Spectral galerkin method for solving helmholtz and laplace dirichlet problems on multiple open arcs

Carlos Jerez-Hanckes; José Pinto

doi:10.1007/978-3-030-39647-3_30

Spectral galerkin method for solving helmholtz and laplace dirichlet problems on multiple open arcs

Carlos Jerez-Hanckes, José Pinto

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We present a spectral numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on an unbounded non-Lipschitz domain ℝ²∖ Γ ¯, where Γ is a finite collection of open arcs. Through an indirect method, a first kind formulation is derived whose variational form is discretized using weighted Chebyshev polynomials. This choice of basis allows for exponential convergence rates under smoothness assumptions. Moreover, by implementing a simple compression algorithm, we are able to efficiently account for large numbers of arcs as well as a wide wavenumber range.

Original language	English
Title of host publication	Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference
Editors	Spencer J. Sherwin, Joaquim Peiró, Peter E. Vincent, David Moxey, Christoph Schwab
Publisher	Springer
Pages	383-393
Number of pages	11
ISBN (Print)	9783030396466
DOIs	https://doi.org/10.1007/978-3-030-39647-3_30
State	Published - 2020
Externally published	Yes
Event	12th International Conference on Spectral and High-Order Methods, ICOSAHOM 2018 - London, United Kingdom Duration: 9 Jul 2018 → 13 Jul 2018

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	134
ISSN (Print)	1439-7358
ISSN (Electronic)	2197-7100

Conference

Conference	12th International Conference on Spectral and High-Order Methods, ICOSAHOM 2018
Country/Territory	United Kingdom
City	London
Period	9/07/18 → 13/07/18

Access to Document

10.1007/978-3-030-39647-3_30

Cite this

Jerez-Hanckes, C., & Pinto, J. (2020). Spectral galerkin method for solving helmholtz and laplace dirichlet problems on multiple open arcs. In S. J. Sherwin, J. Peiró, P. E. Vincent, D. Moxey, & C. Schwab (Eds.), Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference (pp. 383-393). (Lecture Notes in Computational Science and Engineering; Vol. 134). Springer. https://doi.org/10.1007/978-3-030-39647-3_30

Jerez-Hanckes, Carlos ; Pinto, José. / Spectral galerkin method for solving helmholtz and laplace dirichlet problems on multiple open arcs. Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference. editor / Spencer J. Sherwin ; Joaquim Peiró ; Peter E. Vincent ; David Moxey ; Christoph Schwab. Springer, 2020. pp. 383-393 (Lecture Notes in Computational Science and Engineering).

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title = "Spectral galerkin method for solving helmholtz and laplace dirichlet problems on multiple open arcs",

abstract = "We present a spectral numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on an unbounded non-Lipschitz domain ℝ2∖ Γ ¯, where Γ is a finite collection of open arcs. Through an indirect method, a first kind formulation is derived whose variational form is discretized using weighted Chebyshev polynomials. This choice of basis allows for exponential convergence rates under smoothness assumptions. Moreover, by implementing a simple compression algorithm, we are able to efficiently account for large numbers of arcs as well as a wide wavenumber range.",

author = "Carlos Jerez-Hanckes and Jos{\'e} Pinto",

note = "Publisher Copyright: {\textcopyright} 2020, The Author(s).; 12th International Conference on Spectral and High-Order Methods, ICOSAHOM 2018 ; Conference date: 09-07-2018 Through 13-07-2018",

year = "2020",

doi = "10.1007/978-3-030-39647-3_30",

language = "English",

isbn = "9783030396466",

series = "Lecture Notes in Computational Science and Engineering",

publisher = "Springer",

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editor = "Sherwin, {Spencer J.} and Joaquim Peir{\'o} and Vincent, {Peter E.} and David Moxey and Christoph Schwab",

booktitle = "Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference",

}

Jerez-Hanckes, C & Pinto, J 2020, Spectral galerkin method for solving helmholtz and laplace dirichlet problems on multiple open arcs. in SJ Sherwin, J Peiró, PE Vincent, D Moxey & C Schwab (eds), Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference. Lecture Notes in Computational Science and Engineering, vol. 134, Springer, pp. 383-393, 12th International Conference on Spectral and High-Order Methods, ICOSAHOM 2018, London, United Kingdom, 9/07/18. https://doi.org/10.1007/978-3-030-39647-3_30

Spectral galerkin method for solving helmholtz and laplace dirichlet problems on multiple open arcs. / Jerez-Hanckes, Carlos; Pinto, José.
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference. ed. / Spencer J. Sherwin; Joaquim Peiró; Peter E. Vincent; David Moxey; Christoph Schwab. Springer, 2020. p. 383-393 (Lecture Notes in Computational Science and Engineering; Vol. 134).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AB - We present a spectral numerical scheme for solving Helmholtz and Laplace problems with Dirichlet boundary conditions on an unbounded non-Lipschitz domain ℝ2∖ Γ ¯, where Γ is a finite collection of open arcs. Through an indirect method, a first kind formulation is derived whose variational form is discretized using weighted Chebyshev polynomials. This choice of basis allows for exponential convergence rates under smoothness assumptions. Moreover, by implementing a simple compression algorithm, we are able to efficiently account for large numbers of arcs as well as a wide wavenumber range.

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Jerez-Hanckes C, Pinto J. Spectral galerkin method for solving helmholtz and laplace dirichlet problems on multiple open arcs. In Sherwin SJ, Peiró J, Vincent PE, Moxey D, Schwab C, editors, Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018 - Selected Papers from the ICOSAHOM Conference. Springer. 2020. p. 383-393. (Lecture Notes in Computational Science and Engineering). doi: 10.1007/978-3-030-39647-3_30

Spectral galerkin method for solving helmholtz and laplace dirichlet problems on multiple open arcs

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