Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials. ETNA - Electronic Transactions on Numerical Analysis
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6 septembre 2021
- handle: 10670/1.8r1v3r
- issn: 1068-9613
- doi: 10.1553/etna_vol54s534
info:eu-repo/semantics/openAccess
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Luis García Ramos et al., « Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials. ETNA - Electronic Transactions on Numerical Analysis », Elektronisches Publikationsportal der Österreichischen Akademie der Wissenschafte, ID : 10.1553/etna_vol54s534
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Résumé
We introduce a new polynomial preconditioner for solving the discretized Helmholtz equation preconditioned with the complex shifted Laplace (CSL) operator. We exploit the localization of the spectrum of the CSL-preconditioned system to approximately enclose the eigenvalues by a non-convex ‘bratwurst’ set. On this set, we expand the function 1/z into a Faber series. Truncating the series gives a polynomial, which we apply to the Helmholtz matrix preconditioned by the shifted Laplacian to obtain a new preconditioner, the Faber preconditioner. We prove that the Faber preconditioner is nonsingular for degrees one and two of the truncated series. Our numerical experiments (for problems with constant and varying wavenumber) show that the Faber preconditioner reduces the number of GMRES iterations.