Abstract
In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we give estimates on conditioning, and we study the spectral distribution in the Weyl sense using the tools of the theory of Generalized Locally Toeplitz matrix-sequences. Furthermore, we give a concise description of the spectral properties when non-constant coefficients come into play. Several numerical experiments are reported and critically discussed.
Highlights
The study of the fractional Laplace operator was initiated in 1938 by Marcel Riesz in his seminal work [37]
The aim of this paper is twofold: first, we review the spectral properties of finite difference discretizations, which are of Toeplitz type associated with symbols, as in (i) and (ii); second, we investigate the spectral properties of the matrices obtained by the Matrix Transfer Technique (MTT) as fractional powers of both finite difference and finite element approximations of the standard Laplacian
Induced by a different choice of boundary conditions can be represented as a zerodistributed matrix-sequence and GLT5 combined with items GLT1-GLT2 represents the key for the related spectral analysis
Summary
The study of the fractional Laplace operator was initiated in 1938 by Marcel Riesz in his seminal work [37]. We review two main proposals to define the fractional Laplace operator which enable us to construct approximations in two or more spatial variables As it is wellknown, the standard Laplacian in Rd can be formulated as. (a) compute a fractional power of the d-dimensional Laplacian in (1); (b) ‘fractionalize’ each integer order partial derivative in (1) These two strategies coincide in the 1D case, while they lead to different operators in more than one dimension. The aim of this paper is twofold: first, we review the spectral properties of finite difference discretizations, which are of Toeplitz type associated with symbols, as in (i) and (ii); second, we investigate the spectral properties of the matrices obtained by the MTT as fractional powers of both finite difference and finite element approximations of the standard Laplacian.
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