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

摘要

The paper is devoted to the numerical solution of algebraic systems of the type A~αu = f, 0 < α < 1, where A is a symmetric and positive definite matrix. We assume that A is obtained from finite difference or finite element approximations of second order elliptic problems in R~d, d = 1,2 and we have an optimal method for solving linear systems with matrices A + cI. 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-α) (t). Comprehensive numerical experiments on some model problems are used to compare the efficiency of these three algorithms depending on a. 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.