Summary In this paper, we consider efficient algorithms for solving the algebraic equation Aαboldu=boldf, 0<α<1, where scriptA is a properly scaled symmetric and positive definite matrix obtained from finite difference or finite element approximations of second‐order elliptic problems in Rd, d=1,2,3. This solution is then written as boldu=Aβ−αboldF with boldF=A−βboldf with β positive integer. The approximate solution method we propose and study is based on the best uniform rational approximation of the function tβ−α for 0

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Here we study theoretically and compare experimentally with the methods developed in [19,8] an efficient method for solving systems of algebraic equations A α u h = f h , 0 < α < 1, where A is an N × N matrix coming from the discretization of a fractional diffusion operator. More specifically, we focus on matrices obtained from finite difference or finite element approximation of second order elliptic problems in R d , d = 1, 2, 3. The proposed methods are based on the best uniform rational approximation (BURA) r α,k (t) of t α on [0, 1]. Here r α,k is a rational function of t involving numerator and denominator polynomials of degree at most k.The approximation ofWe show that the proposed method is exponentially convergent with respect to k and has some attractive properties. First, it reduces the solution of the nonlocal system to solution of k systems with matrix ( A + cj I) and cj > 0, j = 1, 2, . . . , k. Thus, good computational complexity can be achieved if fast solvers are available for such systems. Second, the original problem and its rational approximation in the finite difference case are positivity preserving. In the finite element case, this valid for schemes obtained by mass lumping under certain mild conditions on the mesh. Further, we prove that the lumped mass schemes still have the expected rate of convergence, at times assuming additional regularity on the right hand side. Finally, we present comprehensive numerical experiments on a number of model problems for various α in one and two spatial dimensions. These illustrate the computational behavior of the proposed method and compare its accuracy and efficiency with that of other methods developed by Harizanov et. al. [19] and Bonito and Pasciak [8] .