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