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

Abstract

We discuss, study, and compare experimentally three methods for solving the system of algebraic equations ${\mathbb{A}}^{\alpha }\mathbf{u}=\mathbf{f}$, $0<\alpha <1$, 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) $r_{\alpha }(t)$ of $t^{1-\alpha }$ for $0\le t\le 1$, is used to get the rational approximation of $t^{-\alpha }$ in the form $t^{-1}{r}_{\alpha }(t)$. Here we develop another method, denoted by R-BURA, that is based on the best rational approximation $r_{1-\alpha }(t)$ of $t^{\alpha }$ on the interval $[0,1]$ and approximates $t^{-\alpha }$ via $r_{1-\alpha }^{-1}(t)$. 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 ${\mathbb{A}}^{\alpha }\mathbf{u}=\mathbf{f}$ to solving a number of equations of the type $(\mathbb{A}+c\mathbb{I})\mathbf{u}=\mathbf{f}$, $c\ge 0$. 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 $\alpha$. The presented results prove the concept of the new R-BURA method, which performs well for $\alpha$ close to $1$ in contrast to BURA, which performs well for $\alpha$ close to $0$. As a result, we show theoretically and experimentally, that they have mutually complementary advantages.

12 pages, 5 tables, 2 figures