A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator

Lea Conen, Victorita Dolean, Rolf Krause, Frédéric Nataf

Research output: Contribution to journalArticle

9 Citations (Scopus)
159 Downloads (Pure)

Abstract

The Helmholtz equation governing wave propagation and scattering phenomena
is difficult to solve numerically. Its discretization with piecewise linear finite
elements results in typically large linear systems of equations. The inherently
parallel domain decomposition methods constitute hence a promising class of
preconditioners. An essential element of these methods is a good coarse space.
Here, the Helmholtz equation presents a particular challenge, as even slight
deviations from the optimal choice can be devastating.

In this paper, we present a coarse space that is based on local eigenproblems
involving the Dirichlet-to-Neumann operator. Our construction is completely
automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. The resulting method is parallel by design and its efficiency is demonstrated on 2D homogeneous and heterogeneous numerical examples.
Original languageEnglish
Pages (from-to)83-99
Number of pages17
JournalJournal of Computational and Applied Mathematics
Volume271
Early online date12 Apr 2014
DOIs
Publication statusPublished - 1 Dec 2014

Fingerprint

Helmholtz equation
Hermann Von Helmholtz
Helmholtz Equation
Dirichlet
Domain decomposition methods
Wave Scattering
Linear system of equations
Domain Decomposition Method
Parameter Tuning
Operator
Piecewise Linear
Wave propagation
Wave Propagation
Linear systems
Convergence Rate
Tuning
Discretization
Scattering
Finite Element
Numerical Examples

Keywords

  • Helmholtz equation
  • Dirichlet-to-Neumann operator
  • domain decomposition
  • coarse space

Cite this

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title = "A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator",
abstract = "The Helmholtz equation governing wave propagation and scattering phenomenais difficult to solve numerically. Its discretization with piecewise linear finiteelements results in typically large linear systems of equations. The inherentlyparallel domain decomposition methods constitute hence a promising class ofpreconditioners. An essential element of these methods is a good coarse space.Here, the Helmholtz equation presents a particular challenge, as even slightdeviations from the optimal choice can be devastating.In this paper, we present a coarse space that is based on local eigenproblemsinvolving the Dirichlet-to-Neumann operator. Our construction is completelyautomatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. The resulting method is parallel by design and its efficiency is demonstrated on 2D homogeneous and heterogeneous numerical examples.",
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author = "Lea Conen and Victorita Dolean and Rolf Krause and Fr{\'e}d{\'e}ric Nataf",
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A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator. / Conen, Lea; Dolean, Victorita; Krause, Rolf; Nataf, Frédéric.

In: Journal of Computational and Applied Mathematics, Vol. 271, 01.12.2014, p. 83-99.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator

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