A Survey on Numerical Methods for Spectral Space-Fractional Diffusion Problems

10/03/2020 ∙ by Stanislav Harizanov, et al. ∙ 0

The survey is devoted to numerical solution of the fractional equation A^α u=f, 0 < α <1, where A is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in ℝ^d. The operator fractional power is a non-local operator and is defined through the spectrum. Due to growing interest and demand in applications of sub-diffusion models to physics and engineering, in the last decade, several numerical approaches have been proposed, studied, and tested. We consider discretizations of the elliptic operator A by using an N-dimensional finite element space V_h or finite differences over a uniform mesh with N grid points. The numerical solution of this equation is based on the following three equivalent representations of the solution: (1) Dunford-Taylor integral formula (or its equivalent Balakrishnan formula), (2) extension of the a second order elliptic problem in Ω× (0,∞)⊂ℝ^d+1 (with a local operator) or as a pseudo-parabolic equation in the cylinder (x,t) ∈Ω× (0,1), (3) spectral representation and the best uniform rational approximation (BURA) of z^α on [0,1]. Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of A^-α. In this paper we present the main ideas of these methods and the corresponding algorithms, discuss their accuracy, computational complexity and compare their efficiency and robustness.



There are no comments yet.


page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.