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

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Abstract

The survey is devoted to numerical solution of the equation Aαu = 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 Rd. The fractional power Aα is a non-local operator and is defined though the spectrum of A. 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 Vh or finite differences over a uniform mesh with N points. In the case of finite element approximation we get a symmetric and positive definite operator Ah: Vh → Vh, which results in an operator equation Aαhuh = fh for uh ∈ Vh.

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, (2.5)), (2) extension of the a second order elliptic problem in Ω×(0,∞) ⊂ Rd+1 [17, 55] (with a local operator) or as a pseudo-parabolic equation in the cylinder (x, t) ∈ Ω×(0, 1), [70, 29], (3) spectral representation (2.6) and the best uniform rational approximation (BURA) of zα on [0, 1], [37, 40]. Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of A−αh. 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.

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Authors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Acad. G. Bontchev Str., Block 25A, 1113, Sofia, Bulgaria

    Stanislav Harizanov & Svetozar Margenov

  2. Department of Mathematics, Texas A&M University, 505 D Blocker Build., College Station, TX, 77843, USA

    Raytcho Lazarov

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  1. Stanislav Harizanov
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Harizanov, S., Lazarov, R. & Margenov, S. A Survey on Numerical Methods for Spectral Space-Fractional Diffusion Problems. Fract Calc Appl Anal 23, 1605–1646 (2020). https://doi.org/10.1515/fca-2020-0080

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