Preconditioning the helmholtz equation with the shifted Laplacian and faber polynomials

被引：0

作者：

Ramos L.G. ^{[1
]}

Sète O. ^{[1
]}

Nabben R. ^{[1
]}

机构：

[1] Technische Universität Berlin, Institute of Mathematics, Straße des 17. Juni 136, Berlin

来源：

Electronic Transactions on Numerical Analysis | 2021年 / 54卷

关键词：

'bratwurst' sets; Faber polynomials; GMRES; Helmholtz equation; Iterative methods; Preconditioning; Shifted Laplace preconditioner;

D O I：

10.1553/ETNA_VOL54S534

中图分类号：

学科分类号：

摘要：

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. Copyright © 2021, Kent State University.

引用

页码：534 / 557

页数：23

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