We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number (Formula presented.), approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a (Formula presented.) preconditioning when the variable coefficient wave number (Formula presented.) is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented.

Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number μ

Adriani A.;Serra Capizzano S.
;
2024-01-01

Abstract

We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number (Formula presented.), approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a (Formula presented.) preconditioning when the variable coefficient wave number (Formula presented.) is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented.
2024
2024
Caputo fractional derivatives; clustering; eigenvalue asymptotic distribution; Generalized Locally Toeplitz sequences; Helmholtz equations; preconditioning; spectral symbol
Adriani, A.; Serra Capizzano, S.; Tablino-Possio, C.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2186691
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