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

We discuss, study, and compare experimentally three methods for solving the system of algebraic equations $\mathbb{A}^α\bf{u}=\bf{f}$, $0< α<1$, where $\mathbb{A}$ is a symmetric and positive definite matrix obtained from finite difference or finite element approximations of second order elliptic problems in $\mathbb{R}^d$, $d=1,2,3$. The first method, introduced by Harizanov et.al, based on the best uniform rational approximation (BURA) $r_α(t)$ of $t^{1-α}$ for $0 \le t \le 1$, is used to get the rational approximation of $t^{-α}$ in the form $t^{-1}r_α(t)$. Here we develop another method, denoted by R-BURA, that is based on the best rational approximation $r_{1-α}(t)$ of $t^α$ on the interval $[0,1]$ and approximates $t^{-α}$ via $r^{-1}_{1-α}(t)$. The third method, introduced and studied by Bonito and Pasciak, is based on an exponentially convergent quadrature scheme for the Dundord-Taylor integral representation of the fractional powers of elliptic operators. All three methods reduce the solution of the system $\mathbb{A}^α\bf{u}=\bf{f}$ to solving a number of equations of the type $(\mathbb{A} +c\mathbb{I})\bf{u}= \bf{f}$, $c \ge 0$. Comprehensive numerical experiments on model problems with $\mathbb A$ obtained by approximation of elliptic equations in one and two spatial dimensions are used to compare the efficiency of these three algorithms depending on the fractional power $α$. The presented results prove the concept of the new R-BURA method, which performs well for $α$ close to $1$ in contrast to BURA, which performs well for $α$ close to $0$. As a result, we show theoretically and experimentally, that they have mutually complementary advantages.