Recent results on domain decomposition preconditioning for the high-frequency Helmholtz equation using absorption

Ivan Graham, Euan Spence, Eero Vainikko

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

In this paper we present an overview of recent progress on the development and analysis of domain decomposition preconditioners for discretised Helmholtz problems, where the preconditioner is constructed from the corresponding problem with added absorption. Our preconditioners incorporate local subproblems that can have various boundary conditions, and include the possibility of a global coarse mesh. While the rigorous analysis describes preconditioners for the Helmholtz problem with added absorption, this theory also informs the development of efficient multilevel solvers for the "pure" Helmholtz problem without absorption. For this case, 2D experiments for problems containing up to about 50 wavelengths are presented. The experiments show iteration counts of order about (n0.2) and times (on a serial machine) of order about (nα), { with α∈[1.3,1.4]} for solving systems of dimension n. This holds both in the pollution-free case corresponding to meshes with grid size (k−3/2) (as the wavenumber k increases), and also for discretisations with a fixed number of grid points per wavelength, commonly used in applications. Parallelisation of the algorithms is also briefly discussed.
Original languageEnglish
Title of host publicationModern Solvers for Helmholtz Problems
EditorsD. Lahaye, J. Tang, K. Vuik
Place of PublicationBasel, Germany
PublisherBirkhäuser
ISBN (Print)9783319288314
DOIs
Publication statusPublished - 18 Feb 2017

Publication series

NameGeosystems Mathematics
PublisherSpringer

Fingerprint

Helmholtz equation
Decomposition
Wavelength
Pollution
Experiments
Boundary conditions

Cite this

Graham, I., Spence, E., & Vainikko, E. (2017). Recent results on domain decomposition preconditioning for the high-frequency Helmholtz equation using absorption. In D. Lahaye, J. Tang, & K. Vuik (Eds.), Modern Solvers for Helmholtz Problems (Geosystems Mathematics). Basel, Germany: Birkhäuser. https://doi.org/10.1007/978-3-319-28832-1

Recent results on domain decomposition preconditioning for the high-frequency Helmholtz equation using absorption. / Graham, Ivan; Spence, Euan; Vainikko, Eero.

Modern Solvers for Helmholtz Problems. ed. / D. Lahaye; J. Tang; K. Vuik. Basel, Germany : Birkhäuser, 2017. (Geosystems Mathematics).

Research output: Chapter in Book/Report/Conference proceedingChapter

Graham, I, Spence, E & Vainikko, E 2017, Recent results on domain decomposition preconditioning for the high-frequency Helmholtz equation using absorption. in D Lahaye, J Tang & K Vuik (eds), Modern Solvers for Helmholtz Problems. Geosystems Mathematics, Birkhäuser, Basel, Germany. https://doi.org/10.1007/978-3-319-28832-1
Graham I, Spence E, Vainikko E. Recent results on domain decomposition preconditioning for the high-frequency Helmholtz equation using absorption. In Lahaye D, Tang J, Vuik K, editors, Modern Solvers for Helmholtz Problems. Basel, Germany: Birkhäuser. 2017. (Geosystems Mathematics). https://doi.org/10.1007/978-3-319-28832-1
Graham, Ivan ; Spence, Euan ; Vainikko, Eero. / Recent results on domain decomposition preconditioning for the high-frequency Helmholtz equation using absorption. Modern Solvers for Helmholtz Problems. editor / D. Lahaye ; J. Tang ; K. Vuik. Basel, Germany : Birkhäuser, 2017. (Geosystems Mathematics).
@inbook{12b576804ea54a61adb0757c94f7a061,
title = "Recent results on domain decomposition preconditioning for the high-frequency Helmholtz equation using absorption",
abstract = "In this paper we present an overview of recent progress on the development and analysis of domain decomposition preconditioners for discretised Helmholtz problems, where the preconditioner is constructed from the corresponding problem with added absorption. Our preconditioners incorporate local subproblems that can have various boundary conditions, and include the possibility of a global coarse mesh. While the rigorous analysis describes preconditioners for the Helmholtz problem with added absorption, this theory also informs the development of efficient multilevel solvers for the {"}pure{"} Helmholtz problem without absorption. For this case, 2D experiments for problems containing up to about 50 wavelengths are presented. The experiments show iteration counts of order about (n0.2) and times (on a serial machine) of order about (nα), { with α∈[1.3,1.4]} for solving systems of dimension n. This holds both in the pollution-free case corresponding to meshes with grid size (k−3/2) (as the wavenumber k increases), and also for discretisations with a fixed number of grid points per wavelength, commonly used in applications. Parallelisation of the algorithms is also briefly discussed.",
author = "Ivan Graham and Euan Spence and Eero Vainikko",
year = "2017",
month = "2",
day = "18",
doi = "10.1007/978-3-319-28832-1",
language = "English",
isbn = "9783319288314",
series = "Geosystems Mathematics",
publisher = "Birkh{\"a}user",
editor = "D. Lahaye and J. Tang and K. Vuik",
booktitle = "Modern Solvers for Helmholtz Problems",
address = "Switzerland",

