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We discuss, study, and compare experimentally three methods for solving thesystem of algebraic equations $\mathbb{A}^\alpha \bf{u}=\bf{f}$, $0< \alpha<1$, where $\mathbb{A}$ is a symmetric and positive definite matrix obtainedfrom finite difference or finite element approximations of second orderelliptic problems in $\mathbb{R}^d$, $d=1,2,3$. The first method, introduced byHarizanov et.al, based on the best uniform rational approximation (BURA)$r_\alpha(t)$ of $t^{1-\alpha}$ for $0 \le t \le 1$, is used to get therational approximation of $t^{-\alpha}$ in the form $t^{-1}r_\alpha(t)$. Herewe develop another method, denoted by R-BURA, that is based on the bestrational approximation $r_{1-\alpha}(t)$ of $t^\alpha$ on the interval $[0,1]$and approximates $t^{-\alpha}$ via $r^{-1}_{1-\alpha}(t)$. The third method,introduced and studied by Bonito and Pasciak, is based on an exponentiallyconvergent quadrature scheme for the Dundord-Taylor integral representation ofthe fractional powers of elliptic operators. All three methods reduce thesolution of the system $\mathbb{A}^\alpha \bf{u}=\bf{f}$ to solving a number ofequations of the type $(\mathbb{A} +c\mathbb{I})\bf{u}= \bf{f}$, $c \ge 0$.Comprehensive numerical experiments on model problems with $\mathbb A$ obtainedby approximation of elliptic equations in one and two spatial dimensions areused to compare the efficiency of these three algorithms depending on thefractional power $\alpha$. The presented results prove the concept of the newR-BURA method, which performs well for $\alpha$ close to $1$ in contrast toBURA, which performs well for $\alpha $ close to $0$. As a result, we showtheoretically and experimentally, that they have mutually complementaryadvantages.

Comparison analysis on two numerical methods for fractional diffusion problems based on rational approximations of $t^{\gamma}, \ 0 \le t \le 1$