## Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials

Luis García Ramos, Olivier Sète, and Reinhard Nabben

### Abstract

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.

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### Key words

Helmholtz equation, shifted Laplace preconditioner, iterative methods, GMRES, preconditioning, Faber polynomials, ‘bratwurst’ sets

### AMS subject classifications

65F08, 65F10, 30C10, 30C20

### Links to the cited ETNA articles

 [18] Vol. 31 (2008), pp. 403-424 Yogi A. Erlangga and Reinhard Nabben: On a multilevel Krylov method for the Helmholtz equation preconditioned by shifted Laplacian [29] Vol. 5 (1997), pp. 62-76 Vincent Heuveline and Miloud Sadkane: Arnoldi-Faber method for large non Hermitian eigenvalue problems

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