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.

Matrices associated to two conservative discretizations of Riesz fractional operators and related multigrid solvers

Donatelli M.;Mazza M.;Semplice M.;Trotti K.
2022-01-01

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.
2022
banded preconditioning; finite volume methods; fractional diffusion equations; multigrid methods; spectral distribution; Toeplitz matrices
Donatelli, M.; Krause, R.; Mazza, M.; Semplice, M.; Trotti, K.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2131202
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