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

The survey is devoted to numerical solution of the fractional equation $A^\alpha u=f$, $0 < \alpha <1$, where $A$ is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain $\Omega$ 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 $\Omega \times (0,\infty)\subset \mathbb R^{d+1}$ (with a local operator) or as a pseudo-parabolic equation in the cylinder $(x,t) \in \Omega \times (0,1)$, (3) spectral representation and the best uniform rational approximation (BURA) of $z^\alpha$ on $[0,1]$. Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of $A^{-\alpha}$. 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.