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Matrices associated to two conservative discretizations of Riesz fractional operators and related multigrid solvers
Author(s) -
Donatelli Marco,
Krause Rolf,
Mazza Mariarosa,
Semplice Matteo,
Trotti Ken
Publication year - 2022
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2436
Subject(s) - discretization , mathematics , multigrid method , toeplitz matrix , context (archaeology) , finite volume method , focus (optics) , mathematical analysis , partial differential equation , pure mathematics , paleontology , physics , mechanics , optics , biology
Abstract In this article, we focus on a two‐dimensional conservative steady‐state Riesz fractional diffusion problem. As is typical for problems in conservative form, we adopt a finite volume (FV)‐based discretization approach. Precisely, we use both classical FVs and the so‐called finite volume elements (FVEs). While FVEs have already been applied in the context of fractional diffusion equations, classical FVs have only been applied in first‐order discretizations. By exploiting the Toeplitz‐like structure of the resulting coefficient matrices, we perform a qualitative study of their spectrum and conditioning through their symbol, leading to the design of a second‐order FV discretization. This same information is leveraged to discuss parameter‐free symbol‐based multigrid methods for both discretizations. Tests on the approximation error and the performances of the considered solvers are given as well.