Comparison Analysis of Two Numerical Methods for Fractional Diffusion Problems Based on the Best Rational Approximations of t on [0, 1]
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Springer Nature Switzerland AG 2019. The paper is devoted to the numerical solution of algebraic systems of the type (forumala presented)., where 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 d, d = 1, 2 and we have an optimal method for solving linear systems with matrices (forumala presented). We study and compare experimentally two methods based on best uniform rational approximation (BURA) of t on [0, 1] with the method of Bonito and Pasciak, (Math Comput 84(295):20832110, 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):115128, 2018) and based on the BURA r(t) of t1 on [0, 1], is used to get the BURA of t on [1, ) through t1r(t). The second method, developed in this paper and denoted by R-BURA, is based on the BURA r1(t) of t on [0, 1] that approximates t on [1, ) via r11(t). Comprehensive numerical experiments on some model problems are used to compare the efficiency of these three algorithms depending on . The numerical results show that R-BURA method performs well for close to 1 in contrast to BURA, which performs well for close to 0. Thus, the two BURA methods have mutually complementary advantages.
