Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials

Electronic Transactions on Numerical Analysis

article

Luis García Ramos¹  Olivier Sète¹  Reinhard Nabben¹ 

[1] Technische Universität Berlin, Institute of Mathematics

关键词: Helmholtz equation;    shifted Laplace preconditioner;    iterative methods;    GMRES;    preconditioning;    Faber polynomials;    ‘bratwurst’ sets;   

DOI  :  10.1553/etna_vol54s534

学科分类：数学（综合）

来源: Kent State University * Institute of Computational Mathematics

【 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.

【 授权许可】

Unknown