On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition

Niall Bootland; Victorita Dolean

doi:10.1007/978-3-030-55874-1_16

On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition

Niall Bootland, Victorita Dolean

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution book

4 Citations (Scopus)

Abstract

We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used.

Original language	English
Title of host publication	Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference
Subtitle of host publication	European Conference, Egmond aan Zee, The Netherlands, September 30 - October 4
Editors	Fred J. Vermolen, Cornelis Vuik
Place of Publication	Cham, Switzerland
Publisher	Springer
Pages	175-184
Number of pages	10
ISBN (Print)	9783030558734
DOIs	https://doi.org/10.1007/978-3-030-55874-1_16
Publication status	Published - 7 Jun 2021
Event	European Numerical Mathematics and Advanced Applications Conference 2019 - Egmond aan Zee, Netherlands Duration: 30 Sept 2019 → 4 Oct 2019 https://www.enumath2019.eu/

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	139
ISSN (Print)	1439-7358
ISSN (Electronic)	2197-7100

Conference

Conference	European Numerical Mathematics and Advanced Applications Conference 2019
Abbreviated title	ENUMATH 2019
Country/Territory	Netherlands
City	Egmond aan Zee
Period	30/09/19 → 4/10/19
Internet address	https://www.enumath2019.eu/

Keywords

Dirichlet-to-Neumann mapping
domain decomposition
Helmholt problem

Access to Document

10.1007/978-3-030-55874-1_16

Cite this

Bootland, N., & Dolean, V. (2021). On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition. In F. J. Vermolen, & C. Vuik (Eds.), Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference: European Conference, Egmond aan Zee, The Netherlands, September 30 - October 4 (pp. 175-184). (Lecture Notes in Computational Science and Engineering; Vol. 139). Springer. https://doi.org/10.1007/978-3-030-55874-1_16

Bootland, Niall ; Dolean, Victorita. / On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition. Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference: European Conference, Egmond aan Zee, The Netherlands, September 30 - October 4. editor / Fred J. Vermolen ; Cornelis Vuik. Cham, Switzerland : Springer, 2021. pp. 175-184 (Lecture Notes in Computational Science and Engineering).

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abstract = "We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used. ",

keywords = "Dirichlet-to-Neumann mapping, domain decomposition, Helmholt problem",

author = "Niall Bootland and Victorita Dolean",

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Bootland, N & Dolean, V 2021, On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition. in FJ Vermolen & C Vuik (eds), Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference: European Conference, Egmond aan Zee, The Netherlands, September 30 - October 4. Lecture Notes in Computational Science and Engineering, vol. 139, Springer, Cham, Switzerland, pp. 175-184, European Numerical Mathematics and Advanced Applications Conference 2019, Egmond aan Zee, Netherlands, 30/09/19. https://doi.org/10.1007/978-3-030-55874-1_16

On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition. / Bootland, Niall; Dolean, Victorita.
Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference: European Conference, Egmond aan Zee, The Netherlands, September 30 - October 4. ed. / Fred J. Vermolen; Cornelis Vuik. Cham, Switzerland: Springer, 2021. p. 175-184 (Lecture Notes in Computational Science and Engineering; Vol. 139).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution book

TY - GEN

T1 - On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition

AU - Bootland, Niall

AU - Dolean, Victorita

PY - 2021/6/7

Y1 - 2021/6/7

N2 - We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used.

AB - We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used.

KW - Dirichlet-to-Neumann mapping

KW - domain decomposition

KW - Helmholt problem

UR - http://arxiv.org/pdf/1912.06053v1

U2 - 10.1007/978-3-030-55874-1_16

DO - 10.1007/978-3-030-55874-1_16

M3 - Conference contribution book

SN - 9783030558734

T3 - Lecture Notes in Computational Science and Engineering

SP - 175

EP - 184

BT - Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference

A2 - Vermolen, Fred J.

A2 - Vuik, Cornelis

PB - Springer

CY - Cham, Switzerland

T2 - European Numerical Mathematics and Advanced Applications Conference 2019

Y2 - 30 September 2019 through 4 October 2019

ER -

Bootland N, Dolean V. On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition. In Vermolen FJ, Vuik C, editors, Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference: European Conference, Egmond aan Zee, The Netherlands, September 30 - October 4. Cham, Switzerland: Springer. 2021. p. 175-184. (Lecture Notes in Computational Science and Engineering). doi: 10.1007/978-3-030-55874-1_16

On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition

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