}

TY - CHAP

T1 - Recent results on domain decomposition preconditioning for the high-frequency Helmholtz equation using absorption

AU - Graham, Ivan

AU - Spence, Euan

AU - Vainikko, Eero

PY - 2017/2/18

Y1 - 2017/2/18

N2 - In this paper we present an overview of recent progress on the development and analysis of domain decomposition preconditioners for discretised Helmholtz problems, where the preconditioner is constructed from the corresponding problem with added absorption. Our preconditioners incorporate local subproblems that can have various boundary conditions, and include the possibility of a global coarse mesh. While the rigorous analysis describes preconditioners for the Helmholtz problem with added absorption, this theory also informs the development of efficient multilevel solvers for the "pure" Helmholtz problem without absorption. For this case, 2D experiments for problems containing up to about 50 wavelengths are presented. The experiments show iteration counts of order about (n0.2) and times (on a serial machine) of order about (nα), { with α∈[1.3,1.4]} for solving systems of dimension n. This holds both in the pollution-free case corresponding to meshes with grid size (k−3/2) (as the wavenumber k increases), and also for discretisations with a fixed number of grid points per wavelength, commonly used in applications. Parallelisation of the algorithms is also briefly discussed.

AB - In this paper we present an overview of recent progress on the development and analysis of domain decomposition preconditioners for discretised Helmholtz problems, where the preconditioner is constructed from the corresponding problem with added absorption. Our preconditioners incorporate local subproblems that can have various boundary conditions, and include the possibility of a global coarse mesh. While the rigorous analysis describes preconditioners for the Helmholtz problem with added absorption, this theory also informs the development of efficient multilevel solvers for the "pure" Helmholtz problem without absorption. For this case, 2D experiments for problems containing up to about 50 wavelengths are presented. The experiments show iteration counts of order about (n0.2) and times (on a serial machine) of order about (nα), { with α∈[1.3,1.4]} for solving systems of dimension n. This holds both in the pollution-free case corresponding to meshes with grid size (k−3/2) (as the wavenumber k increases), and also for discretisations with a fixed number of grid points per wavelength, commonly used in applications. Parallelisation of the algorithms is also briefly discussed.

UR - http://dx.doi.org/10.1007/978-3-319-28832-1

U2 - 10.1007/978-3-319-28832-1

DO - 10.1007/978-3-319-28832-1

M3 - Chapter

SN - 9783319288314

T3 - Geosystems Mathematics

BT - Modern Solvers for Helmholtz Problems

A2 - Lahaye, D.

A2 - Tang, J.

A2 - Vuik, K.

PB - Birkhäuser

CY - Basel, Germany

ER -