Fractional Laplace operator in two dimensions, approximating matrices, and related spectral analysis
- PDF / 2,106,107 Bytes
- 25 Pages / 439.37 x 666.142 pts Page_size
- 91 Downloads / 181 Views
Fractional Laplace operator in two dimensions, approximating matrices, and related spectral analysis Lidia Aceto1 · Mariarosa Mazza2 · Stefano Serra‑Capizzano3,4 Received: 24 October 2019 / Revised: 4 May 2020 / Accepted: 4 July 2020 © The Author(s) 2020
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. Keywords Riesz fractional derivative operator · Toeplitz matrices · matrix transfer technique · spectral analysis · GLT theory Mathematics Subject Classification 47A58 · 34A08 · 15B05 · 65F60 · 65F15 · 65F35
This work was funded by GNCS-INdAM. The authors are members of the INdAM research group GNCS. * Lidia Aceto [email protected] 1
Department of Mathematics, University of Pisa, Via F. Buonarroti, 1/C, 56127 Pisa, Italy
2
Department of Science and High Technology, University of Insubria, via Valleggio 11, 22100 Como, Italy
3
Department of Humanities and Innovation, University of Insubria, via Valleggio 11, 22100 Como, Italy
4
Division of Scientific Computing, Department of Information Technology, Uppsala University, 751 05 Uppsala, Sweden
13
Vol.:(0123456789)
27
Page 2 of 25
L. Aceto et al.
1 Introduction The study of the fractional Laplace operator was initiated in 1938 by Marcel Riesz in his seminal work [37]. However, since then numerous definitions for such operator have been proposed due to the need of describing several distinct types of applications (for a recent survey, refer to [24] and to the references therein). In the present paper, 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
𝛥=
d ∑ 𝜕2 . 2 j=1 𝜕xj
(1)
For generalizing such operator towards the fractional case two different strategies can be followed: (a) compute a fractional power of the d-dimensional Laplacian in (1); (b) ‘fractionalize’ each integer order partial derivative in (1). Obviously, these two strategies coincide in the 1D case, while they lead to different operators in more than one dimension. Formulation (a) is particularly useful when imposing boundary conditions which differ from the standard Dirichlet ones. Indeed, in [21] Ilić et al. showed that the problems involving the fractional Laplacian of the form (a) are well defined on fin
Data Loading...
