Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation

Abstract

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] .

Rational Approximations in Robust Preconditioning of Multiphysics Problems

“…However, the convergence rate of the first BURA method was not robust with respect to the condition number of A, which is correctly noted in [5]. This shortcoming is completely overcome in [10]. Several aspects of the step-by-step development of the BURA method can be traced in the survey paper [8].…”

“…Therefore, methods based on best uniform rational approximation are expected to be the most efficient. From now on, when we talk about BURA, we will have in mind the following variant of the method proposed and analyzed in [10].…”

“…As correctly noted in [5], a disadvantage of the method introduced in 2018 in [11] was that the accuracy depended on the condition number of the matrix A. This drawback has been overcome in the improved BURA approximation developed in [10]; see also the survey paper [8]. It reads as A…”

“…As an alternative, we propose to use the best uniform rational approximation (BURA) method for A −α [8][9][10][11]. The aim is to develop a robust preconditioning algorithm of optimal computational complexity.…”

Approximation of a fractional power of an elliptic operator

“…The use of such technology for solving multidimensional problems of fractional diffusion is discussed. 7,8 For a fractional power of a self-adjoint positive definite operator, the following integral representation holds: 9,10…”