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The survey is devoted to numerical solution of the fractional equation$A^\alpha u=f$, $0 < \alpha <1$, where $A$ is a symmetric positive definiteoperator corresponding to a second order elliptic boundary value problem in abounded domain $\Omega$ in $\mathbb R^d$. The operator fractional power is anon-local operator and is defined through the spectrum. Due to growing interestand 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 usingan $N$-dimensional finite element space $V_h$ or finite differences over auniform mesh with $N$ grid points.  The numerical solution of this equation is based on the following threeequivalent representations of the solution: (1) Dunford-Taylor integral formula(or its equivalent Balakrishnan formula), (2) extension of the a second orderelliptic problem in $\Omega \times (0,\infty)\subset \mathbb R^{d+1}$ (with alocal operator) or as a pseudo-parabolic equation in the cylinder $(x,t) \in\Omega \times (0,1) $, (3) spectral representation and the best uniformrational approximation (BURA) of $z^\alpha$ on $[0,1]$. Though substantiallydifferent in origin and their analysis, these methods can be interpreted assome rational approximation of $A^{-\alpha}$. In this paper we present the mainideas of these methods and the corresponding algorithms, discuss theiraccuracy, computational complexity and compare their efficiency and robustness.

A Survey on Numerical Methods for Spectral Space-Fractional Diffusion Problems