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Matrices associated to two conservative discretizations of Riesz fractional operators and related multigrid solvers. (English) Zbl 07584144

Summary: 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.

MSC:

65F08 Preconditioners for iterative methods
15B05 Toeplitz, Cauchy, and related matrices
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