Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation

In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propos...

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Autores: Darrigrand, V., Rodríguez-Rozas, A., Muga, I., Pardo, D., Romkes, A., Prudhomme, S.
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2017
País:España
Institución:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/841
Acceso en línea:http://hdl.handle.net/20.500.11824/841
http://dx.doi.org/10.1002/nme.5601
Access Level:acceso embargado
Palabra clave:error representation
finite element methods
goal-oriented adaptivity
Helmholtz equation
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spelling Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equationDarrigrand, V.Rodríguez-Rozas, A.Muga, I.Pardo, D.Romkes, A.Prudhomme, S.error representationfinite element methodsgoal-oriented adaptivityHelmholtz equationIn goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method.V. Darrigrand, A. Rodriguez-Rozas and D. Pardo were partially funded by the Projects of the Spanish Ministry of Economy and Competitiveness with reference MTM2013-40824-P, MTM2016-76329-R (AEI/FEDER, EU), MTM2016-81697-ERC and the Basque Government Consolidated Research Group Grant IT649- 13 on “Mathematical Modeling, Simulation, and Industrial Applications (M2SI)”. A. Rodriguez-Rozas and D.Pardo were also partially funded by the BCAM “Severo Ochoa” accreditation of excellence SEV-2013-0323 and the Basque Government through the BERC2014-2017 program. A. Rodriguez-Rozas acknowledges support from Spanish Ministry under Grant No. FPDI- 2013-17098. I. Muga was partially funded by the FONDECYT project 1160774. The first four authors were also partially funded by the European Union’s Horizon 2020, research and innovation program under the Marie Sklodowska-Curie grant agreement No 644202. Serge Prudhomme is grateful for the support by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.info201820182017info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/841http://dx.doi.org/10.1002/nme.5601reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttps://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5601info:eu-repo/grantAgreement/EC/H2020/644202info:eu-repo/grantAgreement/MINECO//SEV-2013-0323info:eu-repo/grantAgreement/MINECO//MTM2016-76329-Rinfo:eu-repo/grantAgreement/MINECO//MTM2016-81697-ERCinfo:eu-repo/grantAgreement/MINECO//MTM2013-40824-Pinfo:eu-repo/grantAgreement/Gobierno Vasco/BERC/BERC.2014-2017Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/embargoedAccessoai:bird.bcamath.org:20.500.11824/8412026-06-19T12:47:47Z
dc.title.none.fl_str_mv Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation
title Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation
spellingShingle Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation
Darrigrand, V.
error representation
finite element methods
goal-oriented adaptivity
Helmholtz equation
title_short Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation
title_full Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation
title_fullStr Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation
title_full_unstemmed Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation
title_sort Goal-oriented adaptivity using unconventional error representations for the multi-dimensional Helmholtz equation
dc.creator.none.fl_str_mv Darrigrand, V.
Rodríguez-Rozas, A.
Muga, I.
Pardo, D.
Romkes, A.
Prudhomme, S.
author Darrigrand, V.
author_facet Darrigrand, V.
Rodríguez-Rozas, A.
Muga, I.
Pardo, D.
Romkes, A.
Prudhomme, S.
author_role author
author2 Rodríguez-Rozas, A.
Muga, I.
Pardo, D.
Romkes, A.
Prudhomme, S.
author2_role author
author
author
author
author
dc.subject.none.fl_str_mv error representation
finite element methods
goal-oriented adaptivity
Helmholtz equation
topic error representation
finite element methods
goal-oriented adaptivity
Helmholtz equation
description In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method.
publishDate 2017
dc.date.none.fl_str_mv 2017
2018
2018
info
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/841
http://dx.doi.org/10.1002/nme.5601
url http://hdl.handle.net/20.500.11824/841
http://dx.doi.org/10.1002/nme.5601
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5601
info:eu-repo/grantAgreement/EC/H2020/644202
info:eu-repo/grantAgreement/MINECO//SEV-2013-0323
info:eu-repo/grantAgreement/MINECO//MTM2016-76329-R
info:eu-repo/grantAgreement/MINECO//MTM2016-81697-ERC
info:eu-repo/grantAgreement/MINECO//MTM2013-40824-P
info:eu-repo/grantAgreement/Gobierno Vasco/BERC/BERC.2014-2017
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
info:eu-repo/semantics/embargoedAccess
rights_invalid_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
eu_rights_str_mv embargoedAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
instname:Basque Center for Applied Mathematics (BCAM)
instname_str Basque Center for Applied Mathematics (BCAM)
reponame_str BIRD. BCAM's Institutional Repository Data
collection BIRD. BCAM's Institutional Repository Data
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