (x, t) ∈ Ω × (0, 1), [70, 29] , (3) 谱表示 (2.6) 和 zα 在 [0, 1], [37,40] 上的最佳均匀有理近似 (BURA)。虽然在起源和分析上有很大不同，这些方法可以解释为 Ah−α $ {\mathcal A}_h^{-\alpha} $ 的一些有理近似。在本文中，我们介绍了这些方法和相应算法的主要思想，讨论了它们的准确性、计算复杂度并比较了它们的效率和鲁棒性。