Comparison Analysis of Two Numerical Methods for Fractional Diffusion Problems Based on the Best Rational Approximations of t ^γ on [0, 1]

Abstract

The paper is devoted to the numerical solution of algebraic systems of the type ${\mathbb A}^\alpha \mathbf {u}=\mathbf {f}$, 0 < α < 1, where ${\mathbb A}$ is a symmetric and positive definite matrix. We assume that ${\mathbb A}$ is obtained from finite difference or finite element approximations of second order elliptic problems in $\mathbb {R}^d$, d = 1, 2 and we have an optimal method for solving linear systems with matrices ${\mathbb A} + c \mathbb {I}$. We study and compare experimentally two methods based on best uniform rational approximation (BURA) of t ^γ on [0, 1] with the method of Bonito and Pasciak, (Math Comput 84(295):2083–2110, 2015), that uses exponentially convergent quadratures for the Dunford-Taylor integral representation of the fractional powers of elliptic operators. The first method, introduced in Harizanov et al. (Numer Linear Algebra Appl 25(4):115–128, 2018) and based on the BURA r _α(t) of t ^1−α on [0, 1], is used to get the BURA of t ^−α on [1, ∞) through t ⁻¹ r _α(t). The second method, developed in this paper and denoted by R-BURA, is based on the BURA r _1−α(t) of t ^α on [0, 1] that approximates t ^−α on [1, ∞) via $r^{-1}_{1-\alpha }(t)$. Comprehensive numerical experiments on some model problems are used to compare the efficiency of these three algorithms depending on α. The numerical results show that R-BURA method performs well for α close to 1 in contrast to BURA, which performs well for α close to 0. Thus, the two BURA methods have mutually complementary advantages.