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

共 59 条

[1]

AIRAKSINEN T., HEIKKOLA E., PENNANEN A., TOIVANEN J., An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation, J. Comput. Phys, 226, pp. 1196-1210, (2007)

[2]

BAYLISS A., GOLDSTEIN C. I., TURKEL E., On accuracy conditions for the numerical computation of waves, J. Comput. Phys, 59, pp. 396-404, (1985)

[3]

BECKERMANN B., REICHEL L., Error estimates and evaluation of matrix functions via the Faber transform, SIAM J. Numer. Anal, 47, pp. 3849-3883, (2009)

[4]

BOLLHOFER M., GROTE M. J., SCHENK O., Algebraic multilevel preconditioner for the Helmholtz equation in heterogeneous media, SIAM J. Sci. Comput, 31, pp. 3781-3805, (2009)

[5]

COCQUET P.-H., GANDER M. J., How large a shift is needed in the shifted Helmholtz preconditioner for its effective inversion by multigrid?, SIAM J. Sci. Comput, 39, pp. A438-A478, (2017)

[6]

COLEMAN J. P., MYERS N. J., The Faber polynomials for annular sectors, Math. Comp, 64, pp. 181-203, (1995)

[7]

COOLS S., VANROOSE W., Local Fourier analysis of the complex shifted Laplacian preconditioner for Helmholtz problems, Numer. Linear Algebra Appl, 20, pp. 575-597, (2013)

[8]

COOLS S., VANROOSE W., On the optimality of shifted Laplacian in a class of polynomial preconditioners for the Helmholtz equation, Modern Solvers for Helmholtz Problems, pp. 53-81, (2017)

[9]

CURTISS J. H., Faber polynomials and the Faber series, Amer. Math. Monthly, 78, pp. 577-596, (1971)

[10]

DUBOIS P. F., GREENBAUM A., RODRIGUE G. H., Approximating the inverse of a matrix for use in iterative algorithms on vector processors, Computing, 22, pp. 257-268, (1979)

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